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Event When Speaker Title Presentation Material
TOD 19th July 2012
11:30 to 12:30
P Boyland Some topological tools in two-dimensional dynamics
TOD Introductory Seminar

We informally survey various topological/dynamical tools which have proved useful in two-dimensional fluid dynamics and other applications. These include braids, isotopy classes, the Thurston-Nielsen trichotomy, pseudo-Anosov maps, topological growth rates and isotopy stability of dynamics. All definitions and hopefully a physical intuition will be provided.

TODW01 23rd July 2012
10:30 to 11:00
K Moffatt Welcome & Opening Remarks
TODW01 23rd July 2012
11:00 to 11:40
Ultimate state of two-dimensional Rayleigh-Bénard convection
Determining the transport properties of high Rayleigh number convection turbulent convection remains a grand challenge for experiment, simulation, theory, and analysis. In this talk, after a brief review of the theory and applications of Rayleigh-Bénard convection we describe recent results for mathematically rigorous upper limits on the vertical heat transport in two dimensional Rayleigh-Bénard convection between stress-free isothermal boundaries derived from the Boussinesq approximation of the Navier-Stokes equations. These bounds challenge some popular theoretical arguments regarding the nature of the asymptotic high Rayleigh number ‘ultimate regime’ of turbulent convection. This is joint work with Jared Whitehead.
TODW01 23rd July 2012
11:45 to 12:05
On helical multiscale characterisation of homogeneous turbulence
The helical properties of five prototypical homogeneous turbulent flows are investigated: statistically steady forced isotropic turbulence, decaying isotropic turbulence, decaying rotating turbulence, growing sheared turbulence, and growing rotating sheared turbulence with a rotation ratio f/S = +0.5. The five turbulent flows were originally studied using direct numerical simulations. An orthogonal wavelet decomposition is used to study the scale-dependent properties of the cases. For comparison, a solenoidal uncorrelated Gaussian random field is included in the analysis as a sixth case. It was found that flows with growing turbulent kinetic energy and turbulent motion at large scales show a maximum in the velocity helicity probability distribution functions (PDFs) at zero, corresponding to a trend to local two-dimensionalization of the flow with vorticity and velocity being perpendicular. Flows with decaying turbulent kinetic energy and turbulent motion at small scales, how ever, show maxima of the velocity helicity PDFs at plus and minus one, indicating a preference for helical motion with alignment or anti-alignment of vorticity and velocity. The PDFs of vorticity helicity always assume maxima at plus and minus for all flows. Joint PDFs of relative velocity helicity and relative vorticity helicity show that the quantities tend to have the same sign for all flows including the random field, indicating that vorticity helicity dissipates velocity helicity.
TODW01 23rd July 2012
12:10 to 12:30
Turbulent vorticity and helicity in stratified atmospheric boundary layer
To measure the spatial derivative of velocity v, it is necessary to possess the sensors, which size much less than internal scale of turbulence. In a surface layer it is estimated by size of an order 1 mm. The concept of the acoustical method of vorticity measurements and the first results of its realization are obtained in IAP [Bovsheverov et al. 1971]. Helicity (a scalar product of the velocity v and the vorticity) is one of the important characteristics of large-scale atmospheric motions [Etling 1985, Moffat, Tsinober 1992; Kurgansky 2002, Chkhetiani 2001]. Direct experiments aimed at the measurement of turbulent helicity are extremely rare. They have been carried out under laboratory conditions in turbulence beyond a grid [Kholmyansky et al. 2001]. First helicity measurements in atmospheric boundary layer were made in IAP Zvenigorod station in 2004 [Koprov et al. 2005]. Experimental estimates of the spectrum of the turbulent helicity in the atmospheric boundary layer give the spectrum slope of about -5/3. Proceeding from the helicity and energy spectra, we obtain for dissipation ? ? 0.0003 m/s3, ? ? 0.003 m2/s3 and ? ? 0.0005 m/s3, ? ? 0.001m2/s3. Helicity components in day conditions shows considerably big intermittency than circulation. Average value for Hz has made 0.2 m/s2. The correlation factor between product factors in a day series at moderately unstable stratification has made 0.344. Similar indicators for Hx: average value of 0.46 m/s2, factor of correlation 0.215. In an evening series average values of both measured helicity components had the same sign, as in the afternoon, but on 1-2 order smaller values. Probability density functions (PDF) for circulations Zz, Zx, vertical velocity w and temperature T have been calculated at unstable and stable stratification. Asymmetry of Zz changes a sign at change of a sign on parameter of stratification whereas asymmetry for Zx is small and keeps a sign at any stratification. PDF of helicity components and complex triple two-point correlations of velocity and vorticity show strong non-gaussian character. Spectrum slope for these correlations is close to f -1. This fact corresponds to the "2/15" law for the helicity cascade [Chkhetiani 1996, 2010]. The data obtained for both the values of turbulent helicity in the atmospheric boundary layer and helicity spectra indicate the existence, at least in the region of the scales that have been considered, of parallel cas-cades of the energy and helicity. The importance of the determination of actual helicity cascades in natural systems is also stimulated by the fact that numerical calculations of the Navier-Stokes equations manifest certain effects of nonzero helicity on the energy transfer over the spectrum. This emphasizes the role of helicity in the formation of large-scale structures.
TODW01 23rd July 2012
14:00 to 14:40
New conservation laws of helical flows
Conservation laws in incompressible fluid dynamics, in particular inviscid motion, constitute an axiomatic basis for fluid mechanics. In 3D, mass and momentum conservation forms the fundamental basis, which is further extended by energy, vorticity and helicity conservation. Interesting enough considering reduced dimensions a much broader set of conserved quantities is observed in particularly for 2D/planar and axisymmetric flows. For the planar case it is well known that any once differential function of the vorticity is a materially conserved quantity and hence an infinite number of additional conservation laws exist. Further, the most simple one, the square of the vorticity, is named enstrophy, and is “weakly conserved” in the viscous case and constitutes a fundamental invariant for 2D turbulence. Recently we have shown that the known set of additional conservation laws may be considerably extended for helical flows which constitute a Lie symmetry induced concaten ation of planar and axisymmetric flows living on the (r, a z + b \phi; t) spatially reduced system with a^2 + b^2 > 0 and r, z, \phi are the classical coordinates in a cylinder coordinate system. Various infinity dimensional new conservations laws have been established including e.g. a generalized helicity. Even for the 2D/planar and axisymmetric flows new conservation laws have been derived not reported in the literature before. The construction of the new results is based on the “direct method”. It relies on two key theorems: (i) the Euler operator applied to a term is always zero if and only if the term is in divergence form; (ii) any non-trivial conservation law of a given set of differential equations can only be constructed by a linear combination of the given equations with some multipliers to be determined by theorem (i). This is a necessary and sufficient condition. The process of finding the new conservation laws was aided by the computer algebra system Maple employing the package GeM by A. Cheviakov.
TODW01 23rd July 2012
14:45 to 15:05
MV Kurgansky Simple models of helical baroclinic vortices
Two distinct asymptotic solutions of inviscid Boussinesq equations for a steady helical baroclinic Rankine-like vortex with prescribed buoyant forcing are considered and critically compared. In both cases the relative distribution of the velocity components is the same across the vortex at all altitudes (the similarity assumption). The first vortex solution demonstrates monotonic growth with height of the vortex core radius, which becomes infinite at a certain critical altitude, and the corresponding attenuation of the vertical vorticity. The second vortex solution schematizes the vortex core as an inverted cone of small angular aperture. These idealized vortices are then embedded in a convectively unstable boundary layer; the resulting approximate vortex solutions have been applied to determine the maximum rotational velocity in vortices. Both models predict essentially the same dependence of the model-inferred peak rotational velocity on the local swirl ratio (the ratio of the maximum swirl velocity to the average vertical velocity in the main vortex updraft). The helicity budget of the vortex flow is analyzed in detail, where applicable.
TODW01 23rd July 2012
15:10 to 15:30
Using fluid variational variables to obtain new analytic solutions with nonzero helicity
Flow equations being nonlinear are notoriously difficult to solve analytically. In this work we show that through the three independent functions variational formalism for stationary barotropic flows one can obtain new analytical solutions of the flow equations. The flows are constructed such that they flow on predetermined Bernoullian surfaces from which the rest of the variational variables are derived. The flow obtained has non zero helicity.
TODW01 23rd July 2012
16:00 to 16:40
P Boyland Exponential growth in two-dimensional topological fluid dynamics
In two-dimensional multi-connected fluid regions the Thurston-Nielsen (TN) theory implies that the essential topological length of material lines grows either exponentially or linearly; the TN theory and subsequent results provide many procedures for determining which growth rate occurs. Our first application is to Euler flows. The main theorem is that there are periodic stirring protocols for which generic initial vorticity yields a solution to Euler's equations which is not periodic and further, the sup norm of the gradient of the vorticity grows exponentially in time. The second application investigates which stirring protocols maximize the efficiency of mixing in the precise, topological sense of the maximal exponential growth of per unit generator of certain push-point mapping classes on the punctured disk. Related Links http://www.math.ufl.edu/~boyland/papers.html - my paper's page
TODW01 23rd July 2012
16:45 to 17:05
S Tanda Topological crystals as a new paradigm
We report the discovery of Mobius, Ring, Figure-8, and Hopf-link Crystals in NbSe3, conventionally grown as ribbons and whiskers. We also reveal their formation mechanisms of which two crucial components are the spherical selenium (Se) droplet, around which a NbSe3 bar wraps due to surface tension, and the monoclinic (P2(1)/m) crystal symmetry inherent in NbSe3, which induces a twist in the strip when bent. Our crystals provide a non-fictitious topological Mobius world governed by a non-trivial real-space topology. We class ed these topological crystals as an intermediary between condensed matter physics and mathematics. References [1]A Mobius strip of single crystals, S. Tanda et al., Nature 417, 397 (2002). [2]Formation and growth of NbSe3 topological crystals, T. Tsuneta and S.Tanda J. Cryst. Growth 267, 223 (2004). [3]Topologically linked crystals, T. Matsuura et al., J. Cryst. Growth 297, 157 (2006). [4]Topological eff ects of the superconducting vortex state in a TaSe3 ring crystal, G. Kumagai et al., Phys. Rev. B81, 184506 (2010). [5]Chiral Charge-Density waves, J. Ishioka et al., Phys. Rev. Lett 105, 176401 (2010). [6]Topology-change surgery for crystals, T. Matsuura et al., Phys. Rev. B83 174113 (2011).
TODW01 23rd July 2012
17:10 to 17:30
Numerical and analytical study of an asymptotic equation for deformation of vortex lattices
It is known that when two-dimensional flows are subject to a suitable background rotation, formation of vortex lattices are observed. We can make use of critical points of the vorticity field and their connectivity (so-called, surface networks) to study reconnection of vorticity contours in 2D turbulence. In this talk we begin by noting how this method applies to the study of formation of vortex lattices. We then study a coarse-grained, asymptotic equation which describes deformation vortex lattices derived by Smirnov and Chukbar, Sov. Phys. JETP vol 93, 126-135(2001). It reads $\phi_t=\phi_{xx} \phi_{yy}-\phi_{xy}^2,$ where $\phi$ denotes displacement of vortex locations. This equation is particularly valid for geostrophic Bessel vortices with a screened interaction. Numerical results are reported which indicate an ill-posed nature of the time evolution. Self-similar blow-up solutions were already given by those authors, which have an infinite total energy. We ask whether finite-time blow-up can take place developing from smooth initial data with a finite energy. More general self-similar blow-up solutions are sought, but all are found to have infinite total energy. Finally, remarks are made in connection with the Tkachenko-type lattice.
TODW01 24th July 2012
09:00 to 09:40
Knotted vortex tubes in the Euler equation
In this talk we will address the problem of the existence of stationary knotted and linked vortex tubes for the Euler equation. The existence of these structures is a very interesting question in Fluid Mechanics, which dates back to Lord Kelvin's studies of thin vortex tubes. We will discuss some contributions to this problem (jointly with Alberto Enciso) using Beltrami fields. This builds upon (and significantly improves) our previous work on linked vortex lines of steady solutions to the Euler equation.
TODW01 24th July 2012
09:45 to 10:05
A Enciso Knots and links in fluid mechanics
In this talk I will discuss the existence of steady solutions to the incompressible Euler equations that have stream and vortex lines of any prescribed knot (or link) type. More precisely, I will show that, given any locally finite link L in R^3, one can transform it by a smooth diffeomorphism F, close to the identity in any C^p norm, such that F(L) is a set of periodic trajectories of a real analytic steady solution u of the Euler equations in R^3. If the link is finite, we shall also see that u can be assumed to decay as 1/|x| at infinity, so that u is in L^p for all p>3. This problem is motivated by the well-known analysis of the structure of steady incompressible flows due to V.I. Arnold and K. Moffatt, among others. Time permitting, we will also very recent results on the topology of potential flows, that is, of steady fluids whose velocity field is the gradient of a harmonic function in R^3. These results are closely related to classic questions in potential theory that were first considered by M. Morse and W. Kaplan in the first half of the XX century and have been revisited several times after that, by Rubel, Shiota and others. The guiding principle of the talk will be that a strategy of "local, analysis-based constructions" + "global approximation methods", fitted together using ideas from differential topology, can be used to shed some light on the qualitative behavior of steady fluid flows. Most of the original results presented in this talk will be based on the papers: A. Enciso, D. Peralta-Salas, Knots and links in steady solutions of the Euler equation, Ann. of Math. 175 (2012) 345-367. A. Enciso, D. Peralta-Salas, Submanifolds that are level sets of solutions to a second-order elliptic PDE, arXiv:1007.5181. A. Enciso, D. Peralta-Salas, Arnold's structure theorem revisited, in preparation.
TODW01 24th July 2012
10:10 to 10:30
F Maggioni & SZ Alamri & CF Barenghi & RL Ricca Velocity, energy and helicity of vortex knots and unknots
In this talk we examine the effect of several geometric and topological aspects on the dynamics and energetics of vortex torus knots and unknots. The knots are given by small-amplitude torus knot solutions [1] to the Localized Induction Approximation (LIA) law. Vortex evolution is thus studied in the context of the Euler equations by direct numerical integration of the Biot-Savart law. Earlier stability results on vortex knots and unknots [2] are here extended [3]-[4], and the velocity, helicity and kinetic energy of different vortex knots and unknots are presented for comparison. Vortex complexity is parametrized by the winding number w given by the ratio of the number of meridian wraps to that of longitudinal wraps. We find that for w 1 knots and poloidal coils have approximately same speed and energy of the reference vortex ring. Kinetic helicity is dominated by writhe contributions and increases with knot complexity. The stabilizing effect of the Biot-Savart law for all knots and unknots tested is also confirmed. Our results provide information on relationships between geometry, topology and dynamics of complex vortex systems and apply to quantized vortices in superfluid 4He.

References

[1] Ricca, R.L. (1993) Torus knots and polynomial invariants for a class of soliton equations. Chaos 3, 83-91. [1995 Erratum. Chaos 5, 346.]

[2] Ricca, R.L., Samuels, D.C. & Barenghi, C.F. (1999) Evolution of vortex knots. J. Fluid. Mech. 391, 29-44.

[3] Maggioni, F., Alamri, S.Z., Barenghi, C.F. & Ricca, R.L. (2009) Kinetic energy of vortex knots and unknots. Il Nuovo Cimento C, 32(1), 133–142.

[4] Maggioni, F., Alamri, S., Barenghi, C.F. & Ricca R.L. (2010) Velocity, energy and helicity of vortex knots and unknots. Phys. Rev. E., 82(2), 026309–026317.

TODW01 24th July 2012
11:00 to 11:40
Intermittency and conditional regularity of solutions of the 3D Navier-Stokes equations
Both numerical and experimental evidence suggests that solutions of the three-dimensional Navier-Stokes equations display a high degree of intermittency, which is manifest in spiky excursions away from averages in the vorticity field. This phenomenon may be intimately bound up with the enduring problem of regularity and is addressed by discussing two new conditional (& unusual) regularity assumptions.
TODW01 24th July 2012
11:45 to 12:05
Quantum vortex reconnections
In superfluid helium and in atomic Bose-Einstein condensates, quantum mechanics constrains the rotational motion to discrete filaments of fixed circulation which is equal to Planck's constant divided by the mass of the relevant boson. Because of their simplicity (no viscosity, vorticity confined to vortex lines, fixed circulation), quantum fluids are ideal systems where to study the topology of vortex flows. In this talk I shall report results on the motion of vortex rings perturbed by Kelvin waves (a classical problem first studied by Lord Kelvin), vortex bundles, vortex knots and turbulent tangles of such discrete vortices. In the case of turbulence, I shall focus on its properties, and the relation between kinetic energy and vortex length.
TODW01 24th July 2012
12:10 to 12:30
An accurate and efficient method to compute steady vortices without symmetry
When considering steady solutions of the Euler equations, it is often of interest to find isovortical flows, that is, solutions that can be obtained from rearrangements of a given vorticity distribution. Since inviscid transitions between such flows are, in principle, possible, these solutions may act as attractors in the unsteady dynamics (e.g. Dritschel 1986, 1995; Flierl & Morrison 2011). The computation of such steady vortex flows still presents some challenges. Existing Newton iteration methods become inefficient as the vortices develop fine-scale features; in addition, these methods do not, in general, find solutions from isovortical rearrangements. On the other hand, available relaxation approaches are more affordable, but their convergence is not guaranteed. In this work, we consider flows that may be approximated by a collection of uniform vortices, and overcome the limitations outlined above by using a discretization, based on an inverse-velocity mapping, which radically increases the efficiency of Newton iteration methods. In addition, we introduce a procedure to enforce the isovortical constraint in the solution method. We illustrate our methodology by exploring the solution structure of a wide range of unbounded flows. We uncover several families of lower-symmetry vortices. While asymmetric point vortex flows have been found by Aref & Vainchtein (1998), it appears that this is the first time that nonsingular, asymmetric steady vortices have been computed. In addition, we discover that, as the limiting vortex state for each flow is approached, each family of solutions traces a clockwise spiral in a bifurcation plot consisting of a velocity-impulse diagram. By the recently introduced ‘‘IVI diagram’’ stability approach (Luzzatto-Fegiz & Williamson 2010, 2011), each turn of this spiral is associated with a loss of stability. Such spiral structure is suggested to be a universal feature of steady, uniform-vorticity Euler flows.
TODW01 24th July 2012
14:00 to 14:40
Instabilities and ill-posedness for the magnetogeostrophic equation
We discuss an active scalar equation that is motivated by a model presented by Keith Moffatt for the geodynamo and magnetogeostrophic turbulence in the Earth's fluid core. We prove that the non-difusive equation is Lipshitz ill-posed in Sobolev spaces. In contrast, the diffusive equation is globally well-posed. In this case we give an example of a steady state that is nonlinearly unstable, and hence produces a dynamo effect in the sense of an exponentially growing magnetic field. This is joint work with Vlad Vicol.
TODW01 24th July 2012
14:45 to 15:05
RM Kerr Dissipation and enstrophy statistics in turbulence: are the simulations and mathematics converging?
This presentation will be based upon the Focus on Fluids article with this title to appear in JFM 700 (2012). The Focus will be on: Yeung, Donzis, & Sreenivasan, 2012 Dissipation, enstrophy and pressure statistics in turbulence simulations at high Reynolds numbers. J. Fluid Mech. 700 and the two themes of the FoF are that Yeung et al resolves a remaining question about the convergence of higher-order statistics and that this result is related to new mathematics on temporal intermittency in turbulence in Gibbon, J.D. 2009 Estimating intermittency in three- dimensional Navier-Stokes turbulence. J. Fluid Mech. 625. What Yeung et al. finds is that even if the fluctuations of the higher-order vorticity and strain statistics are so large that they do not converge individually, their ratios do converge. Gibbon (2009) shows that this type of behaviour is expected and Gibbon (TODW01) will present specific predictions for the ordering of these statistics at any given time and the t ype of maximum growth during the most intermittent periods. However, Yeung et al does not give time variations, so a direct comparison is not possible. My new results are from simulations of the reconnection of anti-parallel vortex tubes, an example of the events assumed by Gibbon (2009), where this time-dependent analysis has been done. This simulation develops, after just two reconnection steps, most of the properties associated with fully-developed turbulence, including a -5/3 spectrum with the proper coefficient and the expected enstrophy production skewness, and the intermittency ratios are consistent with Yeung et al. Turbulence develops after reconnections by: Forming orthogonal vortices, which wrap up as in the Lundgren spiral vortex model. The temporal ordering and growth of the higher-order vorticity statistics obey the bounds of the new mathematical predictions exactly. Thus the connection between the latest high Reynolds number calculations and the latest mathematics is demonstrated.
TODW01 24th July 2012
15:10 to 15:30
D Kolomenskiy & HK Moffatt & M Farge & K Schneider Fluid dynamics of flapping wings associated with change of domain topology
We re-examine the clap-fling-sweep mechanism employed by some insects to increase lift. As argued by Lighthill (J Fluid Mech 60(1):1-17, 1973), this mechanism can create a circulatory motion even in a totally inviscid fluid, due to a topological change of the solid boundary that represents the wings surfaces. During the stroke, the wings first clap together behind the insect's back, then open in a fling motion around the `hinge' formed by the two trailing edges, and finally separate at the hinge and sweep apart. In a two-dimensional approximation, we use two different conformal mappings in simply and doubly connected domains, respectively, to calculate the complex potential at all stages of the process. The results indicate that circulation (equal in magnitude and opposite round the two wings) can be generated in an inviscid fluid, and that this circulation appears when a solid body immersed in the fluid breaks into two pieces (when fling gives way to sweep). Bound vortex sheets produced during fling are still carried by the just-separated wings. This is accompanied by a continuous time evolution of the velocity everywhere in the fluid, although the pressure field jumps instantaneously at the moment of wing separation. In a viscous fluid, the flow during the break is essentially different because, locally, the Reynolds number is very low near the hinge point. We describe it by local similarity solutions to the Stokes equation (J Fluid Mech 676:572-606, 2011). Three-dimensional effects are present in the flow. We study them by performing numerical simulations of the Navier-Stokes equations using a Fourier spectral method with volume penalization. The flow before the break is found to be in a good agreement with the two-dimensional approximation. After the wings move farther than one chord length apart, the three-dimensional nature of the flow becomes essential (J Fluids Struct 27(5-6):784-791, 2011).
TODW01 24th July 2012
15:35 to 15:55
Vortical dynamo in turbulent multiphase flows
Magnetic disturbances are known to be amplified by helical turbulence. The possibility of amplification of large-scale hydrodynamic fields by small-scale helical turbulence is considered. The important difference between hydrodynamic and magnetic theories is that the latter describe the evolution of magnetic field on the background of a given hydrodynamic flow (kinematic dynamo), whereas in hydrodynamics such a situation is more complex. The hydrodynamic problem is self-consistent and non-linear. A generation of large-scale helical vortices resulting from the instability of small-scale helical turbulence with respect to two-scale disturbance is considered. In order to investigate such instability, we consider two cases: (1) an incompressible fluid containing rigid particles; (2) an incompressible fluid containing gas babbles. An equation describing the evolution of mean disturbances is derived and the instability increment is obtained. The analysis revealed that helical turbulence in an incompressible fluid with rigid particles and in incompressible fluid with gas babbles is unstable against vortical disturbances. The generation terms formally coinciding with those in the theory of hydromagnetic dynamo are contained in Reynolds averaged equations derived at the scale of mean motions. It should be noted that only helicity is enough for the process of generation in magnetohydrodynamics. In hydrodynamic theory, because of the mentioned differences, it is also necessary to take into account additional factors. In this paper two such additional factors are the presence of rigid particles or gas babbles whose motions provide the existence of divergence at a turbulent scale and thus provide a non-zero value of the Reynolds stresses in the averaged equations.
TODW01 24th July 2012
15:55 to 16:05
A Libin Coherent Beltrami Structures
TODW01 24th July 2012
16:45 to 17:30
Relative equilibria of point vortices. (Aref Memorial Lecture)
A relative equilibrium of a system of point vortices is a configuration which rotates with constant angular velocity around its centre of vorticity. It is easy to write down the equations for the vortex positions and many simple configurations with symmetry are known. Several asymmetric states have been found numerically, including some surprising ones with some of the vortices being very close. Very little is known analytically about the general problem. Here we consider the case where the vortices are identical and placed on two perpendicular lines which we choose to be the axes of a coordinate system. We define two polynomials p(z) and q(z) whose roots are the vortex positions on each line in the complex plane, and derive a differential equation for p for given q. We discuss how the general solution to the differential equation relates to physical vortex configurations. The main result is that if q has m solutions symmetrically placed relative to the real axis and p is of degree n, it must have at least n-m+2 real roots. For m=2 this is a complete characterisation, and we obtain an asymptotic result for the location of the two vortices on the imaginary axis as the number of vortices on the real axis tends to infinity.
TODW01 25th July 2012
09:00 to 09:40
B Khesin The filament equation in any dimension
We show that the LIA approximation of the incompressible Euler equation describes the skew-mean-curvature flow on vortex membranes in any dimension. This generalizes the classical binormal, or vortex filament, equation in 3D. We present a Hamiltonian framework for higher-dimensional vortex filaments and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively. This framework, in particular, allows one to define the symplectic structures on the spaces of vortex sheets.
TODW01 25th July 2012
09:45 to 10:05
Tackling structural complexity by Jones' polynomials
In this paper we present new results based on applications of knot theory to tackle and quantify structural complexity of vortex dynamics. In the ideal case of Euler equations we show that the topology of any vortex tangle, made by knots and links, can be identified and described by the Jones’s polynomials of the tangle, expressed in terms of kinetic helicity. Explicit calculations of the Jones polynomial for the left-handed and right-handed trefoil knots and for the Whitehead link via the figure-of-eight knot are worked out for illustration. This novel approach contributes to establish a fundamentally new paradigm in topological fluid mechanics, by extending the former interpretation of helicity in terms of linking numbers to the much richer context of knot polynomials, and by opening up new directions of work both in mathematical fluid dynamics and in numerical diagnostics of vortex flows.
TODW01 25th July 2012
10:10 to 10:30
Y Mitsumatsu Helicity in differential topology
The helicity plays many intresting roles in 3-dimensinal diffential topology. One of its earliest appearences in was the question by Dennis Sullivan, asking to express the Godbilon-Vey invariant in terms of linking of fluids. Here the Godbillon-Vey is an invariant for codimension1 foliations which lives in the 3rd de Rham cohomology. The question is well-understood and we know which fluid motion should be taken. If we think of helicity as a quadratic function on the space of incompressible fluids, namely the space of divergence free vector fields, it goes down to a symmetric bilinear form. Some ideas concerning this bilinear form for studies of foliations and contact structures are introduced. For example, in the case of codimmension one foliations the 1st foliated cohomology will appear. If the foliation is deofrmed to contact structures, unexpectedly phenomena which might be related to a quatization procedure is found.
TODW01 25th July 2012
11:00 to 11:40
How much do we know about the problem of global regularity for the three-dimensional Navier-Stokes and Euler equations?
In this talk I will survey the status of, and the most recent advances concerning, the questions of global regularity of solutions to the three-dimensional Navier-Stokes and Euler equations of incompressible fluids. Furthermore, I will also present recent global regularity results concerning certain three-dimensional geophysical flows, including the three-dimensional viscous "primitive equations" of oceanic and atmospheric dynamics.
TODW01 25th July 2012
11:45 to 12:05
Impulse of vortex knots from diagram projections
By using methods based on the analysis of standard plane projections of complex tangles, we can extract geometric information and use it to determine the impulsive forces associated with vortex knots and links. This method relies on the interpretation of linear and angular momentum of ideal vortex filaments in terms of projected areas [1]. An immediate application of this method allows one to make predictive estimates of the evolution and dynamical features of vortex knots and links [2]. This will be illustrated by a number of examples, some related to well-known results from laboratory and numerical experiments, where vortex ring collisions and vortex linking have been studied, and some others, such as the production of a trefoil vortex knot, proposed as "thought" experiments [3].

