Seminars (TOD)

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.

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.

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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.

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."

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.

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

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.

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.

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.

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.

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.

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.

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

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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).

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.

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.

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.

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].

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.)

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.

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.

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.

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.

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).

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.

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.

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.

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.

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.

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).

An introduction to cosmic strings and superstrings including a status report of the observational bounds.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.