[1] Ricca, R.L. (2008) Momenta of a vortex tangle by structural complexity analysis. Physica D 237, 2223-2227.

[2] Ricca, R.L. (2009) Structural complexity and dynamical systems. In Lectures on Topological Fluid Mechanics (ed. R.L. Ricca), pp. 169-188. Springer-CIME Lecture Notes in Mathematics 1973. Springer-Verlag.

[3] Ricca, R.L. (2009) New developments in topological fluid mechanics. Nuovo Cimento C 32, 185-192.
TODW01 25th July 2012
12:10 to 12:30
A Soap-Film Mobius Strip Changes Topology with a Twist Singularity
It is well-known that a soap film spanning a looped wire can have the topology of a Möbius strip and that deformations of the wire can induce a transformation to a two-sided film, but the process by which this transformation is achieved has remained unknown. In this talk I will disucss recent experimental and theoretical work [Goldstein, Moffatt, Pesci, and Ricca, PNAS 107, 21979 (2010)] that has uncovered the dynamics of this transition. We find that this process consists of a collapse of the film toward the boundary that produces a previously unrecognized finite-time twist singularity that changes the linking number of the film's Plateau border and the centerline of the wire. We conjecture that it is a general feature of this type of transition that the singularity always occurs at the surface boundary. The change in linking number is shown to be a consequence of a viscous reconnection of the Plateau border at the moment of the singularity. High-speed imaging of the collapse dynamics of the film's throat, similar to that of the central opening of a catenoid, reveals a crossover between two power laws. Far from the singularity, it is suggested that the collapse is controlled by dissipation within the fluid film surrounding the wire, whereas closer to the transition the power law has the classical form arising from a balance between air inertia and surface tension. Analytical and numerical studies of minimal surfaces and ruled surfaces are used to gain insight into the energetics underlying the transition and the twisted geometry in the neighborhood of the singularity. A number of challenging mathematical questions arising from these observations are posed.
TODW01 26th July 2012
09:00 to 09:40
Topological approaches to problems of stirring and mixing
I will review two topological approaches to stirring and mixing. The first involves constructing systems such that the fluid motion is topologically complex, usually by imposing a specific motion of rods. I will discuss optimization strategies that can be implemented. The second is diagnostic, where flow characteristics are deduced from observations of periodic or random orbits and their topological properties.
TODW01 26th July 2012
09:45 to 10:05
G Hornig & A Wilmot-Smith & D Pontin Relaxation of braided magnetic and vorticity fields
In this talk we will first report on a series of numerical MHD experiments on the turbulent relaxation of braided magnetic fields in plasmas of high magnetic Reynolds numbers (Wilmot-Smith et al. 2009, 2010). These experiments have produced relaxed states which in some cases differ drastically from the predictions of the Taylor hypothesis, that is the assumption that the final state of a turbulent relaxation is a linear force-free field with the same total helicity as the initial state. We present a method to determine the topological degree of the field line mapping which shows that there are further constraints on the relaxation process beyond the conservation of the total helicity (A. Yeates et al., Phys. Rev. Lett. 105, 2010). These constraints can prevent the system from relaxing to a Taylor state and hence limit the energy which can be released. We will then report on a new series of experiments where we test whether similar constraints hold in the hydrodynamic case, that is we investigate the relaxation of incompressible flows with braided vorticity field lines.
TODW01 26th July 2012
10:10 to 10:30
O Velasco Fuentes Stirring vortices with vorticity holes
A vorticity hole is a region with, in absolute value, significantly lower vorticity than its surroundings. Here we discuss the dynamics of a Rankine vortex with either one elliptic hole or two equal circular holes. If we assume symmetry and null vorticity within the holes, the evolution only depends on the hole size and either the aspect ratio of the elliptic hole or the separation of the circular holes. We computed the evolution with a contour-dynamics model and found that it is analogous to either that of the Kirchhoff vortex or that of the vortex pair, but the vorticity holes are additionally affected by their interaction with the boundary of the Rankine vortex. To quantify the stirring of fluid particles, both inside and outside the vortex, we analysed the set of hyperbolic trajectories and associated manifolds of the time-evolving velocity field. The strongest stirring always occurred in the areas of highest vorticity, which contradicts the generally accepted notion that vortices are regions of null to weak stirring.
TODW01 26th July 2012
11:00 to 11:40
Identifying topological chaos using set-oriented methods
Identifying topological chaos using the Thurston-Nielsen classification theorem (TNCT) is a powerful approach to quantifying and predicting chaos in a variety of fluid systems. This approach is most easily applied to systems stirred by physical rods, since the rods can be prescribed to move on sufficiently complex space-time trajectories. In many cases, however, an analysis based solely on the motion of physical rods cannot capture the full complexity of the flow. Consideration of 'ghost rods', or material particles that 'stir' the fluid, can provide the missing information needed for an accurate topological representation. Unfortunately, even when such low-order periodic orbits exist, they can be difficult to identify. We will discuss the use of set-oriented, or mapping-based, statistical methods for identifying periodic regions in the domain having high local residence time. These 'almost-cyclic sets' can reveal the underlying topology of the s ystem, enabling application of the TNCT even in the absence of low-order periodic orbits. Viscous flow examples show that this approach can provide a good representation of system behavior over a range of parameters.
TODW01 26th July 2012
11:45 to 12:05
Y Kimura Mass transport by vortex motions
Mass transport by the motion of 3D vortex filaments is studied. Perhaps the simplest example of such a transport is one by a vortex ring. As another example, we recently demonstrated that a 3D vortex soliton (Hasimoto soliton) can also transport mass along with its propagation. The common features of the transport, one by a vortex ring and the other by a vortex, is that some fluid particles move in closed orbits which make knots with the vortices. For the analysis of the mechanism of this trapping, a model composed of two straight vortex filaments in the 3D space, which is called the chopsticks model, is presented. We report mass transport by this model.
TODW01 26th July 2012
12:10 to 12:30
On the regularity of Lagrangian trajectories in the 3D Navier-Stokes flow
The paper considers suitable weak solutions of the 3D Navier-Stokes equations. Such solutions are defined globally in time and satisfy local energy inequality but they are not known to be regular. However, as it was proved in a seminal paper by Caffarelli, Kohn and Nirenberg, their singular set S in space-time must be ``rather small'' as its one-dimensional parabolic Hausdorff measure is zero. In the paper we use this fact to prove that almost all Lagrangian trajectories corresponding to a given suitable weak solution avoid a singular set in space-time. As a result for almost all initial conditions in the domain of the flow Lagrangian trajectories generated by a suitable weak solution are unique and C^1 functions of time. This is a joint work with James C. Robinson.
TODW01 26th July 2012
14:00 to 14:40
Instability by weak precession of the flow in a rotating sphere
The linear stability analysis is performed of the steady flow in a weakly precessing sphere of rapid rotaion. It is well-known that all the disturbances damp with decay rate proportional to Re^{-1/2} without precession, where Re is the Reynolds number defined by the sphere radius, the the spin angular velococity, and the kinematic viscosity of fluid. We show by an asymptotic analysis for large Re and small Gamma, the ratio of the precession and spin angular velocities, that with weak precession of Gamma of order Re^{-1/2} destabilizes the disturbances by the coupling between an symmetric (with respect to the spin axis) mode and (2,1,1) mode through "the conical shear layers" emanating from the critical circles along the sphere boundary. It is found the critical curve for the instability behaves as Gamma = $7.9 Re^{-0.5}$ asymptotically, which agrees well with an observation in an precessing spheroid of ellipticity $0.9$ by Goto {\it et al.} (2011).
TODW01 26th July 2012
14:45 to 15:05
Evolution of the leading edge vortex over an accelerating rotating wing
Flapping flight is a subject of interest for more than two decades. During this time it has been found that a stable leading edge vortex is responsible for the high lift that flapping and revolving wings can produce. However, many of these studies were limited to Reynolds numbers of few hundred, which characterize insects. Recently, the interest on designing and realizing miniature hovering vehicles requires expanding our understanding of the basic flow mechanism which govern such wing maneuvers at higher Reynolds numbers. In this study the flow field over an accelerating rotating wing model is analyzed in various Reynolds numbers ranging from 250 to 2000 using particle image velocimetry. These experimental results are compared with three-dimensional and time-accurate Navier-Stokes flow simulations. The study depicts the characteristic size and time scales of the leading-edge vortex. The results show that the topology of the leading-edge vortex is Reynolds number dependent; i n comparison to a diffused and detached leading-edge vortex at Reynolds number 250, at Reynolds number 2000 the leading-edge vortex is not stationary and can cover up to about 75 percent of the local wing chord. Furthermore, it is shown that the spanwise velocity component increases considerably at Reynolds number of 1000 and above. Moreover, in Reynolds number 250 the circulation within the leading-edge vortex during wing acceleration exceeds its asymptotic value which develops over steadily revolving wings. At Reynolds number 1000 and above, on the other hand, the circulation within the leading-edge vortex evolves much slower. These findings shed new insights about the differences between the aerodynamic characteristics of steady revolving wings and flapping ones and will be utilized to investigate the stability of the leading-edge vortex in wider range of Reynolds numbers.
TODW01 26th July 2012
15:10 to 15:30
Parallel computation of vortex tube reconnection using graphics card and vortex particle methods
Understanding the dynamics and mutual interaction among various types of vortical motions is a key ingredient in clarifying and controlling fluid motions . One of the most fundamental 3D vortical interactions related to the vortex tube reconnection. In the paper will be present the numerical results of the vortex tube reconnections for different initial configurations like reconnection o the vortex tube with counter-rotating vortices that are initially parallel and was sinusoidal perturbed (Crow instability), the vortex reconnection of the initially straight offset tube and reconnection of the vortex rings . We try to find and demonstrate some universal process for core reconnections. It will be shown the effect of mixing of the fluid by the reconnection. It was done by tracing the passive markers that were initially placed near the space where the reconnection took place. For numerical simulations we use the vortex particle methods. Due to the large time consuming at single processor unit we constructed the numerical code for multiprocessor unit of graphics card. It was proved that vortex particle method in version Vortex -in Cell are very good suited for parallel computation. We carefully tested the method by comparing the numerical results with some theoretical results (the motion of the vortex ring) and with results that were published in literature. The speed-up which we obtained was nearly 50 times grater with comparison to the single processor.
TODW01 26th July 2012
15:35 to 15:55
Cascade of vortex loops initiated by a single reconnection of quantum vortices
We demonstrate that a single reconnection of two quantum vortices can lead to the creation of a cascade of vortex rings. Our analysis involves localized induction approximation, high-resolution Biot-Savart and Gross-Pitaevskii simulations. The latter showed that the rings cascade starts on the atomic scale, with rings diameters orders of magnitude smaller than the characteristic line spacing in the tangle. Vortex rings created in the cascades may penetrate the tangle and annihilate on the boundaries. This provides an efficient decay mechanism for sparse or moderately dense vortex tangle at very low temperatures.
TODW01 26th July 2012
15:55 to 16:05
Point-vortex dynamics and stability of relative equilibria on surfaces
TODW01 26th July 2012
16:45 to 17:30
Finite time singularities for the free boundary incompressible Euler equations
We prove the existence of smooth initial data for the 2D free boundary incompressible Euler equations (also known for some particular scenarios as the water wave problem), for which the smoothness of the interface breaks down in finite time into a splash singularity or a splat singularity. Moreover, we show a stability result together with numerical evidence that there exist solutions of the 2D water wave equation that start from a graph, turn over and collapse in a splash singularity (self intersecting curve in one point) in finite time. Joint work with A. Castro, C. Fefferman, F. Gancedo and J. Gomez-Serrano.
TODW01 27th July 2012
09:00 to 09:40
Three-dimensional vorticity dynamics in miscible Hele-Shaw displacements
We perform three-dimensional DNS simulations of the transient, variable viscosity Navier-Stokes equations in the Boussinesq approximation, coupled to a convection-diffusion equation for a concentration field, to simulate miscible viscous fingers in Hele-Shaw cells. The three-dimensional problem allows for new instabilities and patterns that cannot be captured by traditional gap-averaged modeling. For constant density displacements, the simulations reveal the mechanism by which the initial spanwise vorticity of the base flow, when perturbed, gives rise to the cross-gap vorticity that drives the fingering instability in the classical Darcy sense. Cross-sections at constant streamwise locations reveal the existence of a streamwise vorticity quadrupole that induces fluid transport from the walls of the cell to its center, thereby leading to a new hydrodynamic instability, termed 'inner splitting' that had not been previously reported. If gravity is included, the nature of the two-dimensional base flow and its subsequent instability changes dramatically. The interaction between Saffman-Taylor and Rayleigh-Taylor instabilities can lead to additional splitting events, and it can significantly enhance the mixing rates of the two fluids, thereby altering the overall displacement efficiency.
TODW01 27th July 2012
09:45 to 10:05
Y Hattori & SG Llewellyn-Smith Motion of axisymmetric magnetic eddies with swirl
We consider the motion of axisymmetric magnetic eddies with swirl in ideal MHD (magnetohydrodynamics) flow. The magnetic field is assumed to be toroidal, while the velocity field has both toroidal and poloidal components. First, the contour-dynamics formulation by Hattori and Moffatt (2006) for the case without swirl is extended to include swirl velocity so that the cross helicity does not vanish in general. The strength of vortex sheets which appear on the contours varies with time under the influence of the centrifugal force due to swirl and the magnetic tension due to the Lorentz force. Numerical simulation using the contour-dynamics formulation shows that there exist counter-propagating dipolar structures at the radius of balance between the centrifugal force and the magnetic tension; these structures are well described by the steady solutions obtained by perturbation expansion. The effects of vorticity inside the eddy on the motion of the eddies are also investigated. Next, we seek exact steady spherical solutions of magnetic eddies with swirl. A generalized family of exact solutions is found; it includes Hill’s spherical vortex, the Hicks-Moffatt family which is a non-MHD vortex with swirl and the family of a MHD vortex without swirl found by Hattori and Moffatt (2006).
TODW01 27th July 2012
10:10 to 10:30
Variational method for estimating nonlinear acceleration of collisionless magnetic reconnection
Magnetic reconnection in collisionless plasma is studied by applying a technique that utilizes variational principle. Following the fact that a model of collisionless (namely, dissipationless) plasma constitutes a Hamiltonian system, the corresponding variational principle is formulated, where the displacement field of fluid elements is the dynamical variable to be varied. The effect of so-called "electron inertia" is a singular perturbation that modifies the topological invariant of plasma and, hence, allows magnetic reconnection to occur without any dissipation mechanism. If the potential energy of this variational principle decreases for some displacement field (under the modified topological constraint), such the fluid motion turns out to grow with the release of free energy. A rather simple fluid motion is enough to prove the occurrence of spontaneous magnetic reconnection. Decrease of potential energy in the nonlinear regime (where the magnetic island is larger than the width of boundary layer) is found to be steeper than in the linear regime, resulting in acceleration of the reconnection.
TODW01 27th July 2012
11:00 to 11:40
Singular Casimir elements: their mathematical justification and physical implications
The problem of singular Poisson operator, which occurs in the noncanonical Hamiltonian formulation of equations describing ideal fluid and plasma dynamics, is studied in the context of a Casimir deficit, where Casimir elements constitute the center of the Lie-Poisson algebra underlying the Hamiltonian formulation. The nonlinearity of the evolution equation makes the Poisson operator inhomogeneous on phase space (the function space of the state variable), and it is seen that this creates a singularity where the nullity of the Poisson operator (the "dimension" of the center) changes. Singular Casimir elements stemming from this singularity are unearthed using a generalized functional derivative (which may be regarded as hyperfunctions generated by an "infinite-dimensional partial differential operator" with a singularity). A singular perturbation (introduced by finite dissipation) destroys the leaves foliated by Casimir elements, removing the topological constraint and allowing the state vector to move towards lower-energy state in unconstrained phase space. The dynamics of an ideal plasma is constrained by the helicity as well as helical-flux Casimir elements pertinent to resonant singularities. A finite-resistivity singular perturbation gives rise to negative-energy "tearing modes" by destroying the helical-flux Casimir leaves.
TODW01 27th July 2012
11:45 to 12:05
A unified view of topological invariants of barotropic & baroclinic fluids & their application to formal stability analysis of three-dimensional ideal gas flows
Integrals of an arbitrary function of the vorticity, two-dimensional topological invariants of an ideal barotropic fluid, take different guise from the helicity. Noether's theorem associated with the particle relabeling symmetry group leads us to a unified view that all the topological invariants of a barotropic fluid are variants of the cross helicity. Baroclinic fluid flows admit, as the Casimir invariants, a class of integrals including an arbitrary function of the entropy and the potential vorticity. A consideration is given to them from the view point of Noether's theorem. We then develop a new energy-Casimir convexity method for a baroclinic fluid, and establish a novel linear stability criterion, to three-dimensional disturbances, for equilibria of general rotating flows of an ideal gas without appealing to the Boussinesq approximation. By exploiting a larger class of the Casimir invariants, we have succeeded in ruling out a term including the gradient of a dep endent variable from the energy-Casimir function. For zonally symmetric flows, the resulting criterion is regarded as an extended Richardson number criterion for stratified rotating shear flows with compressibility taken into account.
TODW01 27th July 2012
12:10 to 12:30
Non-stationary boundary layers and energy dissipation in incompressible flows
In fully turbulent incompressible flows, the two presumed culprits for energy dissipation are thin boundary layers on the one hand, and three-dimensional stretching of vortex tubes on the other hand. Although both suspects seem equally important, much more theoretical effort has been invested to study the latter, due to the emphasis that has been put on homogeneous turbulence since the 1930s. We argue that the same effort should be dedicated to interrogate the former, especially since it abides the simpler two-dimensional (2D) setting. Indeed, while the shortcomings of the perfect fluid model, which lead to the d'Alembert paradox, are well understood, the failure of the Prandtl viscous boundary layer theory to predict the right scaling of energy dissipation still lacks a complete explanation. Several possibilities have been explored in recent years, like finite time singularities in the Prandtl equations, ill-posedness of the Prandtl equations, or even earlier breakdown of the asymptotic expansion itself. What happens beyond the Prandtl regime is even more mysterious. According to a criterion proven by Kato, energy should then be dissipated at scales as fine as the inverse of the Reynolds number, but the process by which this could happen remains elusive. We review the main issues at hand, using a 2D dipole-wall interaction as illustrative test case. First we re-derive the Euler-Prandtl equations in the vorticity formulation, solve them numerically, and pinpoint the origin of the discrepancy with a reference Navier-Stokes solution. We then proceed to explore alternative asymptotic expansions, allowing for a localized collapse of the boundary layer to finer scales, emphasizing in particular the importance of the topology of the vorticity field, as well as the consequences regarding the scaling of energy dissipation.
TODW01 27th July 2012
14:00 to 14:40
How fast can vorticity stretch itself?
We consider some examples of computing or bounding the long-time growth of vorticity in Euler flows under various assumptions. Estimates for axisymmetric flow without swirl, obtained under constraints on vorticity volume and and kinetic energy, are discussed. Axisymmetric flow with swirl is treated under a truncation equivalent to a axisymmetric "stretched" version of the Taylor-Green problem. The close relation of flows with swirl to 2D Boussinesq convection is used to give some examples of periodic flows developing arbitrarily large vorticity in a bounded domain. Finally, a moving line problem is introduced which produces finite time singularities. It is noted that locally 2D vortex structures of the kind obtained in the growth estimates without swirl motion could be consistent with the equations of motion of the line, but no computable examples of this are known.
TODW01 27th July 2012
14:45 to 15:05
Absence of singular stretching of interacting vortex filaments
A promising mechanism for generating a finite-time singularity in the incompressible Euler equations is the stretching of vortex filaments. Here, we argue that interacting vortex filaments cannot generate a singularity by analyzing the asymptotic dynamics of their collapse. We use the separation of the dynamics of the filament shape, from that of its core to derive constraints that must be satisfied for a singular solution to remain self consistent uniformly in time. Our only assumption is that the length scales characterizing filament shape obey scaling laws set by the dimension of circulation as the singularity is approached. The core radius necessarily evolves on a different length scale. We show that a self similar ansatz for the filament shapes cannot induce singular stretching, due to the logarithmic prefactor in the self interaction term for the filaments. More generally, there is an antagonistic relationship between the stretching rate of the filaments and the requ irement that the radius of curvature of filament shape obeys the dimensional scaling laws. This suggests that it is unlikely that solutions in which the core radii vanish sufficiently fast to maintain the filament approximation exist.
TODW01 27th July 2012
15:10 to 15:30
Probing fundamental bounds in hydrodynamics using variational optimisation methods
This work demonstrates how the modern methods of PDE-constrained optimization can be used to assess sharpness of a class of fundamental functional estimates in fluid mechanics. These estimates concern bounds on the instantaneous rate of growth and finite-time growth of quadratic quantities such as the enstrophy and palinstrophy in viscous incompressible flows. Sharpness of such estimates is inherently related to the problem of singularity formation in the 3D Navier-Stokes system. In our presentation we will first review earlier results of Lu & Doering (2008) and Ayala & Protas (2011) concerning the maximum growth of enstrophy the 1D Burgers equation. We will then present several new results regarding the maximum growth of palinstrophy in 2D flows and will discuss some questions concerning sharpness of the corresponding analytical estimates. While it is well known that solutions of 1D Burgers equations and 2D Navier-Stokes equation evolving from smooth initial data remain smooth for all times, the question whether the best available estimates for the maximum growth of enstrophy and palinstrophy are sharp is both interesting and relevant. One reason is that such estimates are derived using similar mathematical techniques as in the 3D case where blow-up cannot be ruled out. We will show how new insights regarding these problems can be obtained by formulating them as variational PDE optimization problems which can be solved computationally using suitable adjoint--based gradient descent (or ascent) methods. In particular, we will discuss certain topological features of the families of vorticity fields maximizing the instantaneous rate of growth of palinstrophy in 2D. In offering a systematic approach to finding flow solutions closest to saturating a given analytical bound, the proposed approach provides a bridge between theory and computation.
TOD 2nd August 2012
11:30 to 12:30
Higher norms of vorticity applied to the intermittency and blow-up problems in the 3D Navier-Stokes and Euler equations
Intermittency has been observed in turbulent flows for more than 60 years and yet the challenge remains to show that solutions of the 3D Navier-Stokes equations generically display this phenomenon. Not un-naturally it is intimately bound up with the even longer-standing regularity problem. This talk will display some technical methods, involving ladders of higher Lp-norms of vorticity (for both the NS and Euler equations), that will give some insight into the problems involved.
TOD 7th August 2012
11:30 to 12:30
Development of singularities in relaxing magnetic fields
When a magnetic field relaxes to the state of magnetostatic equilibrium, discontinuities, called current-sheets are likely to form. In a perfectly conducting medium the relaxation process is subject to topological constraints. In a medium of very large, but finite conductivity current sheets play an important role as they are the places where the topological constraints are broken and the medium is heated by strong electrical currents. I discuss magnetic relaxation and current sheet formation in the simplest, one-dimensional configuration and in complex, three-dimensional chaotic fields.
TOD 9th August 2012
11:30 to 12:30
Pollockian Mechanics: Painting with Viscous Jets
Beginning around 1945, American Abstract Expressionist painter Jackson Pollock invented and perfected a new artistic technique based on pouring and dripping liquid pigment onto a canvas stretched horizontally on the floor. Long recognized as important and influential by art historians, Pollock's works, and the tangled webs he created, have recently received attention also from scientists. But although the artist manipulated gravitational flows to achieve his aims, the fluid dynamical aspects of his process remained largely unexplored. I will discuss Pollockian Mechanics -- the physics of lifting paint by viscous adhesion and dispensing it in free jets -- focusing on the role of fluid instability. This technique will be contrasted with flows of pigment employed by other artists. I will conclude with comments on the scaling regularities of the poured patterns and their affinity to the "geometry of nature."
TOD 14th August 2012
11:30 to 12:30
Magnetic braids, Hamiltonian action and selective decay
We explain the relation between magnetic braids and their representation by the action of the corresponding Hamiltonian system. We show how this function captures certain topological properties of the braid and then go on to explore the role it has in the relaxation of magnetic braids under resistive dynamics at high magnetic Reynolds numbers.
TOD 16th August 2012
11:30 to 12:30
R Buniy An algebraic classification of entangled states
We provide a classification of entangled states that uses new discrete entanglement invariants. The invariants are defined by algebraic properties of linear maps associated with the states. We prove a theorem on a correspondence between the invariants and sets of equivalent classes of entangled states. The new method works for an arbitrary finite number of finite-dimensional state subspaces. As an application of the method, we considered a large selection of cases of three subspaces of various dimensions. We also obtain an entanglement classification of four qubits, where we find 27 fundamental sets of classes.
TOD 21st August 2012
11:30 to 12:30
Reactive-infiltration instabilities in fractures and porous rock
A reactive fluid flowing through a porous or fractured rock and dissolving the rock matrix may trigger an instability, leading to spontaneous formation of pronounced channels or wormholes. I will present a linear-stability analysis of this system and show that there are two different instabilites. One is associated with an initial uniform-porosity state and the other with a steadily propagating one-dimensional dissolution front. I will discuss the origin of both instabilities and the physical conditions under which they can be observed. In particular, it is argued that the former 'initial- state' instability is relevant to the dissolution of fractures in carbonate rocks, giving rise to the formation of limestone caves. Finally, I will discuss the later stages of the system evolution, when the channels interact, competing for the available flow; eventually the growth of the shorter ones ceases. This leads to self-similar patterns of growth, with the flow becoming concentrated within a few active channels.
TOD 23rd August 2012
11:30 to 12:30
Knots: from the sailing boat down to the cell's nucleus illustrated by means of simple experiments
Knots are pervasive in our daily life at the macroscopic level as well as at the microscopic level. With the help of some "experiments", I will present few important features of knots starting with macroscopic knots and then go into the cell's nucleus where knots play a role in the functioning of the DNA. Among the presented examples I will discuss some mathematical properties, the hydrodynamic behavior of knots, their resistance to traction and the unknotting of DNA knots by enzymes
TOD 28th August 2012
11:30 to 12:30
On the 2-point problem for the Lagrange-Euler equation
Consider the motion of ideal incompressible fluid in a bounded domain (or on a compact Riemannian manifold). The configuration space of the fluid is the group of volume-preserving diffeomorphisms of the flow domain, and the flows are geodesics on this infinite-dimensional group where the metric is defined by the kinetic energy. The geodesic equation is the Lagrange-Euler equation. The problem usually studied is the initial-value problem, where we look for a geodesic with given initial fluid configuration and initial velocity field. In this talk we consider a different problem: find a geodesic connecting two given fluid configurations. The main result is the following Theorem: Suppose the flow domain is a 2-dimensional torus. Then for any two fluid configurations there exists a geodesic connecting them. This means that, given arbitrary fluid configuration (diffeomorphism), we can "push" the fluid along some initial velocity field, so that by time one the fluid, moving according to the Lagrange- Euler equation, assumes the given configuration. This theorem looks superficially like the Hopf-Rinow theorem for finite-dimensional Riemannian manifolds. In fact, these two theorems have almost nothing in common. In our case, unlike the Hopf-Rinow theorem, the geodesic is not, in general case, the shortest curve connecting the endpoints (fluid configurations). Moreover, the length minimizing curve can not exist at all, while the geodesic always exists. The proof is based on some ideas of global analysis (Fredholm quasilinear maps) and microlocal analysis of the Lagrange-Euler equation (which may be called a ?microglobal analysis?).
TOD 30th August 2012
11:30 to 12:30
T Lipniacki Stochastic travelling waves in bistable biochemical system: Numerical and mathematical analysis
I will discuss stochastic transitions in a bistable biochemical system of trans-activating molecules on a hexagonal lattice. Kinetic Monte Carlo simulations demonstrated that the steady state of the system is controlled by the diffusion, and size of the reactor. In considered example, in small reactor the system remains inactive. In larger domain, however, the system activates spontaneously at some place of the reactor and then the activity wave propagates until whole domain becomes active. The expected time to activation grows exponentially with the diffusion coefficient.

I will interpret these results by analytical considerations of a simpler bistable system, which evolution is equivalent to the one dimensional birth and death process.
TODW02 3rd September 2012
09:10 to 09:50
Chromatin Compaction: A Modeling Exploration of Fiber Hetergeoneity and Linker Histone Influence
TODW02 3rd September 2012
10:10 to 10:50
The topological equivalence of the yeast chromosome centromere and the yeast plasmid partitioning locus
Centromeres are the DNA loci responsible for the faithful partitioning of eukaryotic chromosomes. The elaborate protein assembly, called the kinetochore complex, organized at the centromere is responsible for attaching sister chromosomes to the mitotic spindle, thus ensuring their equal segregation during cell division. Centromeres in nearly all eukaryotes are ‘regional’ centromeres- long DNA segments with no consensus sequence elements- that are established epigenetically. Members of the budding yeast lineage are a stark exception. Their chromosomes harbor ‘point’ centromeres- very short centromeres with three well defined sequence elements- that are genetically determined. The yeast centromere chromatin engenders a positive supercoil, presumably due to a nucleosome containing a variant of the histone H3. Yeast harbors a selfish plasmid whose stability is comparable to that of the chromosomes of its host. The plasmid achieves this fidelity of segregation with the help of two partitioning proteins and a partitioning locus called STB (for stability). Strikingly, the chromatin at STB is also positively supercoiled. When STB function is inactivated, its topology shifts to standard negative supercoiling. The magnitudes of the CEN and STB induced positive supercoiling are comparable. The topological equivalence of CEN and STB, along with other functional similarities between the two, are consistent with the notion that the non-standard point centromere had its origin in the partitioning locus of an ancestral plasmid.
TODW02 3rd September 2012
10:50 to 11:30
How the genome folds
I describe Hi-C, a novel technology for probing the three-dimensional architecture of whole genomes. Developed together with collaborators at the Broad Institute and UMass Medical School, Hi-C couples proximity-dependent DNA ligation and massively parallel sequencing.

Our lab employs Hi-C to construct spatial proximity maps of the human genome. Using Hi-C, it is possible to confirm the presence of chromosome territories and the spatial proximity of small, gene-rich chromosomes. Hi-C maps also reveal an additional level of genome organization that is characterized by the spatial segregation of open and closed chromatin to form two genome-wide compartments. At the megabase scale, the conformation of chromatin is consistent with a fractal globule, a knot-free conformation that enables maximally dense packing while preserving the ability to easily fold and unfold any genomic locus. The fractal globule is distinct from the more commonly used globular equilibrium model. Our results demonstrate the power of Hi-C to map the dynamic conformations of whole genomes.
TODW02 3rd September 2012
11:30 to 12:10
L Mirny The role of topological constraints on condensed polymers and DNA in human cells
Human DNA is two meters long and is folded into a structure that fits in a cell nucleus of just 5 microns in diameter. Recently developed Hi-C technique provides comprehensive information about genome folding. Our analysis of Hi-C data provides and biophysical polymer modeling show that scaling observed in the data is consistent with non-equilibrium and unknotted polymer state – the crumpled (fractal) globule. We demonstrate that the fractal globule emerges as a result of polymer collapse and has a short lifetime, rapidly mixing while remaining largely unknotted. Long-time dynamics of a condensed polymer reveals that spatial and topological equilibration happen at vastly different time scales and that topological constrains have little effect on relatively short and flexible chains.
TODW02 3rd September 2012
13:30 to 14:10
Large scale organization of chromatin
In the cell nucleus chromosomes have a complex architecture serving vital functional purposes. One of the current key open problem concerns the principles which orchestrate their 3D structure. We discuss a model of the molecular mechanisms of chromatin self- organization which we compare against available FISH and Hi-C data.
TODW02 3rd September 2012
14:10 to 14:50
S Levene All Good Things Must Bend: the Energetics of DNA Shape and Flexibility in Biological Assemblies
DNA-loop formation is an essential mechanistic aspect of many biological processes including gene regulation, DNA replication, and recombination. These loops are mediated by proteins bound at specific sites along the contour of a single DNA molecule and are closely coupled to the topological state of DNA domains through supercoiling, knotting, and linking. The complex interplay between DNA topology and the regulation of DNA transactions remains poorly understood and the effects of a chromatin environment on such interactions are essentially unknown. However, new insights can come from novel experimental approaches and computational models of DNA flexibility and folding under geometric and/or topological constraints. Experimental studies of Cre-loxP recombination and lac-repressor-mediated gene regulation will be used to illustrate the problems and general principles of complex nucleoprotein organization. A new method for directly computing the thermodynamic (i.e., free-energy) cost for mesoscopic models of nucleoprotein assemblies will also be discussed.
TODW02 3rd September 2012
15:20 to 15:40
Structure meets function at the mouse X chromosome inactivation center
Characterizing the folding principles of mammalian chromosomes is of capital importance to understand the complexity of gene expression regulation, particularly during the major transcriptional changes occurring in development. This may help elucidating the mechanisms by which regulatory elements contact gene promoters (i.e. by looping out intervening DNA), understand what is the cell-to-cell variability of these interactions and how it does reflect transcriptional variability. We analyzed a 4.5 Mb region of the X chromosome that includes the X-inactivation center by Chromatin Conformation Capture Carbon-Copy (5C), in order to gain insights into how chromatin structure is organized during early mouse embryonic stem cell (ESC) differentiation. We uncovered that chromatin is organized into Topologically Associating Domains (TADs), within which genomic elements preferentially interact. To fully reconstruct the statistical repertoire of chromatin conformations that give rise to these domains, we have used a combination of Monte Carlo simulations of a polymer model, high-resolution DNA FISH and quantitative RNA FISH. We show that in the TAD that contains the Tsix ncRNA (a master regulator of X-chromosome inactivation), enhancer-promoter contacts take place in a subset of cells where the whole domain is compacted, rather than resulting from stable DNA loops. In these cells, the probability of transcribing a gene is higher than in cells where the domain is in an elongated conformation. We thus show a correlation between the spatial proximity of a promoter and an enhancer and their transcriptional activity at the single cell level.
TODW02 3rd September 2012
15:40 to 16:00
F Maggioni Modeling chromatin fibre folding for human embryonic stem cells and cancer cells
All diverse cell types in an organism essentially have an identical genome. Generation of tissue specific cells is through an epigenomic process in which progressive alterations in the chromatin state generates lineage committed cells from pluripotent embryonic stem cells. Ultimately, establishment of terminally differentiated cells results in a stable chromatin state. Chromatin modification can be studied by chromatin immunoprecipitation (ChIP) that identifies regions that are over-represented as transcriptionally active sequences. In this talk we describe chromatin-state maps for pluripotent, cancer and lineagecommitted cells using three-dimensional modelling of fibre conformation. The model takes into account of the local structure of chromatin organised into euchromatin (open chromatin), permissive for gene activation, and heterochromatin (closed chromatin), transcriptionally silenced. Open chromatin is assumed to be modelled by a linker DNA while the closed chromatin by means of a solenoid structure in which DNA winds onto six nucleosome spools per turn with two left-handed superhelical turns around an histone octamer. The model represents a single gyre of a solenoid by means of a torus knot that winds around a torus once in the longitudinal direction and twelve in the meridian direction. Closed and open chromatin is then connected by means of piecewise polynomial transformations based on cubic Hermite spline functions. As reprogramming process is associated both with pluripotency and the neoplastic process, our analyses potentially identify cancer-related epigenetic abnormalities. Chromatin fibre conformation are compared in terms of geometric quantities such as curvature and torsion localization, and relative rates, in relation to filament compaction and packing efficiency. This study provides information on relationships between geometry and the transcriptional regulation in stem cells and cancer cells contributing to pluripotency and self-renewal.
TODW02 3rd September 2012
16:00 to 16:20
A Rosa Three-dimensional conformations of entangled ring polymers in solution
The physical properties of semi-dilute and dense polymer solutions are dominated by the topological, mutual constraints (the so-called "entanglements") between the chains. For linear chains, entanglements effects are captured by the Edwards-DeGennes reptation model [Doi & Edwards, "The Theory of Polymer Dynamics"; DeGennes, J. Chem. Phys. (1971)]. Within this model, single-chain diffusive behavior proceeds as it was effectively constrained to an almost one-dimensional path along its contour length. The model received experimental validation and it is nowadays accepted. Conversely, a consistent physical picture for unlinked, circular (ring) polymers is still lacking. One obvious way of tackling the problem is by resorting to computer simulations (Monte-Carlo or Molecular Dynamics) of coarse-grained polymer models, which however present the major difficulty of waiting over very long equilibration times when dealing with very long chains. Here, we present preliminary results concerning an alternative, possible way of attacking the problem: we construct "by hand" putative, equilibrated states for entangled ring polymers, by using a restricted ensemble of physical parameters which are calibrated on solutions of small ring polymers which equilibrate fast. Then, we use this model to "predict" chain behavior for much larger ring polymers.
TODW02 3rd September 2012
16:20 to 17:00
N Kleckner How E.coli organizes and segregates its chromosome
Live cell imaging of the E.coli nucleoid, illuminated with HupA-mCherry, reveals a well-defined helical ellipsoid that is trapped within the cell radially but not longitudinally. Basic elliposidal shape results from longitudinal density bundling while helicity results from interactions between these bundles and the cell periphery. Unexpectedly, the nucleoid exhibits two distinct types of cyclic dynamic changes, both independent of DNA replication: (1) On time scales of seconds-to-minutes, longitudinal density waves flux through the shape over distances comparable to the length of the nucleoid, resulting in dynamic shape changes. (2) At intervals of ~20min, the nucleoid exhibits ~10min pulses of chromosome elongation. These pulses, which are implemented by elongation-biased density waves, are temporally and functionally linked to step-wise separation of sister chromosomes. The presented findings support a two-component model for sister segregation and the existence of nucleoid stress cycles which, we propose, function to release the nucleoid from linkages that constrain both morphogenetic evolution and separation of sisters. These cycles could comprise a primordial cell cycle, and the same principles could pertain broadly across evolutionary space and time.
TODW02 4th September 2012
08:30 to 09:10
A tangled problem: the structure,function and folding of knotted proteins
Since 2000, when they were first identified by Willie Taylor, the number of knotted proteins within the pdb has increased and there are now nearly 300 such structures. The polypeptide chain of these proteins forms a topologically knotted structure. There are now examples of proteins which form simple 31 trefoil knots, 41, 52 Gordian knots and 61 Stevedore knots. Knotted proteins represent a significant challenge to both the experimental and computational protein folding communities. When and how the polypeptide chain knots during the folding of the protein poses an additional complexity to the folding landscape. We have been studying the structure, folding and function of two types of knotted proteins – the 31-trefoil knotted methyltransferases and 52-knotted ubiquitin C-terminal hydrolases. The first part of the talk will focus on our folding studies on knotted trefoil methyltransferases and will include our work on (i) equilibrium unfolding experiments in chemical denaturants, (ii) kinetic analysis of unfolding/folding pathways, (iii) protein engineering on both the small scale (single point mutants) and large scale (creating N- and C-terminal fusions with a stable beta-grasp domain which are the deepest knotted structures known), (iv) circularisation experiments which establish that the polypeptide chain remains knotted even in the chemically denatured state, and (v) recent in vitro translation work which shows that knotting is rate limiting and also shows how GroEL/GroES play a role in the folding of these proteins in vivo. The second part of the talk will focus on our studies of knotted ubiquitin C-terminal hydrolases – UCH-L1 and UCH-L3. This will include equilibrium and kinetic unfolding and folding studies as well as recent work on the effect of point mutants associated with Parkinson’s Disease on the structure, folding and dynamics of UCH-L1. Recent work on the effect of oxidative damage on the structure of UCH-L1 will also be described and evidence that this protein adopts a partially unfolded form (PUF) on modification with the reactive aldehyde and by-product of cellular oxidative stress, HNE, will be presented. The possible cellular effects of this PUF will be discussed.
TODW02 4th September 2012
09:10 to 09:50
Knotted and unknotted proteins: a comparative study (*)
Knotted proteins, because of their ability to fold reversibly in the same topologically entangled conformation, are the object of an increasing number of experimental and theoretical studies. The aim of the present investigation is to assess, on the basis of presently available structural data, the extent to which knotted proteins are isolated instances in sequence or structure space, and to use comparative schemes to understand whether specific protein segments can be associated to the occurrence of a knot in the native state. A significant sequence homology is found among a sizeable group of knotted and unknotted proteins. In this family, knotted members occupy a primary sub-branch of the phylogenetic tree and differ from unknotted ones only by additional loop segments. These "knot-promoting" loops, whose virtual bridging eliminates the knot, are found in various types of knotted proteins. Valuable insight into how knots form, or are encoded, in proteins could be obtained by targeting these regions in future computational or excision experiments. Results of recent related simulations will be presented.

(*) in collaboration with R. Potestio (MPI Mainz) and H. Orland (CEA Saclay); ref: R. Potestio et al. Plos Comp. Biol. 6, e1000864 (2010), http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000864
TODW02 4th September 2012
10:10 to 10:50
M Cieplak Topological features in stretching of proteins
TODW02 4th September 2012
10:50 to 11:30
Conservation of complex knotting and slipknotting patterns in proteins
While analyzing all available protein structures for the presence of knots and slipknots we detected a strict conservation of complex knotting patterns within and between several protein families despite their large sequence divergence. Since protein folding pathways leading to knotted native protein structures are slower and less efficient than those leading to unknotted proteins with similar size and sequence, the strict conservation of the knotting pattern indicates an important physiological role of knots and slipknots in these proteins. Although little is known about the functional role of knots, recent studies have demonstrated a protein-stabilizing ability of knots and slipknots. Some of the conserved knotting patterns occur in proteins forming transmembrane channels where the slipknot loop seems to strap together the transmembrane helices forming the channel. This is joint work with Ken Millett, Andrzej Stasiak and Jose Onuchic.
TODW02 4th September 2012
11:30 to 11:50
Amino acid patterns for protein folding
Understanding protein sequence-structure relationship is a key to solving many problems of molecular biology, such as annotation of genome sequences, protein structure prediction, protein-protein interaction, and protein evolution, among others. These problems are convenient to consider on the level of super-secondary structures, which define arrangement of strands and helices in 3D structures. In this presentation, first, the strict rule which describe how structure elements - beta-strands come together into super-secondary structures of sandwich-like proteins will be presented. Then the main sequence regularities (a specific set of residues at particular positions), which dictate the folding of amino acid sequence will be described. Residues at certain positions constitute the characteristic residue pattern of specific super-secondary structures. The pattern can be viewed as an amino acid 'tag' that brands a sequence as having a particular super-secondary structure.

References

Alexander Kister and Israel Gelfand, (2009) PNAS, 106, 18996-19000 Alexander Kister (2012) in "Protein super-secondary structures" Ed. A. Kister in "Methods in Molecular Biology' series Humana Press (in press)
TODW02 4th September 2012
11:50 to 12:10
P Szymczak How to untie it-translocation of a knotted protein through a pore
Proteins need to be unfolded when translocated through the pores in mitochondrial and other cellular membranes. Knotted proteins, however, might get stuck during this process since the diameter of the pore is smaller than the size of maximally tightened knot. We report the result of computer simulations of knotted protein translocation, which show how the protein can avoid the topological traps and untie its knot during the translocation
TODW02 4th September 2012
13:30 to 14:10
Computer simulations of knotted DNA and proteins
When mature bacteriophages such as P2 or P4 are assembled in infected cells, a long linear DNA molecule is loaded into the phage capsid and arranges itself in a toroidal, nematic phase. Intriguingly, experiments show that the DNA is not only highly knotted, but also exhibits a rather uncommon knot spectrum. Observation that DNA molecules in bacteriophage capsids preferentially form torus knots provide a sensitive gauge to evaluate various models of DNA arrangement in phage heads. We demonstrate with computer simulations of a simple bead-spring model that an increasing chain stiffness not only leads to nematic ordering and a (somewhat counter-intuitive) increase of knottedness, it is also the decisive factor in promoting formation of DNA torus knots in phage capsids. In the second part of my presentation I will review recent and not so recent advances in the understanding and modelling of protein knots.
TODW02 4th September 2012
14:10 to 14:50
Knots confined to tubes in the simple cubic lattice
Here a polymer is modeled as a self-avoiding polygon in the simple cubic lattice. All knots can be confined to a slab, which is an area bounded by two parallel planes. However not all knots can be constructed in a tube, which is an area bounded by two pairs of parallel planes.

In this talk, we discuss knots and links in a tube and estimate the minimum length required for such a polygon to realize a knot type.
TODW02 4th September 2012
15:20 to 15:40
Simulations of cyclic and linear DNA chains moderately and strongly confined in nanochannels
Structural properties of flexible and semiflexible cyclic and linear chains confined in nanochannels were studied by molecular simulations mimicking single molecule experiments in microfuidic channels used to analyze genomic macromolecules. Experiments (1) with linear and ring macromolecules showed differences in the response to confinement. Simulations of rings (comprising few persistence lengths per chain, as in plasmid rings (2)) and their linear analogs confirmed these trends (3). The radius of gyration Rg of chains satisfactorily represents the stretching of both chain topologies along the channel. Apart from focus on moderate confinement we show that strong confinement applies also for semiflexible rings, though unanticipated for rings in contrast to linear chains where it is known as Odijk regime. Similar response of chain elongation to the confinement Rg(D) is obtained in the case of rings compared to linear chains. However, the relative chain extension in channel is larger for rings, the strong confinement regime extends to larger channel diameters D and under moderate confinement the extension declines less steeply for rings. These findings are explained (3) in terms of a strong self-avoidance of confined rings relative to their linear analogs, stemming from the increased local density in channel due to looping of cycle. The extension of rings is governed by the same analytical function as for linear chain provided half of the contour length for a cyclic chain is considered at full extension. Orientation correlation function and static structure factor for both topologies point out features responsible for recognition of ring architecture.

(1) F. Persson, P. Utko,W. Reisner, N.B. Larsen, A. Kristensen, Nano Lett. 9, 1382-1385, 2009, (2) P. Cifra, Z. Benková, Macrom. Theory & Simul., 20, 65-74, 2011, (3) Z. Benková, P. Cifra, Macomolecules, 45, 2597-2608, 2012
TODW02 4th September 2012
15:40 to 16:00
Diffusion dynamics of circular DNA mini-rings in solution
In gel electrophoresis it was observed in Ref. [1] that underwound DNA mini-rings migrate slower than overwound DNA mini-rings in EDTA solution. Motivated with the novel observation we have evaluated the diffusion constant of a closed ladder-shaped polymer in solution via Brownian dynamics with hydrodynamic interaction. It gives a model of a supercoiled DNA mini-ring consisting of DNA double strands. Here, the topology of the whole molecule such as the linking number (Lk) of the double DNA strands is conserved in time. By introducing the bending rigidity, we have found that the diffusion constant of a ladder-shaped molecule with base flipping becomes smaller and much more dependent on the linking number, Lk [2]. We thus suggest that the numerical observations should explain the different migration sppeds between the underwound and overwound DNA mini-rings. Here we also recall an interesting observation that for knotted DNAs the migration speeds in gel are proportional to th e diffusion constants in a good solvent.

[1] J.M. Fogg, D.J. Catanese, Jr., G.L. Randall, M.C. Swick, and L. Zechiedrich, Proc. Institute Math. App. Vol. 150 (2009), 73-121. [2] N. Kanaeda, T. Deguchi and L. Zechiedrich, Bussei-Kenkyu Vol. 92 (2009) pp. 145-146.
TODW02 4th September 2012
16:00 to 16:20
Influence of topology in coarse-graining of polymer solutions
We employ computer simulations and integral equation theory techniques to perform coarse-graining of self-avoiding ring polymers with different knotedness and to derive effective interaction potentials [1] between the centers of mass (CM) of these macromolecular entities. Different microscopic models for the monomer-monomer interactions and bonding are employed, bringing about an insensitivity of the effective interactions on the microscopic details and a convergence to a universal form for sufficiently long molecules. The pair effective interactions are shown to be accurate up to within the semidilute regime with additional, many-body forces becoming increasingly important as the polymer concentration grows. The dramatic effects of topological constraints in the form of interaction potentials (see figure) are going to be brought forward and critically discussed [2].

We will also show the big impact of topology on the size scaling of a polymer chain in good/poor solvent conditions. This is accomplished calculating the theta temperature for specific topologies and sizes, of a single chain with two complementary methods: scaling law for radius of gyration [3,4] and second virial coefficient calculation [5]. In addition, we investigate the dependence of shape parameters with topology in good/poor solvent conditions.

[1] C. N. Likos, Physics Reports 348(4-5): 267 (2001)

[2] A. Narros, A. J. Moreno, and C. N. Likos, Soft Matter 9(11):2435 (2010)

[3] M. O. Steinhauser, J. Chem. Phys. 122:094901 (2005)

[3] S. S. Jang, Tahir Ça¡gin and W. A. Goddard, J. Chem. Phys. 119:1843 (2005)

[5] V. Krakoviak, J. P. Hansen and A. A. Louis, Phys. Rev. E 67:041801 (2003)
TODW02 4th September 2012
16:20 to 17:00
Locating knots and slipknots in open and closed macromolecules
Once it was imagined that proteins could be knotted, it was necessary to find procedures to identify the presence and precise location of knots and, later, slipknots in both open and closed macromolecules. While some of the initially proposed strategies were largely effective they often depended upon choices that gave rise to problems. In addition to a review of the historical development of these methods, we will review methods developed in collaboration with Akos Dobay, Andrzej Stasiak, Ben Sheldon and, later, with Rawdon in connection with work with Joanna Sułkowska and Jose Onuchic on protein structures. The relationship of this approach with that of other contemporay researcher will be described.
TODW02 4th September 2012
17:00 to 17:40
E Rawdon Knotting of open chains, closed chains and proteins
Some proteins are now classified as being knotted. However, proteins have free ends and knotting, mathematically, is only defined for closed curves. Defining knotting in open chains is tricky and ambiguous. We will show one definition of open knotting and search for knotted arcs within knotted open, closed chains, and proteins. This is joint work with Ken Millett, Andrzej Stasiak, and Joanna Sulkowska.
TODW02 5th September 2012
08:30 to 09:10
Superhelically Driven Structural Transitions in Genomic DNA - Theoretical Analyses, Genomic Distributions and Roles in Regulation
DNA is known to be a highly polymorphic molecule, capable of assuming several alternate conformations in addition to the standard Watson-Crick B-form. These include states of strand separation, left handed helices, cruciforms, and three- and four-stranded structures. Although the B-form is its default conformation in vivo, regions within genomic DNA can be driven into alternate structures by the superhelicity imposed on the molecule by enzymatic activities and by transcription. This talk will present theoretical analyzes of several types of transitions, and of competitions among them. The predicted genomic distributions of locations susceptible to different types of superhelical transitions will be shown, and their statistical significances will be assessed. Several situations will be described where transitions to alternate structures serve biological roles in either normal or pathological processes.
TODW02 5th September 2012
09:10 to 09:50
Co-operative roles for DNA supercoiling and nucleoid-associated proteins in the regulation of bacterial transcription
DNA supercoiling and nucleoid-associated proteins (NAPs) contribute to the regulation of transcription of many bacterial genes. The horizontally-acquired Salmonella pathogenicity island (SPI) genes respond positively to DNA relaxation, to the Fis and H-NS nucleoid-associated proteins and to the OmpR global regulatory protein. The ompR gene is autoregulated and responds positively to DNA relaxation. Binding of the Fis and OmpR proteins to DNA in the SPI-1 and SPI-2 islands and at the ompR gene promoter is differentially sensitive to the topological state of the DNA while H-NS binds regardless of the topological state of the DNA. These data illustrate the overlapping and complex nature of NAP and DNA topological contributions to transcription control in bacteria. They also show that these properties are shared by the core and the horizontally-acquired components of the bacterial genome.
TODW02 5th September 2012
10:10 to 10:50
Chromosomes as topological machines-the role of DNA thermodynamics
The chirality of the DNA molecule underpins its ability to partition superhelicity between twist and writhe. We argue that manipulation of superhelical density and of partitioning by topological devices and processive ATP-dependent motors (DNA and RNA polymerases and topoisomerases) is a fundamental property of both bacterial and eukaryotic chromosomes. On this view chromosomes act as machines in which topological transitions operate at several functional levels - local (e.g. transcription initiation sites), regional (constrained superhelical domains) and global (chromosomes) levels.

The partition between twist and writhe is dependent in part on the sequence of DNA. We have shown that in the E. coli chromosome gradients of DNA gyrase binding sites from the origin to the terminus of DNA replication along both replichores correlate with temporal patterns of gene expression during the growth cycle such that genes expressed during exponential growth are preferentially located in the Ori-proximal region. These observations imply that during exponential growth there exist gradients of superhelical density from the origin to the terminus. Intriguingly the chromosomal DNA sequences exhibit, on average, a gradient of DNA stacking energy in the same direction. We argue that this gradient in the physicochemical properties of DNA integrates the functional response to changes in superhelical density and to regulation by abundant nucleoid-associated proteins.

We further show that the genetic and chromatin organisation in yeast chromatin assembled both in vitro and in vivo is highly dependent on, the stacking/melting energies of DNA sequences. The regions of chromosomes that are sites for topological manipulation (such as transcription and replication initiation sites and preferred sites for topoisomerase II) correlate strongly with low stacking energies and high flexibility. Such regions concomitantly exhibit low nucleosome occupancy. We conclude that the most flexible DNA sequences are, counter-intuitively, poor substrates for octamer deposition. In contrast high nucleosome occupancy correlates with DNA sequences of moderately high stacking energies. In such relatively stiff sequences positioned nucleosomes can often be related to a bending anisotropy appropriate for nucleosome formation.
TODW02 5th September 2012
10:50 to 11:30
How DNA topology and DNA length affect the body's defense against nucleic acids of invading organisms in the blood
It has long been known that human blood contains enzymes that digest DNA to protect the body against invasion by foreign organisms. We set out to determine how DNA length and supercoiling affected DNA vector survival in human serum. Closed circular, supercoiled vectors ranging from ~300 to ~4,000 bp were incubated at 37°C in human serum. Aliquots were taken over several days and were analyzed by gel electrophoresis. We found that digestion in human serum strongly correlated with increasing DNA length. To our surprise, we also uncovered a trend by which serum proteins bound and protected DNA. We recently published that the compaction by DNA supercoiling protected small ( This work was supported by NIH RO1AI054830, Human Frontier Science Program, and Seattle's Children's Hospital Research Foundation, part of NGEC, to L.Z. T.J.B. was supported by NIH Grant T32 GM88129.
TODW02 5th September 2012
11:30 to 11:50
Twisted paths in Euclidean groups: Keeping track of total orientation while traversing DNA
This talk introduces a new mathematical structure for modeling global twist in DNA. The relative rigid-body motion between reference frames attached either to a backbone curve, bi-rods, or individual bases in DNA, can be described well using elements of the Euclidean motion group, SE(n). However, the group law for Euclidean motions does not keep track of overall twist. In the planar case, the universal covering group of SE(2) identifies orientation angle as a quantity on the real line rather than on the circle, and hence keeps track of ``global'' rotations (not modulo 360 degrees). However, in the three-dimensional case, no such structure exists since the the orientational part of the universal cover of SE(3) can be identified with the quaternion sphere. In this talk a new mathematical structure for ``adding'' framed curves and extracting global twist is present. Though reminiscent of the group operation in braid theory and in homotopy theory, this structure is distinctly different, as it is geometric in nature, rather than topological. The motivation for this mathematical structure and its applications to DNA conformation will be presented.
TODW02 5th September 2012
11:50 to 12:10
R Cortini Chiral effects in DNA supercoiling
Supercoiling is a topological property of DNA which is known to be crucially important in the genetic regulation of virtually every living cell. Electrostatic interactions play a fundamental role in determining the conformation of DNA molecules. They are generally taken into account assuming that the charge is homogeneously smeared on the surface of DNA molecules. We developed a theory that instead takes into account the helical pattern of charge on the DNA molecular surfaces. We find that the intrinsic chirality of the charge structures gives rise to important and non trivial phenomena. Crucially, it determines an asymmetry in the energetics of DNA-DNA crossovers: right-handed crossings, occurring in positively supercoiled molecules are more stable than left-handed ones, which occur in negatively supercoiled molecules. We explored the consequences of this fact first by developing a theory of spontaneous DNA braid formation, and then applying it to closed loop DNA supercoilin g and single-molecule DNA micromanipulation experiments. The theory can give an account of some yet unexplained observations and biological facts. It gives a plausible explanation for the occurrence of tight supercoiling of DNA loops observed in cryo-EM and AFM images in high ionic strength environment. It can shed light on the preference for positive supercoiling in hyperthermophylic bacteria and archea. Finally, it induces to reinterpret classical experiments that show that divalent metal ions overwind DNA. The biological implications of these important facts could be very important, and are yet to be fully explored.
TODW02 5th September 2012
13:30 to 14:10
Sequence-Dependent Coarse-Grain Descriptions of DNA: models, methods and simulations
TODW02 5th September 2012
14:10 to 14:50
K Zakrzewska DNA recognition studied by molecular simulations
In order to fulfill its biological role DNA has to interact with other molecules, ranging from small ligands to protein complexes. In many cases these molecules intercalate, at least partially, into DNA. This intercalation can involve the conjugated rings of a drug, or the hydrophobic side chains of a protein. How different DNA binding molecules find their target sites, and what role intercalation plays in this mechanism, is still not understood. We try to answer these questions by analyzing the energetic and mechanistic aspects of recognition using molecular simulations at the atomic level. Results will be presented for binding a small drug, daunomycin, and for binding a protein, SRY, a mammalian transcription factor. We will show that daunomycin intercalates into DNA by a complex, multistep process, starting with an intermediate, minor groove bound state. In the case of SRY, the mechanism of DNA sequence specificity, via deformation of the double helix, will be discussed. References: A systematic molecular dynamics study of nearest-neighbor effects on base pair and base pair step conformations and fluctuations in B-DNA, R.Lavery et al. Nucl. Acids Res. (2010) 38(1): 299-313 Protein–DNA Recognition Triggered by a DNA Conformational Switch, 
B. Bouvier et al, Angew. Chem. Int. Ed. 2011, 50, 6516 –6518 Multistep Drug Intercalation: Molecular Dynamics and Free Energy Studies of the Binding of Daunomycin to DNA, M. Wilhelm et al, JACS (2012), published on line.
TODW02 5th September 2012
15:20 to 15:40
J Baxter The yeast Pif1 family helicase RRM3 promotes DNA unwinding during replisome swivelling
During termination of DNA replication, replisomes overcome the topological tension that occurs as forks converge by coupling final unwinding with fork swivelling. Here I show that cells lacking the DNA helicase RRM3 accumulate terminating late replication intermediates (LRI) in plasmids both with and without characterised pause sites. Rrm3 deletion does not alter the level of swivelling that occurs during termination but its depletion extends the lifetime of LRI while it’s over-expression leads to the rapid unwinding of the LRI. Therefore RRM3 promotes DNA unwinding when the replisome swivels during termination. Potentially, this activity is also generally utilised during DNA replication to bypass topological blocks.
TODW02 5th September 2012
15:40 to 16:00
Denaturation transition of stretched DNA in the presence of DNA-binding ligands
Stretching experiments on DNA in the presence of DNA binding ligands have been shown to reveal insight into biological processes of DNA-ligand binding. We generalize the Poland-Scheraga model to consider DNA denaturation in the presence of an external stretching force and DNA-binding ligands which bind to double-stranded DNA by intercalating between two adjacent base pairs. We obtain the phase diagram of DNA denaturation as a function of temperature, stretching force, and the chemical potential of the DNA-binding ligand. Force-extension relations are compared with recent DNA stretching experiments in the presence of DNA intercalating ethidium and ruthenium complexes.
TODW02 5th September 2012
16:00 to 16:20
Twist neutrality and topological aspects of nucleosomal DNA
TODW02 5th September 2012
16:20 to 17:00
D Swigon Dynamics of DNA supercoiling and knotting
Recent experiments on electrostatically induced migration of DNA in nanochanels reveal an intricate phenomenon of compaction of migrating DNA that promotes knotting of the molecule. Subsequent relaxation of the molecule proceeds along several distinct kinetic regimes. The structural details of DNA configurations in different stages of the process are yet unknown. We investigate this and other related phenomena of DNA dynamics using a model in which DNA is represented by a charged elastic rod immersed in a viscous incompressible fluid and the governing equations of the system are solved numerically using the generalized immersed boundary method. The equations of motion of the rod include the fluid–structure interaction, rod elasticity and a combination of two interactions that prevent self-contact, namely the electrostatic interaction and hard-core repulsion. Presented will be results on the effects of electrostatics, steric repulsion, and thermal fluctuations on DNA supe rcoiling and knotting dynamics.
TODW02 5th September 2012
17:00 to 17:40
W Olson Simulated looping propensities of protein-decorated DNA
Although the genetic messages in DNA are stored in a linear sequence of base pairs, the genomes of living species do not function in a linear fashion. Gene expression is regulated by DNA elements that often lie far apart along the genomic sequence but come close together during genetic processing. The intervening residues form loops, which are organized by the binding of various proteins. For example, in E. coli the Lac repressor protein assembly binds two DNA operators, separated by 92 or 401 base pairs, and suppresses the formation of gene products involved in the metabolism of lactose. The system also includes several highly abundant architectural proteins, such as Fis and HU, which, upon binding, bend a double-helical turn of DNA by 45 degrees or more. In order to gain a better understanding of the mechanics of DNA looping, we have investigated the effects of various proteins on the configurational properties of fragments of DNA, treating the DNA with elastic potentials t hat consider the intrinsic structure and deformability of successive base pairs and incorporating the known three-dimensional structural effects of various proteins on DNA double-helical structure. The presentation will highlight some of the new models and computational techniques that we have developed to generate the three-dimensional configurations of protein-mediated DNA loops and illustrate new insights gained from this work about the effects of various proteins on DNA topology and the apparent contributions of non-specific binding proteins to gene expression.
TODW02 6th September 2012
09:10 to 09:50
A Maxwell DNA topology and the mechanism of type II DNA topoisomerases
The topology of DNA plays vital roles in its biological function, particularly in the control of gene expression; DNA topoisomerases are enzymes that control DNA topology in all cells. They are divided into two types, I and II, depending on whether their reactions proceed via single- or double-strand breaks in DNA. The type II enzymes (such as bacterial DNA gyrase and DNA topoisomerase IV) cleave DNA in both strands and transport another double-stranded segment of DNA through this break. This process can lead to DNA relaxation, decatenation and unknotting, and, in the case of gyrase, DNA supercoiling, in reactions coupled to the hydrolysis of ATP. Although it is clear to see why supercoiling by gyrase (an endergonic reaction) requires ATP, this is less obvious with other type II enzymes that can only carry out relaxation. One potential role for ATP in the non-supercoiling topoisomerases is in the simplification of DNA topology: generating steady-state distributions of topoiso mers that are simpler than those seen at thermodynamic equilibrium. However, the energetic requirements for topology simplification are very small compared with the free energy available from ATP hydrolysis. Instead we propose that this energy is used to disrupt protein-protein interfaces in the enzyme, which are very stable in order to prevent unwanted DNA cleavage and accidental double-strand breaks in the chromosome. We suggest that this proposed role for ATP in disrupting protein interfaces may apply to other biological systems.

In related experiments, we are aiming to understand the ways in which DNA topology controls gene expression by investigating how DNA recognition is influenced by DNA supercoiling, using a combined approach involving molecular dynamics and biochemical/biophysical methods. In this approach we are using small DNA circles of varying superhelicity and examining DNA triplex formation as model system to study the thermodynamics of supercoiling-dependent DNA recognition.
TODW02 6th September 2012
10:10 to 10:50
Mechanistic studies of type IA topoisomerases
Type I topoisomerases are enzymes that change the topology of DNA by breaking one DNA strand and passing another DNA strand through the break before resealing it. They are subdivided into three groups based on sequence, structural, and mechanistic similarities. Biochemical, biophysical, and structural studies have provided atomic level understanding of the mechanism of action of these enzymes but many unanswered questions regarding their mechanism of action remain. E. coli topoisomerases I and III (Topo I and Topo III) relax negatively supercoiled DNA and also catenate/decatenate DNA molecules. Although these enzymes share the same mechanism of activity and have similar structures, they participate in different cellular processes: Topo I helps maintain the topological state of DNA, whereas Topo III helps resolve recombination and replication intermediates. In bulk experiments Topo I is more efficient at DNA relaxation whereas Topo III is more efficient at catenation/decatenation. To understand the differences in activity by these two highly related type IA topoisomerases, single molecule magnetic tweezers studies were conducted on several different DNA substrates. The experiments show differences in the way the two proteins work at the single molecule level, while also recovering observations from the bulk experiments. Surprisingly, the experiments show that Topo III relaxes DNA very efficiently, but with long pauses between relaxation events, whereas Topo I relaxes DNA more steadily and slowly. The results provide insights into the mechanism of both proteins and suggest reasons why Topo I is more efficient than Topo III at relaxing negatively supercoiled DNA.
TODW02 6th September 2012
10:50 to 11:30
C Soteros Knot statistics and knot reduction for a lattice polygon model of local strand passage
From DNA experiments, it is known that type II topoisomerases can reduce the fraction of knots in DNA over that found in randomly cyclized DNA; the amount that the fraction of knots is reduced is one measure of "knot reduction". These enzymes act locally in the DNA by transiently breaking one strand of DNA to allow another strand to pass through (strand passage). Szafron and Soteros have used a self-avoiding polygon model on the simple cubic lattice to model this strand passage action. The details of the combinatorial and topological theory behind this model will be reviewed. Also our Monte Carlo results on how knot reduction depends on the local juxtaposition structure at the strand passage site will be reviewed. We have found correlations between knot reduction and the crossing-sign and crossing-angle at the strand passage site. New results on the effect of the strand passage structure and solvent quality on the knot transition probabilities and knot reduction will also be presented.
TODW02 6th September 2012
11:30 to 11:50
Topo IV is the topoisomerase that knots and unknots sister duplexes during DNA replication
DNA topology plays a crucial role in all living cells. In prokaryotes, negative supercoiling is required to initiate replication and either negative or positive supercoiling assists decatenation. The role of DNA knots, however, remains a mystery. Knots are very harmful for cells if not removed efficiently, but DNA molecules become knotted in vivo. If knots are deleterious, why then does DNA become knotted? Here, we used classical genetics, high resolution two-dimensional agarose gel electrophoresis and atomic force microscopy to show that topoisomerase IV (Topo IV), one of the two type-II DNA topoisomerases in bacteria, is responsible for the knotting and unknotting of sister duplexes during DNA replication. We propose that when progression of the replication forks is impaired, sister duplexes become loosely intertwined. Under these conditions, Topo IV inadvertently makes the strand passages that lead to the formation of knots and removes them later on to allow their correct segregation.
TODW02 6th September 2012
11:50 to 12:10
P Sulkowski Topological recursion and classification of multi-stranded biopolymer configurations
In this talk I will present the formalism of so-called "topological recursion" and demonstrate how it can be applied to provide a complete classification of multi-stranded configurations of biomolecules. The "topological recursion" is a beautiful and rather sophisticated method arising from random matrix theory, which already found many applications and is currently under very active study in random matrix / statistical physics / high energy physics communities. In particular, I will present how to use this formalism to classify and compute all topologically inequivalent configurations of biomolecules, consisting of arbitrary number of strands, connected by arbitrary number of bonds or basepairs. This solution provides a new application of random matrix theory in the context of biophysics. Our solution has also an independent interpretation in pure mathematics, i.e. it provides certain important characteristics of moduli spaces of Riemann surfaces with boundaries.
TODW02 6th September 2012
13:30 to 14:10
DNA in confined geometries: topology effects
DNA can be found in confined geometries in different situations: for examples in viruses, in the chromosome or in artificial structures like in microfluidic devices. It is therefore interesting to study the statistical properties of the DNA depending on its length, concentration and topology. We present here data on circular DNA deposited in 2 dimensions with different concentration up to the overlap concentration c* and study also the effect of the length of the circular DNA. We characterize the statistical properties by determining the end-to-end distance as a function of the contour length, the directional correlation function, the area covered by the plasmids, the shape parameters like the sphericity as a function of the concentration.
TODW02 6th September 2012
14:10 to 14:50
A generic non-specific mechanism for the clustering of DNA and chromosome binding proteins
In this talk I will present a generic mechanism which leads to the clustering of DNA-binding proteins, on either prokaryotic DNA or eukaryotic chromatin. The clustering is driven by DNA-mediated interactions, and remarkably, does not require any affinity between the proteins, which we assume interact purely by steric repulsion. On naked DNA, small proteins cluster to form rows, whereas larger, histone-like proteins, form more disordered aggregates. On flexible fibres such as chromatin, the clustering leads to the formation of quasi-spherical foci.

Additionally I will present results from a simulation of an ensemble of Pol II polymerases and of p65 proteins (a well-characterised transcription factor) interacting with chromosomes 5, 8, 14 and 17. These interactions lead to the formation of foci, or factories, where active regions in the DNA come together. The statistics of the contacts compare favourably with 3C data on contacts made by SAMD4 on chromosome 14 and EXT1 on chromosome 8.
TODW02 7th September 2012
08:30 to 09:10
Topology of Xer site-specific recombination
Xer site-specific recombination at cer and psi converts bacterial plasmid multimers to monomers so that they can be efficiently segregated to both daughter cells at cell division. Recombination is catalysed by the XerCD recombinase acting at 30 bp core sites, and is regulated by the action of accessory proteins on accessory DNA sequences adjacent to the core sites. Recombination normally occurs between sites in direct repeat in a negatively supercoiled circular DNA molecule, and yields two circular products linked together in a right-handed 4-noded catenane with anti-parallel sites. Intramolecular recombination, and recombination between sites in inverted repeat on an unknotted circular substrate do not normally take place. However, recombination between sites in inverted repeat on right-handed torus knots with 5 or more nodes, or between anti-parallel sites on the two rings of a torus catenane with 6 or more nodes is efficient. In each case, recombination adds one additional node to the catenated or knotted substrate. These results are consistent with a model in which the accessory DNA sequences are interwrapped around the accessory proteins so that 3 interdomainal nodes are trapped by synapsis of the core sites by the XerCD recombinase. Recombination between directly repeated sites on an unknotted circular substrate formed the 4-noded catenane product with a linkage change ( ΔLk) of +4. This result is consistent with current models for strand exchange by the XerCD and other tyrosine recombinases, which predict recombination via a Holliday junction intermediate with no net change in total linkage ( ΔLg = Δcat + ΔLk = 0).
TODW02 7th September 2012
09:10 to 09:50
I Grainge Simplification of topology during Xer-mediated recombination
The chromosome of the bacterium Escherichia coli is a single circular DNA molecule of about 4.6 Mbp. Various recombinational repair processes that act upon the DNA during or following its replication can lead to a genetic crossover occurring, which in turn leads to a chromosome dimer being formed. This requires resolution back to two un-catenated single chromosomes before the cell can divide. The dimer resolution reaction is catalysed by the two recombinase proteins, XerC and XerD. Catalysis by XerD is in turn controlled by protein-protein interaction with the gamma subdomain of the FtsK DNA translocase. Recent in vitro data has shown that the interaction between XerD and FtsK-gamma is sufficient to stimulate recombination, but that in the absence of the DNA translocation motor of FtsK, then complex toplogical products result. If the motor is added back to these reactions, then simple, unlinked circular products are seen. DNA translocation is required for efficient chromosome dimer resolution in vivo too, mirroring the other experimental results. The focus of current research is into how XerD catalytic activity is stimulated, and how DNA translocation by FtsK leads to products with a simple toplogy.
TODW02 7th September 2012
10:10 to 10:50
Tangle analysis of protein-DNA complexes
Just like local knots can occur in long extension cords, such knots can also appear in DNA. DNA can be be either linear or circular. Some proteins will cut DNA and change the DNA configuration before resealing the DNA. Thus, if the DNA is circular, the DNA can become knotted. Protein-DNA complexes were first mathematically modeled using tangles in Ernst and Sumners seminal paper, "A calculus for rational tangles: applications to DNA recombination" (Math Proc Camb Phil Soc, 1990). A tangle consists of arcs properly embedded in a 3-dimensional ball. In order to model protein-bound DNA, the protein is modeled by the 3D ball while the segments of DNA bound by the protein can be thought of as arcs embedded within the protein ball. This is a very simple model of protein-DNA binding, but from this simple model, much information can be gained. The main idea is that when modeling protein-DNA reactions, one would like to know how to draw the DNA. For example, are there any cr ossings trapped by the protein complex? How do the DNA strands exit the complex? Is there significant bending? Tangle analysis cannot determine the exact geometry of the protein-bound DNA, but it can determine the overall entanglement of this DNA, after which other techniques may be used to more precisely determine the geometry.
TODW02 7th September 2012
10:50 to 11:30
DNA unknotting and unlinking
DNA replication is the basis for biological inheritance. In bacteria, reproduction starts with replication of the chromosome into two identical daughter molecules, followed by segregation of the newly replicated chromosomes and division of the parent cell into two daughter cells. In circular chromosomes, problems of entanglement during DNA linking complicate the process of chromosome segregation. In Escherichia coli, DNA unlinking is typically mediated by the type II topoisomerase topoIV. In the absence of topo IV, the site-specific recombination system XerCD mediates sister chromosome unlinking. This reaction is activated at the division septum by a powerful translocase FtsK, which coordinates the last stages of chromosome segregation. The mechanism by which the XerCD-FtsK complex simplifies the topology of DNA remains unclear. Techniques from knot theory and low-dimensional topology, aided by computational tools, are used to study the action of such enzymes. Understanding D NA unlinking by Xer recombination will provide a more complete picture of the chromosome segregation process.

This is joint work with Kai Ishihara, David Sherratt, Koya Shimokawa and Christine Soteros.
TODW02 7th September 2012
11:30 to 11:50
R Scharein TopoICE and other software tools for investigating topological problems in DNA
Mathematicians and biologists studying knotting and tangling issues in DNA can benefit from powerful interactive software tools. These can be used to explore what is mathematically possibly in certain kinds of reactions that DNA may undergo (crossing changes or recombination, for example). This information may be helpful in refining biological models or even in suggesting new experiments. In this talk, we will examine several software tools starting with the Topological Interactive Construction Engine (TopoICE). TopoICE has been available as part of the KnotPlot package for several years, in two versions: TopoICE-X (crossing changes) and TopoICE-R (recombination). Here we present these in a revamped, extended and more usable form. Also, we introduce TopoICE-S, intended to investigate knot distances via smoothings. This will be accompanied by newly obtained smoothing distance information. This talk will also unveil TopoICE and other DNA topology tools as free standalone applications (independent of KnotPlot) and running on mobile devices.

This is joint work with Isabel Darcy.
TOD 11th September 2012
11:30 to 12:30
Knots in light and fluids
To tie a shoelace into a knot is a relatively simple affair. Tying a knot in a field is a different story, because the whole of space must be filled in a way that matches the knot being tied at the core. The possibility of such localized knottedness in a space-filling field has fascinated physicists and mathematicians ever since Kelvin?s 'vortex atom' hypothesis, in which the atoms of the periodic table were hypothesized to correspond to closed vortex loops of different knot types. An intriguing physical manifestation of the interplay between knots and fields is the possibility of having knotted dynamical excitations. I will discuss some remarkably intricate and stable topological structures that can exist in light fields whose evolution is governed entirely by the geometric structure of the field. A special solution based on a structure known as a Robinson Congruence that was re-discovered in different contexts will serve as a basis for the discussion. I will then turn to hydrodynamics and discuss topologically non-trivial vortex configurations in fluids.

My lab's website can be found at http://irvinelab.uchicago.edu

TOD 13th September 2012
11:30 to 12:30
Nanoindentation of virus capsids
A coarse-grained model is used to study the mechanical response of 35 virus capsids of symmetries: T1, T2, T3, pT3, T4, and T7. The model is based on the native structure of the proteins that constitute the capsids and is described in terms of the Calpha atoms. The number of these atoms ranges between 8460 (for SPMV -- satellite panicum mosaic virus) and 135780 (for NBV -- nudaureli virus). Nanoindentation by a broad AFM tip is modeled as compression between two planes: either both flat or one flat and one curved. Plots of the compressive force versus plate separation show a variety of behaviors, but in each case there is an elastic F_c results in a drop in the force and emergence of irreversibility. Across the 35 capsids studied, both F_c and the elastic constant are observed to vary by a factor of 20. We argue that for a given linear size of the capsid the elastic constant and F_c depend on the average coordination number of an amino acid in the capsid.
OFB014 18th September 2012
15:00 to 15:30
DNA topology: where maths meets molecular biology
OFB014 18th September 2012
15:30 to 16:00
A new non-viral, non-toxic DNA minivector for basic and clinical research: how basic DNA topology research opened doors for gene therapy
OFB014 18th September 2012
16:30 to 17:05
New approaches to understanding and treating neurodegenerative diseases
OFB014 18th September 2012
17:05 to 17:15
Calcium hypothesis of Alzheimer's disease
OFB014 18th September 2012
17:15 to 18:00
G Winter The business of science; building therapeutic drugs based on proteins
TOD 20th September 2012
11:30 to 12:30
The geometry and topology of random polygons
Here is a natural question in statistical physics: What is the expected shape of a polymer with n monomers in solution? The corresponding mathematical question is equally interesting: Consider the space of n-gons in three dimensional space with length 2, modulo translation. This is a compact manifold. What is the natural metric (and corresponding probability measure) on this manifold? And what are the statistical properties of n-gons in 3-space sampled uniformly from this probability measure?

In this talk, we describe a natural probability measure on length 2 n-gon space pushed forward from the standard measure on the Stiefel manifold of 2-frames in complex n-space. The pushforward map comes from a construction of Hausmann and Knutson from algebraic geometry.

We will be able to explicitly and exactly compute the expected value of the radius of gyration for polygons sampled from our measure, and also give a fast algorithm for directly sampling the space of closed polygons. The talk describes joint work with Malcolm Adams (University of Georgia, USA), Tetsuo Deguchi (Ochanomizu University, Japan), and Clay Shonkwiler (University of Georgia, USA).
TOD 25th September 2012
11:30 to 12:30
On bright pattern of flakes in flow visualization
Tiny and thin reflective flakes, such as aluminum powders, mica flakes or Kalliroscope, are widely used to visualize the flow structure in a closed container. From their brightness distribution we may obtain useful information on the flow, such as the occurrence of instability, the location of turbulent/non-turbulent boundaries, etc. However, it is not straightforward to identify which properties of the flow are reflected in the visualized patterns. We should note that it is not the orientation itself of the flake surface but its time-derivative that responds instantaneously to the velocity gradient. The orientation of flake surface has history effect and may not represent the local flow structure. In this talk we consider the mechanism of formation of brightness distribution of reflective flakes in flows in a precessing spherical cavity.
TOD 26th September 2012
11:30 to 12:30
A Leonard A study of structures with intense vorticity in isotropic turbulence
The characteristics of vortex structures having vorticity occupying the high amplitude tail of the distribution of vorticity amplitudes in homogeneous, isotropic turbulence are investigated. The geometries and vorticity distributions within these structures are determined by use of data obtained from the results of a 1024 cubed DNS at Re_lamda = 433 residing in a Johns Hopkins web-based public database. Of particular interest are (1) the connection between the observed superexponential tail for the vorticity amplitude distribution and these local intense structures and (2) the dynamics that yields such structures.
TOD 27th September 2012
11:30 to 12:30
The average geometric and topological properties of open and closed equilateral polygonal chains
The average properties of open and closed equilateral polygonal chains depend upon geometrical and topological constraints. Several new rigorous results and numerical studies of lattice and off-lattice polygons will be presented. These include ergodic algorithms for thick 3D open chains and lattice random walks as well as the consideration of the presence and scale of entanglements such as knots and slipknots.
TOD 2nd October 2012
11:30 to 12:30
C Shonkwiler Homotopy, link homotopy and (higher?) helicity
I will outline a program for relating link homotopy classes of links in Euclidean 3-space to homotopy classes of certain associated maps. This seems worthwhile since interpreting the linking number - which is the simplest link homotopy invariant - as a homotopy invariant leads directly to the famous Gauss linking integral. Indeed, this approach has already yielded a generalized Gauss integral for Milnor's triple linking number.

The overarching goal is to find invariants of vector fields which will be relevant in, e.g., plasma physics. Specifically, these invariants should be "higher" analogues of helicity, meaning invariants of vector fields which are preserved under volume-preserving diffeomorphisms isotopic to the identity and which provide lower bounds for the field energy. Since helicity can be interpreted as an asymptotic version of the Gauss linking integral, the hope is that higher helicities can be defined as asymptotic versions of generalized Gauss integrals for higher linking invariants.
TOD 4th October 2012
11:30 to 12:30
C Parnell On the topology of complex magnetic fields of the Sun and the magnetosphere
With reference to several magnetohydrodynamic experiments, I will illustrate the typical topological structures found on the Sun and in the Earth's magnetosphere and discuss their importance for key phenomena such as solar flares and aurora. In particular, it has been found that the topological evolution of even relatively simple magnetic field structures are much more complex than expected with much of the complexity confined to thin layers. These thin layers are amazingly intricate and highly dynamic. Understanding not only the magnetic topological evolution, but also the energetic response and the plasma dynamics in these layers, is key to explaining many of the major unanswered questions of not only solar physics, but also of other stars and astrophysical bodies and even laboratory based plasma physics.
TOD 9th October 2012
11:30 to 12:30
Word representation of streamline topologies for structurally stable vortex flows in multiply connected domains
The instantaneous flow field of the inviscid and incompressible fluids in two-dimensional multiply connected exterior domains is described by a Hamiltonian vector field whose Hamiltonian becomes the stream function, which is the imaginary part of the complex potential. The flow is characterized topologically by the streamline pattern, which consists of the contour lines of the stream function. The present research provides us with a new topological classification procedure of the structurally stable streamline patterns generated by many vortex points in the presence of the uniform flow. The procedure allows us to assign a word representation to each structurally stable Hamiltonian vector field in the multiply connected domains. In the present talk, I will explain how to assign the word to the structurally stable streamline patterns and show all the streamline patterns in multiply connected domains of low genus with their word representations.
TOD 10th October 2012
11:30 to 12:30
D Weaire & A Mughal Packing Spheres in a Cylinder
What is the densest packing of spheres of diameter d in a cylinder of diameter D? We have explored this problem computationally for a wide range of D/d, and adduced some analytical results to explain the findings.
TOD 11th October 2012
11:30 to 12:30
A Yahalom Magnetohydrodynamics as a field theory: topological and group theoretical aspects
The combination of electrodynamic fields and flow fields yields the novel field theory of magnetohydrodynamics [2]. This field theory has unique topological and symmetry properties which are absent in each of one of its ingredients. Although the standard equations of magnetohydrodynamics depend on seven quantities: the magnetic vector field B, the velocity vector field V and the density, mathematical analysis [2] shows that only four scalar functions are needed to describe magnetohydrodynamics. This analysis is based on previous work of Yahalom & Lynden-Bell [1]. The four functions include two surfaces whose intersections consist the magnetic field lines, the part of the velocity field not defined by the co-moving magnetic field and the density. The Lagrangian describing magnetohydrodynamics admits a novel group of diffeomorphism [3]. Moreover, the conservation of the topology of magnetic fields, leads to effects which are classical analogous of the quantum Aharonov-Bohm effect [4].

Bibliography

[1] Asher Yahalom and Donald Lynden-Bell "Simplified Variational Principles for Barotropic Magnetohydrodynamics". [Los-Alamos Archives - physics/0603128], Journal of Fluid Mechanics, Volume 607, pages 235-265 (2008). [2] Asher Yahalom "A Four Function Variational Principle for Barotropic Magnetohydrodynamics". EPL 89 (2010) 34005, doi: 10.1209/0295-5075/89/34005 [Los-Alamos Archives - arXiv: 0811.2309]. [3] Asher Yahalom, "A New Diffeomorphism Symmetry Group of Magnetohydrodynamics" Proceedings of the 9th International Workshop "Lie Theory and Its Applications in Physics" (LT-9), 20-26 June 2011, Varna, Bulgaria. [4] Asher Yahalom "Aharonov - Bohm Effects in Magnetohydrodynamics" submitted to EPL [arXiv: 1005.3977].
TOD 23rd October 2012
11:30 to 12:30
P Smolarkiewicz EULAG-MHD: Simulation of the global solar dynamo
An MHD extension of the hydrodynamical solver EULAG [1] has been recently shown to simulate regular magnetic cycles and dynamo action, unprecedented in global simulations of solar magneto-convection [2, 3]. The lecture will highlight the mathematical and numerical features distinguishing EULAG from standard anelastic solar codes, including generalized time-dependent coordinate transformation and nonoscillatory forward-in-time integration schemes. The results illustrate advantages of the implicit large-eddy-simulation (ILES) approach, together with evidence that treatment of small scales may be critical for the production of cyclic behaviour and regular polarity reversals in this type of global simulations.

References: [1] Prusa et al., Comput. Fluids 37, 1193 (2008); [2] Ghizaru et al., ApJL 715, L133 (2010); [3]Racine et al., ApJ 735, 46 (2011) .
TOD 25th October 2012
11:30 to 12:30
J Parsley Helicity, cohomology, and configuration spaces
We realize helicity as an integral over the compactified configuration space of 2 points on a domain M in R^3. This space is the appropriate domain for integration, as the traditional helicity integral is improper along the diagonal MxM. Further, this configuration space contains a two-dimensional cohomology class, which we show represents helicity and which immediately shows the invariance of helicity under SDiff actions on M. This topological approach also produces a general formula for how much the helicity of a 2-form changes when the form is pushed forward by a diffeomorphism of the domain. We classify the helicity-preserving diffeomorphisms on a given domain, finding new ones on the two-holed solid torus and proving that there are no new ones on the standard solid torus.

(This is joint work with Jason Cantarella.)
TOD 30th October 2012
11:30 to 12:30
F Maggioni Optimal kinematics of supercoiled filaments

In this talk we propose kinematics of supercoiling of closed filaments as solutions of the elastic energy minimization [2]. The analysis is based on the thin rod approximation of linear elastic theory, under conservation of self-linking number with elastic energy evaluated by means of bending and torsional influence. Time evolution functions are described by means of piecewise polynomial transformations based on cubic spline functions. In contrast with traditional interpolation, the parameters defining the cubic splines representing the evolution functions are considered as the unknowns in a non-linear optimization problem. We show how the coiling process is associated with conversion of mean twist energy into bending energy through the passage by an inflexional configuration [3] in relation to geometric characteristics of the filament evolution. Geometric models for chromatin fibre folding are finally presented, compared in terms of geometric quantities and tested on ChIP Data (Chromatin ImmunoPrecipitation) generated from human pluripotent embryonic stem cells. These results provide new insights on the folding mechanism [1] and associated energy contents and may find useful applications in folding of macromolecules and DNA packing in cell biology [4].

References: [1] F. Maggioni & R.L. Ricca (2006) Writhing and coiling of closed filaments. Proc. R. Soc. A 462, 3151-3166. [2] Maggioni, F., Potra, F. & Bertocchi, M. Optimal kinematics of a looped filament (under referee reviewing). [3] H.K. Moffatt & R.L. Ricca (1992) Helicity and the C?lug?reanu invariant. Proc. R. Soc. Lond A 439, 411- 429. [4] R.L. Ricca & F. Maggioni (2008) Multiple folding and packing in DNA Computer & Mathematics with Applications, 55, 1044-1053

TOD 1st November 2012
11:30 to 12:30
Knot polynomial invariants in terms of helicity for tackling topology of fluid knots
A new method based on the derivation of the Jones polynomial invariant of knot theory to tackle and quantify structural complexity of vortex filaments in ideal fluids is presented. First, we show that the topology of a vortex tangle made by knots and links can be described by means of the Jones polynomial expressed in terms of kinetic helicity. Then, for the sake of illustration, explicit calculations of the Jones polynomial for the left-handed and right-handed trefoil knot and for the Whitehead link via the figure-of-eight knot are considered. The resulting polynomials are thus function of the topology of the knot type and vortex circulation and we provide several examples of those. While this approach extends the use of helicity in terms of linking numbers to the much richer context of knot polynomials, it offers also new tools to investigate topological aspects of mathematical fluid dynamics and, by directly implementing them, to perform new real-time numerical diagnostics of complex flows.
TOD 6th November 2012
11:30 to 12:30
T Kephart The tight knot spectrum in quantum chromodynamics
In 1867 Lord Kelvin suggested that elementary particles can be knotted fluid vortices in the aether. While his idea was revolutionary for his time, our present experimental knowledge does not agree with his conjecture. Nevertheless, the idea is attractive for its simplicity of relating fundamental physical and mathematical objects. In that spirit we model the observed J^{++} mesonic mass spectrum in terms of energies for tightly knotted and linked chromoelectric QCD flux tubes. The data is fitted with one and two parameter models. We predict one new state at 1200 MeV and a plethora of new states above 1700 MeV.
TOD 8th November 2012
11:30 to 12:30
Quantum information, the Jones polynomial and the Fibonacci model
An (abstract) quantum computer is a unitary transformation U defined on a complex vector space coupled with the preparation of states |psi> for U to act upon and a probabilistic range of measurements with probability ||^2. Designing quantum algorithms U|psi> means finding 'good' unitary transformations U for the sake of certain tasks. Remarkably, it turns out that the Jones polynomial, an invariant of knots and links discovered by Vaughan Jones in the 1980's, has an associated mathematical technology that allows the construction of a sufficient set of such unitary transformations. These occur by making a braided version of an abstract Fibonacci Particle P that can act upon itself to produce itself P,P ---> P or it can act upon itself to produce a null particle P,P ----> 1 that we will denote by 1. Writing a superposition of these two possibilities we have PP = P + 1, and iteration of this equation shows why P is called a Fibonacci particle. This talk will explain how the bracket model for the Jones polynomial gives a natural way to braid the Fibonacci particles and yields a basic structure for topological quantum computing.

>/p> There are many related avenues to the basic direction of this talk and we hope to touch on some of them: quantum computation of the Jones polynomial, the quantum Hall effect (where one expects to find physical analogs of the Fibonacci anyons), Khovanov homology, quantum knots, topological entanglement and quantum entanglement.

TOD 13th November 2012
11:30 to 12:30
E Panagiotou A study of the entanglement in polymer melts
Polymer melts are dense systems of macromolecules. In such dense systems the conformational freedom and motion of a chain is significantly affected by entanglement with other chains which generates obstacles of topological origin to its movement. In this talk we will discuss methods by which one may quantify and extract entanglement information from a polymer melt configuration using tools from knot theory. A classical measure of entanglement is the Gauss linking integral which is an integer topological invariant in the case of pairs of disjoint oriented closed chains in 3-space. For pairs of open chains, we will see that the Gauss linking integral can be applied to calculate an average linking number. In order to measure the entanglement between two oriented closed or open chains in a system with three-dimensional periodic boundary conditions (PBC) we use the Gauss linking number to define the periodic linking number. Using this measure of linking to assess the extend of entanglement in a polymer melt we study the effect of CReTA (Contour Reduction Topological Analysis) algorithm on the entanglement of polyethylene chains. Our results show that the new linking measure is consistent for the original and reduced systems.

For a collection of open or closed chains in 3-space or in PBC, we define the linking matrix. The eigenvalues of this matrix provide insight into the character of the entanglement.
TOD 14th November 2012
11:30 to 12:30
E Kats Dynamics of pore formation in membranes
How long do fluid stressed membranes last before rupture? Conversely, how high must the surface tension be to rupture a membrane? To answer these challenging questions we have developed a theoretical framework that allows description and reproduction of Dynamic Tension Spectroscopy (DTS) observations. The kinetics of membrane rupture is described as a combination of initial pore formation followed by a Brownian process of the pore radius crossing a time-dependent energy barrier.
TOD 14th November 2012
16:30 to 17:30
Symmetric quadratic dynamical systems
We will discuss dynamical systems that arise in classical and modern problems of mathematics, mechanics and biology. For such systems we introduce the notion of algebraic integrability. We will describe a wide class of algebraically integrable systems. This class contains Kovalevskaya, Lotka-Volterra type, and Darboux-Halphen systems and modern generalizations of these systems.
TOD 15th November 2012
11:30 to 12:30
R Kerner Self-assembly of icosahedral viral capsids: the combinatorial analysis approach
An analysis of all possible icosahedral viral capsids is proposed. It takes into account the diversity of coat proteins and their positioning in elementary pentagonal and hexagonal configurations, leading to definite capsid size. We show that the self-organization of observed capsids during their production implies a definite composition and configuration of elementary building blocks. The exact number of different protein dimers is related to the size of a given capsid, labeled by its $T$-number. Simple rules determining these numbers for each value of $T$ are deduced and certain consequences concerning the probabilities of mutations and evolution of capsid viruses are discussed.
TOD 20th November 2012
11:30 to 12:30
The space of soap bubbles
Understanding the space of all 'soap bubbles' - that is, complete embedded constant mean curvatures (CMC) surfaces in $R^3$ - is a central problem in geometric analysis. These CMC surfaces are highly transcendental objects; the topology and smooth structure of their moduli spaces are understood only in some special cases. In this talk we will describe the formal 'Lagrangian embedding' of CMC moduli space into the 'space of asymptotes' and discuss where this is smooth, namely, at a surface with no nontrivial square-integrable Jacobi fields. This nondegeneracy condition has now been established for all coplanar CMC surfaces of genus zero; this allows them to serve as 'building blocks' for more complicated CMC surfaces. There is also a surprising connection with complex projective structures and holomorphic quadratic differentials on $C$ obtained by taking the Schwarzian of the developing map for the projective structure. This assigns each coplanar CMC surface a 'classifying' complex polynomial, and lets us explicitly work out the smooth topology of their moduli spaces.
TOD 21st November 2012
11:30 to 12:30
A Mal'tsev Fermi surface topology and topological numbers in conductivity of normal metals
We consider the geometry of the quasiclassical electron trajectories on complicated Fermi surfaces in the presence of a strong magnetic field. Using rigorous topological theorems we present the classification of different topological types of open electron trajectories on the Fermi surfaces and consider the corresponding conductivity regimes in the limit $B \rightarrow \infty$. It is shown that the presence of the regular non-closed electron trajectories always leads to the presence of 'topological numbers' observable in the conductivity behaviour. On the other hand, the appearance of unstable non-closed electron trajectories may lead to rather interesting behaviour of conductivity, in particular, to the freezing of the longitudinal conductivity in the limit $B \rightarrow \infty$. The full picture of different regimes of magneto-conductivity behaviour can be considered as an important characteristic of the dispersion relation in metals.

(Joint work with S.P. Novikov).
TOD 22nd November 2012
11:30 to 12:30
Classification of rough surfaces using Schramm-Loewner evolution
The theory of Schramm-Loewner Evolution (SLEk) was developed originally as the theory of random curves with conformally invariant probability distribution, describing domain interfaces at criticality. I argue that iso-height lines on rough surfaces can also be regarded as SLEk. This may be regarded as evidence of conformal invariance in systems far from equilibrium. Perhaps this connection provides a new avenue for classification of self-affine surfaces in 2+1 dimensions. In this talk I present numerical evidence that SLE curves exist on rough surfaces. In particular I analyze the KPZ surface, having the same exponent as the self avoiding walk (SAW). I also present evidence that a physically grown Deposited WO3 surface has k=3, i.e. it is in the Ising class. Finally I discuss the Abelian Sandpile model and the Explosive Percolation (Watersheds).
TOD 23rd November 2012
11:30 to 12:30
Some aspects of iceberg mechanics
We review the factors governing the stability, dynamics and decay of icebergs and describe areas where current models are inadequate. These include questions such as draft changes in capsizing icebergs; iceberg trajectory modelling; the melt rate of the ice underside and ways of reducing it; and wave-induced flexure and its role in the break-up of tabular icebergs. In July 2012 the authors worked on a very large (42 sq km) tabular iceberg in Baffin Bay, which had calved from the Petermann Glacier in NW Greenland. We measured incoming swell spectrum and the iceberg response; also the role of buoyancy forces due to erosion of a waterline wave cut and the creation of an underwater ram. The iceberg broke up while we were on it, allowing an instrumental measurement of the calving event. The experiments were included in the BBC-2 film "Operation Iceberg" shown on Nov 1 and repeated on Nov 18.
TOD 23rd November 2012
16:00 to 17:00
Some aspects of iceberg mechanics
TOD 27th November 2012
11:30 to 12:30
Two amusing problems in geometry and topology
This lecture will provide two extended discussions of mathematical problems with strong physical content. The first concerns the dynamics of topological rearrangements of soap films under slow deformation of their boundaries. Through a combination of theory and experiment we illustrate some fascinating new phenomena involving reconnection of Plateau borders and finite-time singularities. Some conjectures are advance on the simplest dynamical models for such behaviour. The second problem of interest is the shape of a ponytail, which we show involves understanding the statistical physics of quenched random curvatures. A density functional theory of hair fibre bundles is constructed that leads to a 'Ponytail Shape Equation' whose solutions describe the envelope of a hair bundle in terms of an equivalent single fibre subject to a radial pressure arising from those random curvatures. Comparison with experimental observations allows for the extraction of the 'equation of state' of hair.
TOD 29th November 2012
11:30 to 12:30
The dynamics of conservative charged molecular strands
The equations of motion are derived for the dynamical folding of charged molecular strands, modeled as flexible continuous filamentary distributions of interacting rigid charge conformations. These equations are nonlocal when the screened Coulomb interactions, or Lennard-Jones potentials between pairs of charges, are included. The nonlocal dynamics is derived in the convective representation of continuum motion by using modified Euler-Poincaré and Hamilton-Pontryagin variational formulations. In the absence of nonlocal interactions, the equations recover the classical Kirchhoff theory of elastic rods. The motion equations in the convective representation are shown to arise by a classical Lagrangian reduction associated to the symmetry group of the system. This approach uses the process of affine Euler-Poincaré reduction initially developed for complex fluids. On the Hamiltonian side, the Poisson bracket of the molecular strand is obtained by reduction of the canonical symplectic structure on the phase space. Time permitting, the dynamics of multibouquets will also be presented.
TODW04 3rd December 2012
10:30 to 11:10
The properties of quantum turbulence: the homogeneous isotropic case
Quantum mechanics constrains the rotational motion of superfluid helium to discrete, vortex filaments of fixed circulation and atomic thickness (quantum vortices). A state of "quantum turbulence" can be easily created by stirring the liquid helium thermally or mechanically. In this talk I shall review recent experiments and numerical calculations which have revealed remarkable similarities between this form of turbulence and turbulence in ordinary fluids. Classical behaviour (such as the celebrated Kolmogorov energy spectrum) seems to arise from the coherence of many quanta of elementary circulation.
TODW04 3rd December 2012
11:10 to 11:30
Quantum and classical turbulence: Alike or different?
The question of how alike and different quantum and classical turbulence are will be addressed using simulations of anti-parallel vortex reconnection. The equations solved are: For quantum fluids the Gross-Pitaevskii equation and for classical turbulence the incompressible Navier-Stokes equation. For the two cases the initial attraction of the vortices, before the first reconnection, are quite similar. And for both, the final states are composed of a stack of vortex rings from which a -5/3 kinetic energy spectrum appears. However, almost everything in between is different, starting with the differences in the underlying physics of the circulation during reconnection. This presentation will describe these differences.
TODW04 3rd December 2012
11:50 to 12:10
Thermally and mechanically driven quantum turbulence in helium II
In most experiments with superfluid helium, turbulence is generated thermally (by applying a heat flux, as in thermal counterflow) or mechanically (by stirring the liquid). By modeling the superfluid vortex lines as reconnecting space curves with fixed circulation, and the driving normal fluid as a uniform flow (for thermal counterflow) and a synthetic turbulent flow (for mechanically driven turbulence), we determine the difference between thermally and mechanically driven quantum turbulence. We find that in mechanically driven turbulence, the energy is concentrated at the large scales, the spectrum obeys Kolmogorov scaling, vortex lines have large curvature, and the presence of coherent vortex structures induces vortex reconnections at small angles. On the contrary, in thermally driven turbulence, the energy is concentrated at the mesoscales, the curvature is smaller, the vorticity field is featureless, and reconnections occur at larger angles. Our results suggest a method t o experimentally detect the presence of superfluid vortex bundles.
TODW04 3rd December 2012
12:10 to 12:30
Generating and Classifying Turbulence in Bose-Einstein condensates
Vortices are a hallmark signature of a turbulent flow. Quantum vortices differ from their classical counterparts because of the quantization of circulation in superfluid flow. This means that the rotational motion of a superfluid is constrained to discrete vortices which all have the same core structure. Turbulence in superfluid Helium has been the subject of many recent experimental and theoretical investigations recently reviewed by Skrbek and Sreenivasan [1]. Recently, experimentalists have been able to visualise individual vortex lines and reconnection events using tracer particles[2]. Weakly interacting Bose-Einstein condensates present a unique opportunity to resolve the structure of vortices and in turn study the dynamics of a vortex tangle (as has recently been created in an atomic cloud[3]).

We investigate ways of generating turbulence in atomic systems by numerically stirring the condensate using a Gaussian 'spoon' (analogous to a laser beam in the experiments), and study the isotrophy of the resulting vortex tangle depending on when the path the spoon stirs is circular or random. We model the system using the Gross-Pitaevskii Equation.

[1] L. Skrbek and K.R. Sreenivasan, PoF 24, 011301 (2012) [2] G.P. Bewley et al. PNAS 105, 13707 (2008). [3] E.A.L. Henn et al. PRL 103, 04301 (2009).
TODW04 3rd December 2012
14:00 to 14:40
Topological Models for Elementary Particles
The talk will be a survey of topological models for elementary particles including the work of Lord Kelvin, Herbert Jehle, Thomas Kephart, Jack Avrin, Sundance Bilson-Thompson and recent work of the speaker with Sundance Bilson-Thompson and Jonathan Hackett and work of the speaker on the Fibonacci model in quantum information theory. Lord Kelvin suggested that atoms (the elementary particles of his time) are knotted vortices in the luminiferous aether. Jehle (much later on) suggested that elemenary particles should be quantized knotted electromagnetic flux. Kephart and Buiny suggest that closed loops of gluon field can be knotted particles -- knotted glueballs. Avrin notes that a three half-twisted Mobius band could be like a proton composed of three quarks mutually bound. Sundance Bilson-Thompson has a theory of framed three-braids that is a topological version of preons. In this theory we can think of particles as topological defects in networks of surfaces and some properties of embedded surfaces may sort out the matter. In the Fibonacci model for topological quantum computing, the system is generated by a braided anyonic abstract particle P that interacts with itself to produce itself (PP -----> P) or iteracts with itself to produce a neutral particle (PP ------> 1). The elementary particle P of the Fibonacci model is a structure that can be seen as a logical particle, underlying all the mathematical structures that we know. Since this talk surveys such a range of ideas, it will be up to the speaker to find a way to summarize these diseparate views at the time of the talk.
TODW04 3rd December 2012
14:40 to 15:00
The spectrum of tightly knotted flux tubes in QCD
TODW04 3rd December 2012
15:20 to 16:00
Quantized black hole charges and the Freudenthal dual
It is well-known that the quantized charges x of 4D black holes may be assigned to elements of an integral Freudenthal triple system (FTS). The FTS is equipped with a quartic form q(x) whose square root yields the lowest order black hole entropy. We show that a subset of these black holes, for which q(x) is necessarily a perfect square, admit a ``Freudenthal dual'' with integer charges ~x, for which ~~x=-x and q(~x)=q(x). [1] ''Black holes admitting a Freudenthal dual'', L. Borsten, D. Dahanayake, M.J. Duff, W. Rubens, Phys.Rev. D80 (2009) 026003 e-Print: arXiv:0903.5517 [hep-th]

[2] '' Freudenthal dual invariant Lagrangians'', L. Borsten, M.J. Duff, S. Ferrara ana A, Marrani (to appear).

TODW04 3rd December 2012
16:00 to 16:20
H Päs Knotted strings and leptonic flavor structure
Tight knots and links arising in the infrared limit of string theories may provide an interesting alternative to flavor symmetries for explaining the observed flavor patterns in the leptonic sector. As an example we consider a type I seesaw model where the Majorana mass structure is based on the discrete length spectrum of tight knots and links. It is shown that such a model is able to provide an excellent fit to current neutrino data and that it predicts a normal neutrino mass hierarchy as well as a small mixing angle $\theta_{13}$.
TODW04 3rd December 2012
16:20 to 16:40
Designing fibred knots in optical fields
TODW04 3rd December 2012
17:00 to 18:00
Rothschild lecture: Superoscillations and weak measurement
Band-limited functions can oscillate arbitrarily faster than their fastest Fourier component over arbitrarily long intervals. Where such ‘superoscillations’occur, functions are exponentially weak. In typical monochromatic optical fields, substantial fractions of the domain (one-third in two dimensions) are superoscillatory. Superoscillations have implications for signal processing, and raise the possibility of sub-wavelength resolution microscopy without evanescent waves. In quantum mechanics, superoscillations correspond to weak measurements, suggesting ‘weak values’ of observables (e.g photon momenta) far outside the range represented in the quantum state. A weak measurement of neutrino speed could lead to a superluminal result without violating causality, but the effect is too small to explain the speed recently claimed in a recent (and now-discredited) experminent.
TODW04 4th December 2012
08:50 to 09:30
Cosmic strings and other flux tubes
An introduction to cosmic strings and superstrings including a status report of the observational bounds.
TODW04 4th December 2012
09:30 to 09:50
Cosmic Strings and our Cosmos?
The talk will be about observational constraints on cosmic strings, notably those from searching for evidence of their signatures in the cosmic microwave sky. I will review current constraints on cosmic strings and future prospects from experiments like Planck.
TODW04 4th December 2012
09:50 to 10:30
Cosmic strings - relativistic vortices in the early universe
In many models of the very early universe, a tangle of relativistic vortices - cosmic strings - are spontaneously formed at a phase transition. I will briefly review the models and observational constraints, compare and contrast cosmic strings with superconductor flux tubes and superfluid vortices, and describe the latest large-scale simulations of their formation and decay.
TODW04 4th December 2012
10:50 to 11:10
L Pogosian Scaling configurations of cosmic superstring networks and their cosmological implications
Cosmic superstring networks contain strings of multiple tensions and Y-junctions. Depending on the magnitude of the fundamental string coupling, the stress-energy power spectrum of the scaling network is dominated either by populous light F strings, or rare heavy D strings. It may be possible to distinguish between these two regimes with future observations of CMB polarization.
TODW04 4th December 2012
11:10 to 11:30
Scaling properties of cosmic superstring networks
I will use a combination of state-of-the-art numerical simulations and analytic modeling to characterize the scaling properties of cosmic superstring networks. Particular attention will be given to the role of extra degrees of freedom in the evolution of these networks. Compared to the 'plain vanilla' case of Goto-Nambu strings, three such extensions play important but distinct roles in the network dynamics: the presence of charges/currents on the string worldsheet, the existence of junctions, and the possibility of a hierarchy of string tensions. I will also comment on insights gained from studying simpler defect networks, including Goto-Nambu strings themselves, domain walls and semilocal strings.
TODW04 4th December 2012
11:30 to 11:50
String networks with junctions: evolution and kinks
I will discuss a number of properties of string networks with junctions, focusing particularly on their evolution and the number of kinks in such networks. These properties are crucial in order to determine the observational consequences of networks with junctions, for instance in Gravitational Waves and CMB polarization.
TODW04 4th December 2012
11:50 to 12:10
Constraints on cosmic superstring formation and their decay mechanism
I will discuss some aspects of cosmic superstrings, related to their formation, nature, evolution and decay mechanisms. In particular, I will discuss the successful and theoretically consistent brane inflation models leading to cosmic superstrings, the dynamics of junctions and the channels of decay mechanisms of cosmic superstrings, which are very important for cosmological/observational consequences.
TODW04 4th December 2012
12:10 to 12:30
Cosmic strings, effective number of neutrinos, gravity waves and the CMB
We will report about recent work concerning cosmic defects, gravity waves and the CMB. On one hand, we simulate a scaling network of cosmic defects, and predict the background gravity wave emission coming from them. On the other, we study degeneracies in the CMB data between cosmic strings and gravity waves. When one tries to fit the CMB data with the effective number of neutrinos as a free parameter, the data prefers a higher number effective neutrinos. There have been works suggesting that the extra relativistic signal could come from gravity waves, and those waves could be created by cosmic defects. We perform a full analysis of the system taking into account the signal that such a network would create, its effect into the effective number of neutrinos, and how those parameters interplay in trying to fit different probes of CMB and other cosmological data.
TODW04 4th December 2012
13:30 to 14:10
JW Fleischer Superfluid and Berezinskii-Kosterlitz-Thouless dynamics of light
There are many advantages to interpreting coherent light as a superfluid. From an optics perspective, fluid language gives new insight into old problems and leads to new physics, e.g. instabilities. From a physics perspective, the ability to control input conditions and directly image the output means that optical experiments enable the observation of features that are difficult, if not impossible, to see in other fields. This is particularly true of coherence dynamics, as phase relationships are relatively easy to uncover through interference. Here, we review our recent results on vortex dynamics and optical thermodynamics, with an emphasis on condensation phenomena and the BKT transition. In the former process, the approach to thermal equilibrium drives the largest-scale mode of the system to become macroscopically occupied. In the latter process, vortex generation becomes more favorable than entropy production, and attempts at long-range order are destroyed. These two processes compete yet can co-exist, with many aspects of their many-body physics still outstanding. Optical experiments are beginning to map the dynamical phase space, through direct measurement of energy and momentum density, vortex number, and coherence properties. While still in their early stages, the results demonstrate condensed matter physics using only light and reinforce the use of photonic systems as an experimental testbed for fluid and statistical physics.
TODW04 4th December 2012
14:10 to 14:30
N Berloff Vortices in non-equilibrium condensates
TODW04 4th December 2012
14:30 to 14:50
Vorton solutions in 2D and 3D
We will present strong evidence for the existence of ring solutions in 2D and 3D known as vortons. In 2D we have are able to find analytic solutions for a twisted domain wall solutions which we use with a semi-analytic model to asset the existence of vortons and this is tested against numerical field theory solutions. We go on to the more difficult problem of vortons in 3D. We present solutions both in the cases of global and local U(1)xU(1) symmetries.
TODW04 4th December 2012
14:50 to 15:10
Log-Analytic Uncertainty Relation
I address the question of what characteristics of the quantum wave function phase can be measured. In particular I am interested in those phase aspects that can be deduced from the amplitude of the wave function. This will be shown to be connected to the topological characteristics of the Logarithm of the wave function in the complex time domain through a relation which is the temporal equivalent to the Kramers - Kronig formulae in the frequency domain. In particular for a wave packet which is reflected from an infinite potential barrier certain characteristics of the phase can be deduced provided that the momentum of the particle time the distance from the barrier is smaller than Planck constant over two and cannot be deduced otherwise. That is the phase characteristics cannot be deduced if the momentum of the particle time the distance from the barrier is bigger than Planck's constant over two which creates an experimental uncertainty in the phase.
TODW04 4th December 2012
15:30 to 15:50
MS Volkov Vortons and their stability
We present classical field theory solutions describing stationary vortex loops carrying persistent currents and balanced against contraction by the centrifugal force. Objects of this type, sometimes called vortons, may exist in various physical contexts, ranging from domains of condensed matter physics described by the non-linear Schroedinger equation and/or Ginzburg-Landau equations, till models of high energy physics such as Witten's theory of superconducting cosmic strings. We also analyze the vorton stability, both at the linear and non-linear levels.
TODW04 4th December 2012
15:50 to 16:10
Investigating Pure Quantum Turbulence in Superfluid 3He and a Means of Directly Inferring the Turbulent Energy Content
The lack of a general solution to the governing Navier-Stokes equations means that there is no fundamental theory of turbulence. Simpler pure quantum turbulence, the tangle of identical singly-quantized vortices in superfluids at T~0 may provide a deeper understanding of turbulence in general. In the present context this is especially relevant to the measurement of the turbulent energy. While the well-known Kolmogorov theory predicts the energy distribution of turbulence and how it decays, in normal systems the turbulent energy is generally only a small perturbation on the total thermal energy of the supporting medium. In quantum turbulence, however, the turbulent energy is accessible. A stationary condensate is necessarily in its ground state with zero enthalpy. Thus any added quantum turbulence accounts for the entire free energy of the superfluid and there are no other contributions.

In superfluid 3He we can generated and detect quantum turbulence and how it evolves with time, and using bolometric methods, we can measure the energy released as previously-generated turbulence decays, which seems to be unique to these systems.
TODW04 4th December 2012
16:10 to 16:50
Wave/Particle Duality via Silicon Droplets?
TODW04 4th December 2012
16:50 to 17:10
Topology of Quantum Liquids
TODW04 5th December 2012
09:00 to 09:40
Analytic and topological aspects of Menger curvatures for curves and submanifolds
We discuss various types of geometric curvature energies based on the concept of Menger curvature. These energies exhibit self-avoidance and regularizing effects on curves and submanifolds, and they control their topology.
TODW04 5th December 2012
09:40 to 10:00
M Mastin Symmetric Criticality for Ropelength
The ropelength of a link embedded in $R^3$ is the ratio of the curve's length to its thickness. Jason Cantarella, Joe Fu, Rob Kusner, and John Sullivan have developed a theory of first order criticality for ropelength. We will discuss an extension of this work for the case of link conformations with rigid rotational symmetry. As an application we will prove that there is an infinite class of knots for which there are geometrically distinct ropelength critical conformations. This work is joint with Jason Cantarella, Jennifer Ellis, and Joe Fu.
TODW04 5th December 2012
10:00 to 10:20
Regularity theory for knot energies
In the past two decades, the introduction of several knot-based geometric functionals has greatly contributed to the field of geometric curvature energies.

The general aim is investigating geometric properties of a given knotted curve in order to gain information on its knot type. More precisely, the original idea was to search a "nicely shaped" representative in a given knot class having strands being widely apart. This led to modeling self-avoiding functionals, so-called knot energies, that blow up on embedded curves converging to a curve with a self-intersection.

Due to the singularities which guarantee the self-repulsion property all these functionals lead to interesting analytical problems which in many cases almost naturally involve fractional Sobolev spaces.

In this talk we consider stationary points of knot energies. To this end we compute the Euler-Lagrange equation and derive higher regularity via a bootstrapping process.
TODW04 5th December 2012
10:20 to 10:40
Critical Links and Unlinks
The configuration spaces Hopf_k or McCool_k of k-component Hopf-links or unlinks can be understood using extensions of Hatcher's proof of the Smale Conjecture. We will describe low-energy critical configurations for Möbius energy or Ropelength on Hopf_k or McCool_k, and sketch a picture of the bottom of the Morse-Smale complex for each.

[This represents part of an ongoing project with Ryan Budney and John M. Sullivan]
TODW04 5th December 2012
11:00 to 11:40
Renormalized potential energies and their asymptotics
Energy of a knot was originally defined as the integration of the renormalized potential of a certain kind. Here, the renormalization can be done as follows: Suppose we are interested in a singular integral $\int_\Omega\omega$, which blows up on a subset $X\subset\Omega$. Remove an $\delta$-tubular neighbourhood of $X$ from $\Omega$, consider the integral over the complement, expand it in a Laurent series of $\delta$, and take the constant term. This idea gave rise to a M\"obius invariant surface energy in the sense of Auckly and Sadun, and recently, to generalization of Riesz potential of compact domains. If we integrate this generalized Riesz potential over the domain, we may need another renormalization around the boundary, according to the order of the generalized Riesz potential.

In this talk I will give "baby cases" of the application of the above story to the study of knots or surfaces.

TODW04 5th December 2012
11:40 to 12:00
Higher order topological invariants from the Chern-Simons action
TODW04 5th December 2012
12:00 to 12:20
Helical Organization of Tropical Cyclones
Recently we found (Levina and Montgomery, 2010) that a tropical cyclone (TC) formation is accompanied by generation of essential nonzero and persistently increasing integral helicity.

In this contribution we consider a helical flow organization on small and large space scales in a forming TC and offer a quantitative analysis for early stages in evolution of large-scale helical vortex based on diagnosis of a set of integral helical and energetic characteristics. Using the data from a near cloud-resolving numerical simulation, a key process of vertical vorticity generation from horizontal components and its amplification by special convective coherent structures – Vortical Hot Towers (VHTs) – is highlighted. The process is found to be a pathway for generation of a velocity field with linked vortex lines of horizontal and vertical vorticity on local and system scales. Based on these results, a new perspective on the role of VHTs in the amplification of the system-scale circulation is emphasized. They are THE CONNECTERS of the primary tangential and secondary overturning circulation on the system scales and are elemental building blocks for the nonzero system-scale helicity of the developing vortex throughout the TC evolution from genesis to the mature hurricane state.

Calculation and analyses of helical and energetic characteristics together with hydro- and thermodynamic flow fields allow the diagnosis of tropical cyclogenesis as an event when the primary and secondary circulations become linked on system scales.

We discuss also how these ideas may be combined with a recent paradigm of ‘Marsupial Pouch’ that allows predicting and tracking the location of tropical cyclogenesis in an easterly wave by means of global operational weather models.

TODW04 5th December 2012
12:20 to 12:35
Vortex knots in a Bose-Einstein condensate
I will present a method for numerically building a quantum vortex knot state in the single scalar field wave function of a Bose-Einstein condensate. I will show how the two topologically simplest vortex knots wrapped over a torus evolve and may preserve their shapes by reporting results of the integration in time of the governing Gross-Pitaevskii equation.

In particular, I will focus on how the velocity of a vortex knot depends on the ratio of poloidal and toroidal radius: in a first approximation it is linear and, for smaller ratio, the knot travels faster. Finally, I will display mechanisms of vortex breaking by reconnections which produce simpler vortex rings whose number depends on initial knot topology.
TODW04 5th December 2012
13:30 to 13:50
Solitons and Breathers on Quantized Superfluid Vortices
It is well known that quantized superfluid vortices can support excitations in the form of helical Kelvin waves. These Kelvin waves play an important role in the dynamics of these vortices and their interactions are believed to be the key mechanism for transferring energy in the ultra low temperature regime of superfluid turbulence in $^4$He. Kelvin waves can be ascribed to low amplitude excitations on vortex filaments. In this talk I will show that larger amplitude excitations of the vortices can be attributed to solitons propagating along the vortex filament. I will review the different class of soliton solutions that can arise as determined analytically from a simplified vortex model based on the localized induction approximation. I will show, through numerical simulations, that these solutions persist even in more realistic models based on a vortex filament model and the Gross-Pitaevskii equation. As a generalisation of these soliton solutions, I also consider the breathe r solutions on a vortex filament and illustrate how, under certain conditions, large amplitude excitations that are localized in space and time can emerge from lower amplitude Kelvin wave like excitations. The results presented are quite generic and are believed to be relevant to a wide class of systems ranging from classical to superfluid vortices. I will also interpret our results on these nonlinear vortex excitations in the context of the cross-over regime of scales in superfluid turbulence.
TODW04 5th December 2012
13:50 to 14:10
Branes, strings and boojums; topological defects in helium-3 and the cosmos
The order parameter of the superfluid helium-3 condensate exhibits broken symmetries that show analogs with those broken in the various transitions undergone by the Universe after the Big Bang. Fortunately for us, the helium-3 order parameter is also sufficiently complex that the superfluid may exist in several phases, the two most stable being the A and B phases. At Lancaster we have developed techniques to investigate the properties of the interface between the A and B phases in the pure condensate limit, far below the superfluid transition temperature. The order parameter transforms continuously across the A–B boundary, making this interface the most coherent two-dimensional structure to which we have experimental access. It has been argued that this ordered 2-d surface in a 3-d bulk matrix, separating the two phases, can provide a good analog of a cosmological brane separating two distinct quantum vacuum states. In superfluid helium-3 the creation of such 2-branes mu st lead to the formation of point and line defects in the texture of the 3-d bulk, simply as a result of the constraints imposed by the interplay of the order parameter symmetries and the geometry of the container. Furthermore, our experiments have shown that removing the 2-branes from the bulk, in a process analogous to brane annihilation, creates new line defects in large quantities. Such observations may provide insight into the formation of topological defects such as cosmic strings arising from brane interactions in the early Universe. Up to now our experimental techniques have only allowed us to infer the properties of the interface and defects by measuring how they impede the transport of quasiparticle excitations in the superfluid, which is essentially a remote measurement. Our new experiments allow us to directly probe the interface region.
TODW04 5th December 2012
14:10 to 14:30
Dynamics of Hopfions
Several materials, such as ferromagnets, spinor Bose-Einstein condensates and some topological insulators, are now believed to support knotted structures. One of the most successful base-models having stable knots is the Faddeev-Skyrme model and it is expected to be contained in some of these experimentally relevant models. The taxonomy of knotted topological solitons (Hopfions) of this model is known. In this talk, we describe the basic properties of static Hopfions, known for quite a long time before discussing some aspects of the dynamics of Hopfions, how the static properties survive in the dynamical situation, and show that they indeed behave like particles: during scattering the Hopf charge is conserved and bound states are formed when the dynamics allows it.
TODW04 6th December 2012
09:00 to 09:20
J Parsley Cohomology reveals when helicity is a diffeomorphism invariant
We consider the helicity of a vector field, which calculates the average linking number of the field’s flowlines. Helicity is invariant under certain diffeomorphisms of its domain – we seek to understand which ones.

Extending to differential (k+1)-forms on domains R^{2k+1}, we express helicity as a cohomology class. This topological approach allows us to find a general formula for how much helicity changes when the form is pushed forward by a diffeomorphism of the domain. We classify the helicity-preserving diffeomorphisms on a given domain, finding new ones on the two-holed solid torus and proving that there are no new ones on the standard solid torus. This approach also leads us to define submanifold helicities: differential (k+1)-forms on n-dimensional subdomains of R^m.
TODW04 6th December 2012
09:20 to 09:40
C Shonkwiler The geometry of random polygons
What is the expected shape of a ring polymer in solution? This is a natural question in statistical physics which suggests an equally interesting mathematical question: what are the statistics of the geometric invariants of random, fixed-length n-gons in space? Of course, this requires first answering a more basic question: what is the natural metric (and corresponding probability measure) on the compact manifold of fixed-length n-gons in space modulo translation?

In this talk I will describe a natural metric on this space which is pushed forward from the standard metric on the Stiefel manifold of 2-frames in complex n-space via the coordinatewise Hopf map introduced by Hausmann and Knutson. With respect to the corresponding probability measure it is then possible to prove very precise statements about the statistical geometry of random polygons.

For example, I will show that the expected radius of gyration of an n-gon sampled according to this measure is exactly 1/(2n). I will also demonstrate a simple, linear-time algorithm for directly sampling polygons from this measure. This is joint work with Jason Cantarella (University of Georgia, USA) and Tetsuo Deguchi (Ochanomizu University, Japan).
TODW04 6th December 2012
09:40 to 10:00
The Expected Total Curvature of Random Polygons
We consider the expected value for the total curvature of a random closed polygon. Numerical experiments have suggested that as the number of edges becomes large, the difference between the expected total curvature of a random closed polygon and a random open polygon with the same number of turning angles approaches a positive constant. We show that this is true for a natural class of probability measures on polygons, and give a formula for the constant in terms of the moments of the edgelength distribution.

We then consider the symmetric measure on closed polygons of fixed total length constructed by Cantarella, Deguchi, and Shonkwiler. For this measure, the expected total curvature of a closed n-gon is asymptotic to n pi/2 + pi/4 by our first result. With a more careful analysis, we are able to prove that the exact expected value of total curvature is n pi/2 + (2n/2n-3) pi/4. As a consequence, we show that at least 1/3 of fixed-length hexagons and 1/11 of fixed-length heptagons in 3-space are unknotted.
TODW04 6th December 2012
10:00 to 10:20
On the energy spectrum of magnetic knots and links
The groundstate energy spectrum of the first 250 zero-framed prime knots and links is studied by using an exact analytical expression derived by the constrained relaxation of standard magnetic flux tubes in ideal magneto-hydrodynamics (Maggioni & Ricca, 2009) and data obtained by the RIDGERUNNER tightening algorithm (Ashton et al., 2011). The magnetic energy is normalized with respect to the reference energy of the tight torus and is plotted against increasing values of ropelength. A remarkable generic behavior characterizes the spectrum of both knots and links. A comparative study of the bending energy reveals that curvature information provides a rather good indicator of magnetic energy levels.

(2009) Maggioni F and Ricca RL. On the groundstate energy of tight knots. Proc. R. Soc. A 465, 2761–2783. (2011) Ashton T, Cantarella J, Piatek M and Rawdon E. Knot tightening by constrained gradient descent. Experim. Math. 20, 57-90.
TODW04 6th December 2012
10:20 to 10:40
Criticality theory for ropelength and related problems
We consider certain new results related to the criticality theory for ropelength developed with Cantarella, Fu and Kusner.
TODW04 6th December 2012
11:10 to 11:30
Topological and Geometric Properties of tightly confined random polygons
Consider equilateral random polygons in a confinement sphere of radius R. In this talk we describe how geometric properties of the random polygons such a curvature or torsion change when the radius of confinement decreases. We also describe how the knot spectrum changes as R decreases.
TODW04 6th December 2012
11:30 to 11:45
Equilibrium configurations of elastic torus knots (n,2)
We study equilibria of braided structures made of two elastic rods with their centrelines remaining at constant distance from each other. The model is geometrically exact for large deformations. Each of the rods is modelled as thin, uniform, homogeneous, isotropic, inextensible, unshearable, intrinsically twisted, and to have circular cross-section. The governing equations are obtained by applying Hamilton's principle to the action which is a sum of the elastic strain energies and the constraints related to the inextensibility of the rods. Hamilton's principle is equivalent to the second-order variational problem for the action expressed in reduced strain-like variables. The Euler-Lagrange equations are derived partly in Euler-Poincare form and are a set of ODEs suitable for numerical solution.

We model torus knots (n,2) as closed configurations of the 2-strand braid. We compute numerical solutions of this boundary value problem using path following. Closed 2-braids buckle under increasing twist. We present a bifurcation diagram in the twist-force plane for torus knots (n,2). Each knot has a V-shaped non-buckled branch with its vertex on the twist axis. There is a series of bifurcation points of buckling modes on both sides of each of the V-branches. The 1st mode bifurcation points for n and n±4 are connected by transition curves that go through (unphysical) self-crossing of the braid. Thus, all the knots turn out to be divided into two classes: one of them may be produced from the right-handed trefoil and the other from the left-handed. Higher-mode post-buckled configurations lead to cable knots.

It is instructive to see how close our elastic knots can be tightened to the ideal shape. For the trefoil knot the tightest shape we could get has a ropelength of 32.85560666, which is remarkably close to the best current estimate. Careful examination reveals that the solution is free from self-intersections though the contact set remains a distorted circle.
TODW04 6th December 2012
11:45 to 12:05
S Blatt The gradient flow of O'Hara's knot energies
All of us know how hard it can be to decide whether the cable spaghetti lying in front of us is really knotted or whether the knot vanishes into thin air after pushing and pulling at the right strings.

In this talk we approach this problem using gradient flows of a family of energies introduced by O'Hara in 1991-1994.We will see that this allows us to transform any closed curve into a special set of representatives - the stationary points of these energies - without changing the type of knot. We prove longtime existence and smooth convergence to stationary points for these evolution equations.
TODW04 6th December 2012
12:05 to 12:25
S Tanda Topological Crystals and the quantum effects
We report the discovery of Mobius, Ring, Figure-8, Hopf-link Crystals in NbSe3, conventionally grown as ribbons and whiskers.

We also reveal their formation mechanisms of which two crucial components are the spherical selenium (Se) droplet, which a NbSe3 ber wraps around due to surface tension, and the monoclinic (P2(1)/m) crystal symmetry inherent in NbSe3, which induces a twist in the strip when bent. Our crystals provide a non-fictious topological Mobius world governed by a non-trivial real-space topology. We classified these topological crystals as an intermediary between condensed matter physics and mathematics. Moreover, we observed Aharonov-Bohm effect of charge-density wave and Frolich type superconductor as electronic properties using topological crystals.
TODW04 6th December 2012
14:00 to 14:20
Quantum vortex reconnections
TODW04 6th December 2012
14:20 to 14:40
Knots in light and fluids
To tie a shoelace into a knot is a relatively simple affair. Tying a knot in a field is a different story, because the whole of space must be filled in a way that matches the knot being tied at the core. The possibility of such localized knottedness in a space-filling field has fascinated physicists and mathematicians ever since Kelvin’s 'vortex atom' hypothesis, in which the atoms of the periodic table were hypothesized to correspond to closed vortex loops of different knot types. An intriguing physical manifestation of the interplay between knots and fields is the possibility of having knotted dynamical excitations. I will discuss some remarkably intricate and stable topological structures that can exist in light fields whose evolution is governed entirely by the geometric structure of the field. A special solution based on a structure known as a Robinson Congruence that was re-discovered in different contexts will serve as a basis for the discussion. I will th en turn to hydrodynamics and discuss topologically non-trivial vortex configurations in fluids.

My lab's website can be found at http://irvinelab.uchicago.edu

TODW04 6th December 2012
14:40 to 14:55
N Proukakis Vortex Dynamics and Turbulence in Confined Quantum Gases
Quantised vortices are known to arise in ultra-low temperature quantum gases as a result of targeted vortex generation (e.g. via phase imprinting or a 'quantum stirrer') or intrinsic system fluctuations. Such vortices interact dynamically, reconnect and can form regular ('lattices') or irregular (turbulent) structures, depending on the system conditions. Focusing initially on the issue of tangled vorticity, we show that the velocity statistics provides a unique identifier of 'quantum' vs. 'ordinary' turbulence, in agreement with related studies in helium. As quantum gas experiments typically feature harmonic confinement, one does not have access to the broad lengthscales relevant for helium, with the total number of vortices typically constrained from a few to a few hundred. In a first attempt to probe 'turbulence' in such systems, we go beyond the usual procedure of looking at the energy spectrum to discuss methods to quantify and ana lyze the amount of clustering of vortices using information extracted from their position and winding, focusing here on the two-dimensional regime. As realistic cold atom experiments are conducted at non-zero temperatures, where the condensate co-exists with a thermal cloud, we also study how temperature modifies the motion of vortices in such systems.

This work has been generously funded by EPSRC.
TODW04 6th December 2012
14:55 to 15:10
Interpretation of quasiparticle scattering measurements in $^3$He-B: a three dimensional numerical analysis
Present research is concerned with numerical modelling of Andreev scattering technique used for detection of quantized vortices in $^3$He-B. The results of numerical analysis of Andreev reflection by three-dimensional turbulent structures are reported. We analyse the Andreev scattering by a dense vortex tangle and calculate the spectral characteristics of the retroreflected beam of thermal excitations. The obtained results are in agreement with experimental observations.
TODW04 6th December 2012
15:30 to 15:45
Macroscopic bundles of vortex rings in superfluid helium
It is well known that two co-axial vortex rings can leap-frog about each other. By direct numerical simulation, we show that in superfluid helium the effect can be generalised to a large number of vortex rings, which form a toroidal bundle. The bundle can be shown to be robust, travelling a significant distance compared to its diameter, whilst simultaneously becoming linked and eventually turbulent. We also discuss the effect of friction at non-zero temperatures, and show how in this case the presence of normal fluid rotation is necessary for the stability of the bundle.
TODW04 6th December 2012
15:45 to 16:05
Knots and links of disclination lines in chiral nematic colloids
Nematic braids formed by disclination lines entangling colloidal particles in nematic liquid crystal are geometrically stabilized and restricted by topology. Experiments with nematic braids show rich variety of knotted and linked disclinations loops that can be manipulated and rewired by laser light. We describe a simple rewiring formalism and demonstrate how the self-linking number of nematic ribbons enables a classification of entangled structures. Controlled formation of arbitrary microscopic links and knots in nematic colloids provides a new route to the fabrication of soft matter with special topological features.

References:

[1] M. Ravnik, M. Škarabot, S. Žumer, U. Tkalec, I. Poberaj, D. Babič, N. Osterman, I. Muševič, Phys. Rev. Lett. 99, 247801 (2007). [2] U. Tkalec, M. Ravnik, S. Žumer, I. Muševič, Phys. Rev. Lett. 103, 127801 (2009). [3] S. Čopar, S. Žumer, Phys. Rev. Lett. 106, 177801 (2011). [4] U. Tkalec, M. Ravnik, S. Čopar, S. Žumer, I. Muševič, Science 333, 62 (2011). [5] S. Čopar, T. Porenta, S. Žumer, Phys. Rev. E 84, 051702 (2011). [6] G. P. Alexander, B. G. Chen, E. A. Matsumoto, R. D. Kamien, Rev. Mod. Phys. 84, 497 (2012).
TODW04 6th December 2012
16:05 to 16:25
Experiments with tangles of quantized vortex lines in superfluid 4He in the T=0 limit
In our experiments, we can create dense ensembles of quantized vortex lines of various degrees of polarization and entanglement, and monitor either their free decay or steady state whilst forcing continuously. The superfluid is forced either by macroscopic bodies that generate large-scale vortex bundles or by microscopic particles (injected ions) that generate uncorrelated vortices. Steady net polarization can be introduced by conducting the experiment in a rotating container. The characterization of vortex tangles is done via measurements of the transport of injected ions through them. The following types of tangles will be reviewed: homogeneous random tangles (no large-scale polarization), homogeneous quasi-classical turbulence (tangles in which the dominant energy is concentrated in large-scale bundles-eddies), steadily polarized anisotropic tangles of either high or low polarization, beams of parallel vortex rings. There is no viscous dissipation in the T=0 limit; the dyn amics of individual vortex lines is conservative except for the Kelvin waves of extremely small wavelengths. The scenario and rate of the evolution of different vortex ensembles largely depend on the mutual polarization of vortioces that affects the frequency of their reconnections. Further plans to investigate the microscopic processes of the quantum cascade (Kelvin wave cascade) and visualization of individual vortex cores will be outlined.
TODW04 6th December 2012
16:25 to 16:45
J Bohr Torus Knots and Links, Twist Neutrality and Biological Applications
We present mathematical restrictions for torus knots and links, and for bent helices. The concept of twist neutrality is developed for bent and coiled structures and biological applications are reviewed [1,2].

[1] K. Olsen and J. Bohr, Geometry of the toroidal N-helix: optimal-packing and zero-twist. New Journal of Physics 14, 023063 (2012).

[2] J. Bohr and K. Olsen, Twist neutrality and the diameter of the nucleosome core particle. Phys. Rev. Lett. 108, 098101 (2012).
TODW04 7th December 2012
09:00 to 09:40
Magnetic Fields in the Early Universe -- Chiral Effects and Topology
I will briefly review cosmic magnetic fields and discuss some ideas to generate them. Special emphasis will be given to the possible generation of helical magnetic fields, and the possible role of chirality in the universe. As a by-product, the discussion will hint at processes that might lead to the production of magnetic monopoles.
TODW04 7th December 2012
09:40 to 10:20
P Akhmet'ev Asymptotic higher ergodic invariants of magnetic lines
V.I.Arnol'd in 1984 formulated the following problem: "To transform asymptotic ergodic definition of Hopf invariant of a divergence-free vector field to Novikov's theory, which generalizes Withehead product in homotopy groups"'.

We shall call divergence-free fields by magnetic fields. Asymptotic invariants of magnetic fields, in particular, the theorem by V.I.Arnol'd about asymptotic Gaussian linking number, is a bridge, which relates differential equitations and topology. We consider 3D case, the most important for applications.

Asymptotic invariants are derived from a finite-type invariant of links, which has to be satisfied corresponding limit relations. Ergodicity of such an invariant means that this invariant is well-defined as the mean value of an integrable function, which is defined on the finite-type configuration space $K$, associated with magnetic lines.

At the previous step of the construction we introduce a simplest infinite family of invariants: asymptotic linking coefficients. The definition of the invariants is simple: the helicity density is a well-defined function on the space $K$, the coefficients are well-defined as the corresponding integral momentum of this function. Using this general construction, a higher asymptotic ergodic invariant is well-defined. Assuming the the magnetic field is represented by a $\delta$-support with contains 3 closed magnetic lines equipped with unite magnetic flows, this higher invariant is equal to the corresponding Vassiliev's invariant of classical links of the order 7, and this invariant is not a function of the pairwise linking numbers of components. When the length of generic magnetic lines tends to $\infty$, the asymptotic of the invariant is equal to 12, this is less then twice order $14$ of the invariant.

Preliminary results arXiv:1105.5876 was presented at the Conference "`Entanglement and Linking"' (Pisa) 18-19 May (2011).
TODW04 7th December 2012
10:40 to 11:20
Topological Solitons from Geometry
Solitons are localised non-singular lumps of energy which describe particles non perturbatively. Finding the solitons usually involves solving nonlinear differential equations, but I shall show that in some cases the solitons emerge directly from the underlying space-time geometry: certain abelian vortices arise from surfaces of constant mean curvature in Minkowski space, and skyrmions can be constructed from the holonomy of gravitational instantons.
TODW04 7th December 2012
11:20 to 11:40
M Nitta Creating Vortons and Knot Solitons via Domain Wall Pair Annihilation in BEC and Field Theory
We show that when a vortex-string is stretched between a pair of a domain wall and an anti-domain wall in two component Bose-Einstein condensates, there remains a vorton after the pair annihilation. We also show that the same configuration in the mass deformed Faddeev-Skyrme model results in a knot soliton (Hopfion) after the pair annihilation.
TODW04 7th December 2012
11:40 to 12:00
T Tchrakian Abelian and non-Abelian Hopfions in all odd dimensions
Hopfions are field configurations of scalar matter systems characterised prominently by the fact that they describe knots in configuration space. Like the 'usual solitons', e.g. Skyrmions, monopoles, vortices and instantons, Hopfions are static and finite energy solutions that are stabilised by a topological charge, which supplies the energy lower bound.

In contrast to the 'usual solitons' however, the topological charge of Hopfions is not the volume integral of a total divergence. While the topological charge densities of the 'usual solitons', namely the Chern-Pontryagin (CP) densities or their descendants, are total divergence, the corresponding quantities for Hopfions are the Chern-Simons (CS) densities which are not total divergence. Subject to the appropriate symmetries however, these CS densities do reduce to total divergence and become candidates for topological charges. Thus, Hopfion field are necessarily subject to the appropriate symmetry to decsribe knots, excluding spherically symmetry, in contrast to the 'usual solitons'.

The construction of these CS densities is enabled by employing complex nonlinear sigma models, which feature composite connections. The CS densities are defined in terms of these connections and their curvatures. (In some dimensions the complex sigma model can be equivalent to a real sigma model, e.g. in D=3 Skyrme-Fadde'ev O(3) model and the corresponding CP^1 model.) It is natural to propose Hopfion fields in all odd space dimensions where a CS density can be defined. This covers both Abelian and non-Abelian theories, namely empolying projective-complex and Grassmannian models, respectively. It is in this sense that we have used the terminology of Abelian and non-Abelian Hopfions.

Explicit field configurations displaying the appropriate symmetries and specific asymptotic behaviours in several (higher) dimensions are proposed, and it is verified that for these configurations the CS densities do indeed become total divergence.
TODW04 7th December 2012
12:00 to 12:20
E Babaev Skyrmions and Hopfions in exotic superconductors
TODW04 7th December 2012
13:30 to 13:50
D Harland Modelling Hopf solitons with elastic rods
I will review recent progress in modelling knotted solitons in the Skyrme-Faddeev model using elastic rods. The effective elastic rod model is simple to use, and can in some cases be solved analytically. It has enabled the discovery of new solitonic states which had eluded direct numerical simulations of the field theory. This suggests more generally that elasticity theory could be a useful tool in the study of solitons.
TODW04 7th December 2012
13:50 to 14:10
Fermionic quantization of knot solitons
Knot solitons arise as global energy minimizers in field theories such as the Faddeev-Skyrme model. Such field theories, when quantized, are inherently bosonic because the fundamental fields represent scalar bosons. Nonetheless, the solitons they support can be given fermionic exchange statistics, provided the classical field configuration space has the right algebraic topology. In this talk I will review a computation of the fundamental group of the Faddeev-Skyrme configuration space, and show how this allows a consistent fermionic quantization of knot solitons. This is based on (separate) collaborations with Dave Auckly and Steffen Krusch.
TODW04 7th December 2012
14:30 to 14:50
Folding and collapse in string-like structures
We argue that the the physics of folding and collapse of string-like structures can be described in terms of topological solitons. For this we use extrinsic geometry of filamental curves in combination of general geometrical arguments, to derive a universal form of energy function, which we propose is essentially unique. We then show that the ensuing equations of motion support topological solitons that are closely related to the solitons in the discrete nonlinear Schrodinger equation. We then argue that with proper parameters, a soliton supporting filament can describe proteins, which are mathematically one dimensional piecewise linear polygonal chains. As an example we show a movie of a simulation, how to model the folding of a medium length protein. The precision we reach is around 40 pico-meters root-mean-square distance form the experimentally constructed structure. Our result proposes that there are at least some 10^20 topological solitons in each human body.
TODW04 7th December 2012
14:50 to 15:10
S Krusch Fermions coupled to Hopf Solitons
Solitons in the Skyrme and the Faddeev-Skyrme model share many similarities. While no topologically non-trivial exact solutions are known in flat space there is a minimal energy charge one soliton on the 3-sphere of sufficiently small radius in both models. The charge one Skyrmion is given by the identity map, whereas the charge one Hopf soliton is given by the Hopf map. Also, the solitons in both models can be semi-classically quantized as fermions, by defining the wave function of the covering space of configuration space and imposing Finkelstein-Rubinstein constraints. When fermions are chirally coupled to Skyrmions the resulting Dirac equation can be solved explicitly on the 3-sphere in the background of a charge one Skyrmion. In this talk, I describe how to couple fermions to Hopf solitons.
TODW04 7th December 2012
15:10 to 15:25
J Garaud Topological solitons in multi-component superconductors : from baby-Skyrmions to vortex loops
The crucial importance of topological excitations in the physics of superconductivity made Ginzburg--Landau vortices one of the most studied example of topological defects. Multi-band/multi-component superconductors extends usual Ginzburg--Landau theory by considering more than one scalar field (several superconducting order parameters).

Family materials, where superconductivity is multi-band/multi-component, has recently been growing. Because of additional fields and new broken symmetries, the zoo of topological defects is much richer in multi-component systems (e.g. Line-like vortices, fractional vortices, baby skyrmions...). For entropic reasons, thermal fluctuations will induce much more complicated three dimensional solitons, in particular vortex loops.

I will discuss various aspects of topological excitations in multi-component/multi-band superconductors.
TOD 11th December 2012
11:30 to 12:30
Generic free surface singularities
Many important partial differential equations used in engineering or theoretical physics describe the motion of lines or surfaces. Locations of particular interest are those where the motion becomes (nearly) singular, and the line or surface forms a cusp or a tip with a very large curvature. Physical examples occur when viscous fluid is poured into a beaker, or when caustics form at the bottom of a coffee cup.

Recent work suggests the structure of many such singularities can be understood using a local geometrical description. The local shape corresponds exactly to a solution of the partial differential equation. Building on this insight, we propose a preliminary classification of such singular points, including higher dimensions of both the surface and the space it lives in. We also investigate how the observed singularity can be understood from the point of view of the dynamical equation, and how the geometrical description could be used in more complex situations.
TOD 13th December 2012
10:00 to 11:00
Surfaces and curvature in Euler and Navier-Stokes simulations. Where next?
This presentation will summarise where the analysis of my latest Euler and Navier-Stokes calculations has gone since I arrived with raw data at TODW01: Topological Fluid Dynamics II. This includes stronger evidence for singularities on the edge of the analytic bound, questions of why the growth is so constrained, and what all of this might be telling us about turbulence.
TOD 13th December 2012
11:30 to 12:30
Modern trends in dynamo theory
Dynamo action is the process by which magnetic fields in astrophysical bodies (and recently, laboratory fluids) are maintained against resistive losses by Faraday induction. For many years a favoured model of this process, known as mean-field electrodynamics, has been widely used to produce tractable models. I shall present a critique of this theory and contrast it it with another dynamo process (small scale dynamo action) that does not, unlike mean-field electrodynamics, rely on broken reflection symmetry or scale separation. Finally, I shall talk about very recent rigorous results concerning the Archontis dynamo, in which the magnetic and velocity fields are closely aligned.
TOD 18th December 2012
11:30 to 12:30
Vikings in shades: navigating by skylight polarization
The hypothesis that Vikings navigated at sea using 'sunstones' to detect the direction of skylight polarization has excited controversy for almost 50 years. It is possible to understand the necessary crystal optics, and light scattering in daylight, with only a minimum of linear algebra. In this talk I will describe the viking navigation hypothesis, and in doing so reveal the deeper and surprising relationship between polarization in birefringent crystals and in the blue sky, based on a topological understanding of the tensors involved in the propagation and scattering of light.
TOD 20th December 2012
11:30 to 12:30
Quadratic invariants for clusters of resonant wave triads
We consider clusters of interconnected resonant triads arising from the Hamiltonian three-wave equation. A cluster consists of N modes forming a total of M connected triads. We investigate the problem of constructing a linearly independent set of quadratic constants of motion. We show that this problem is equivalent to an underlying basic linear problem, consisting of finding the null space of a rectangular M × N matrix A with entries 1, -1 and 0. In particular, we prove that the number of independent quadratic invariants is equal to J = N - M* >= N - M, where M* is the number of linearly independent rows in A. We formulate an algorithm for decomposing large clusters of complicated topology into smaller ones and show how various invariants are related to certain parts and linking types of a cluster, including the basic structures leading to M*
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons