Quantised Flux in Tightly Knotted and Linked Systems
Monday 3rd December 2012 to Friday 7th December 2012
09:00 to 10:15  Registration  
10:15 to 10:30  Opening remarks by Deputy Director, Christie Marr, and Keith Moffatt  INI 1  
10:30 to 11:10 
The properties of quantum turbulence: the homogeneous isotropic case Session: Topological evolution in quantum fluids Chair: Keith Moffatt
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.

INI 1  
11:10 to 11:30 
Quantum and classical turbulence: Alike or different? Session: Topological evolution in quantum fluids Chair: Keith Moffatt
The question of how alike and different quantum and classical turbulence are will be addressed using simulations of antiparallel vortex reconnection. The equations solved are: For quantum fluids the GrossPitaevskii equation and for classical turbulence the incompressible NavierStokes equation. For the two cases the initial attraction of the vortices, before the first reconnection, are quite similar. And for both, the final states are composed of a stack of vortex rings from which a 5/3 kinetic energy spectrum appears. However, almost everything in between is different, starting with the differences in the underlying physics of the circulation during reconnection. This presentation will describe these differences.

INI 1  
11:30 to 11:50  Morning Coffee  
11:50 to 12:10 
Thermally and mechanically driven quantum turbulence in helium II Session: Topological evolution in quantum fluids Chair: Keith Moffatt
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.

INI 1  
12:10 to 12:30 
Generating and Classifying Turbulence in BoseEinstein condensates Session: Topological evolution in quantum fluids Chair: Keith Moffatt
Vortices are a hallmark signature of a turbulent flow. Quantum vortices differ from their classical counterparts because of the quantization of circulation in superfluid flow. This means that the rotational motion of a superfluid is constrained to discrete vortices which all have the same core structure. Turbulence in superfluid Helium has been the subject of many recent experimental and theoretical investigations recently reviewed by Skrbek and Sreenivasan [1]. Recently, experimentalists have been able to visualise individual vortex lines and reconnection events using tracer particles[2]. Weakly interacting BoseEinstein condensates present a unique opportunity to resolve the structure of vortices and in turn study the dynamics of a vortex tangle (as has recently been created in an atomic cloud[3]).
We investigate ways of generating turbulence in atomic systems by numerically stirring the condensate using a Gaussian 'spoon' (analogous to a laser beam in the experiments), and study the isotrophy of the resulting vortex tangle depending on when the path the spoon stirs is circular or random. We model the system using the GrossPitaevskii Equation.
[1] L. Skrbek and K.R. Sreenivasan, PoF 24, 011301 (2012) [2] G.P. Bewley et al. PNAS 105, 13707 (2008). [3] E.A.L. Henn et al. PRL 103, 04301 (2009).

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 14:40 
Topological Models for Elementary Particles Session: Topological gauge theories and particle physics Chair: Roman Buniy
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 BilsonThompson and recent work of the speaker with Sundance BilsonThompson 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 halftwisted Mobius band could be like a proton composed of three quarks mutually bound. Sundance BilsonThompson has a theory of framed threebraids 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.

INI 1  
14:40 to 15:00 
The spectrum of tightly knotted flux tubes in QCD Session: Topological gauge theories and particle physics Chair: Roman Buniy 
INI 1  
15:00 to 15:20  Afternoon Tea  
15:20 to 16:00 
Quantized black hole charges and the Freudenthal dual Session: Topological gauge theories and particle physics Chair: Roman Buniy
It is wellknown 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
ePrint: arXiv:0903.5517 [hepth] [2] '' Freudenthal dual invariant Lagrangians'', L. Borsten, M.J. Duff, S. Ferrara ana A, Marrani (to appear). 
INI 1  
16:00 to 16:20 
H Päs ([TU Dortmund]) Knotted strings and leptonic flavor structure Session: Topological gauge theories and particle physics Chair: Mark Hindmarsh
Tight knots and links arising in the infrared limit of string theories may provide an interesting alternative to flavor symmetries for explaining the observed flavor patterns in the leptonic sector. As an example we consider a type I seesaw model where the Majorana mass structure is based on the discrete length spectrum of tight knots and links. It is shown that such a model is able to provide an excellent fit to current neutrino data and that it predicts a normal neutrino mass hierarchy as well as a small mixing angle $\theta_{13}$.

INI 1  
16:20 to 16:40 
Designing fibred knots in optical fields Session: Topological gauge theories and particle physics Chair: Mark Hindmarsh 
INI 1  
17:00 to 18:00 
Rothschild lecture: Superoscillations and weak measurement Session: Topological gauge theories and particle physics Chair: Mark Hindmarsh
Bandlimited functions can oscillate arbitrarily faster than their fastest Fourier component over arbitrarily long intervals. Where such ‘superoscillations’occur, functions are exponentially weak. In typical monochromatic optical fields, substantial fractions of the domain (onethird in two dimensions) are superoscillatory. Superoscillations have implications for signal processing, and raise the possibility of subwavelength resolution microscopy without evanescent waves. In quantum mechanics, superoscillations correspond to weak measurements, suggesting ‘weak values’ of observables (e.g photon momenta) far outside the range represented in the quantum state. A weak measurement of neutrino speed could lead to a superluminal result without violating causality, but the effect is too small to explain the speed recently claimed in a recent (and nowdiscredited) experminent.

INI 1  
18:00 to 18:30  Welcome Wine Reception 
08:50 to 09:30 
Cosmic strings and other flux tubes Session: Topological evolution in gauge theories and cosmology Chair: Tanmay Vachaspati
An introduction to cosmic strings and superstrings including a status report of the observational bounds.

INI 1  
09:30 to 09:50 
Cosmic Strings and our Cosmos? Session: Topological evolution in gauge theories and cosmology Chair: Tanmay Vachaspati
The talk will be about observational constraints on cosmic strings, notably those from searching for evidence of their signatures in the cosmic microwave sky. I will review current constraints on cosmic strings and future prospects from experiments like Planck.

INI 1  
09:50 to 10:30 
Cosmic strings  relativistic vortices in the early universe Session: Topological evolution in gauge theories and cosmology Chair: Tanmay Vachaspati
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 largescale simulations of their formation and decay.

INI 1  
10:30 to 10:50  Morning Coffee  
10:50 to 11:10 
L Pogosian (Simon Fraser University) Scaling configurations of cosmic superstring networks and their cosmological implications Session: Topological evolution in gauge theories and cosmology Chair: Tanmay Vachaspati
Cosmic superstring networks contain strings of multiple tensions and Yjunctions. Depending on the magnitude of the fundamental string coupling, the stressenergy power spectrum of the scaling network is dominated either by populous light F strings, or rare heavy D strings. It may be possible to distinguish between these two regimes with future observations of CMB polarization.

INI 1  
11:10 to 11:30 
Scaling properties of cosmic superstring networks Session: Topological evolution in gauge theories and cosmology Chair: Tanmay Vachaspati
I will use a combination of stateoftheart 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 GotoNambu 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 GotoNambu strings themselves, domain walls and semilocal strings.

INI 1  
11:30 to 11:50 
String networks with junctions: evolution and kinks Session: Topological evolution in gauge theories and cosmology Chair: Tanmay Vachaspati
I will discuss a number of properties of string networks with junctions, focusing particularly on their evolution and the number of kinks in such networks. These properties are crucial in order to determine the observational consequences of networks with junctions, for instance in Gravitational Waves and CMB polarization.

INI 1  
11:50 to 12:10 
Constraints on cosmic superstring formation and their decay mechanism Session: Topological evolution in gauge theories and cosmology Chair: Tanmay Vachaspati
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.

INI 1  
12:10 to 12:30 
Cosmic strings, effective number of neutrinos, gravity waves and the CMB Session: Topological evolution in gauge theories and cosmology Chair: Tanmay Vachaspati
We will report about recent work concerning cosmic defects, gravity waves and the CMB. On one hand, we simulate a scaling network of cosmic defects, and predict the background gravity wave emission coming from them. On the other, we study degeneracies in the CMB data between cosmic strings and gravity waves. When one tries to fit the CMB data with the effective number of neutrinos as a free parameter, the data prefers a higher number effective neutrinos. There have been works suggesting that the extra relativistic signal could come from gravity waves, and those waves could be created by cosmic defects. We perform a full analysis of the system taking into account the signal that such a network would create, its effect into the effective number of neutrinos, and how those parameters interplay in trying to fit different probes of CMB and other cosmological data.

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
13:30 to 14:10 
JW Fleischer (Princeton University) Superfluid and BerezinskiiKosterlitzThouless dynamics of light Session: Topology of condensed matter and quantum gas systems Chair: Renzo Ricca
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 largestscale mode of the system to become macroscopically occupied. In the latter process, vortex generation becomes more favorable than entropy production, and attempts at longrange order are destroyed. These two processes compete yet can coexist, with many aspects of their manybody 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.

INI 1  
14:10 to 14:30 
N Berloff (University of Cambridge) Vortices in nonequilibrium condensates Session: Topology of condensed matter and quantum gas systems Chair: Renzo Ricca 
INI 1  
14:30 to 14:50 
Vorton solutions in 2D and 3D Session: Topology of condensed matter and quantum gas systems Chair: Renzo Ricca
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 semianalytic 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.

INI 1  
14:50 to 15:10 
LogAnalytic Uncertainty Relation Session: Topology of condensed matter and quantum gas systems Chair: Renzo Ricca
I address the question of what characteristics of the quantum wave function phase can be measured. In particular I am interested in those phase aspects that can be deduced from the amplitude of the wave function. This will be shown to be connected to the topological characteristics of the Logarithm of the wave function in the complex time domain through a relation which is the temporal equivalent to the Kramers  Kronig formulae in the frequency domain. In particular for a wave packet which is reflected from an infinite potential barrier certain characteristics of the phase can be deduced provided that the momentum of the particle time the distance from the barrier is smaller than Planck constant over two and cannot be deduced otherwise. That is the phase characteristics cannot be deduced if the momentum of the particle time the distance from the barrier is bigger than Planck's constant over two which creates an experimental uncertainty in the phase.

INI 1  
15:10 to 15:30  Afternoon Tea  
15:30 to 15:50 
MS Volkov ([University of Tours]) Vortons and their stability Session: Topology of condensed matter and quantum gas systems Chair: Renzo Ricca
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 nonlinear Schroedinger equation and/or GinzburgLandau 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 nonlinear levels.

INI 1  
15:50 to 16:10 
Investigating Pure Quantum Turbulence in Superfluid 3He and a Means of Directly Inferring the Turbulent Energy Content Session: Topology of condensed matter and quantum gas systems Chair: Renzo Ricca
The lack of a general solution to the governing NavierStokes equations means that there is no fundamental theory of turbulence. Simpler pure quantum turbulence, the tangle of identical singlyquantized vortices in superfluids at T~0 may provide a deeper understanding of turbulence in general. In the present context this is especially relevant to the measurement of the turbulent energy. While the wellknown Kolmogorov theory predicts the energy distribution of turbulence and how it decays, in normal systems the turbulent energy is generally only a small perturbation on the total thermal energy of the supporting medium. In quantum turbulence, however, the turbulent energy is accessible. A stationary condensate is necessarily in its ground state with zero enthalpy. Thus any added quantum turbulence accounts for the entire free energy of the superfluid and there are no other contributions.
In superfluid 3He we can generated and detect quantum turbulence and how it evolves with time, and using bolometric methods, we can measure the energy released as previouslygenerated turbulence decays, which seems to be unique to these systems.

INI 1  
16:10 to 16:50 
Wave/Particle Duality via Silicon Droplets? Session: Topology of condensed matter and quantum gas systems Chair: Renzo Ricca 
INI 1  
16:50 to 17:10 
Topology of Quantum Liquids Session: Topology of condensed matter and quantum gas systems Chair: Renzo Ricca 
INI 1 
09:00 to 09:40 
Analytic and topological aspects of Menger curvatures for curves and submanifolds Session: Knots in mathematics: Knot energies Chair: Jason Cantarella
We discuss various types of geometric curvature energies based on the concept of Menger curvature. These energies exhibit selfavoidance and regularizing effects on curves and submanifolds, and they control their topology.

INI 1  
09:40 to 10:00 
M Mastin (University of Georgia) Symmetric Criticality for Ropelength Session: Knots in mathematics: Knot energies Chair: Jason Cantarella
The ropelength of a link embedded in $R^3$ is the ratio of the curve's length to its thickness. Jason Cantarella, Joe Fu, Rob Kusner, and John Sullivan have developed a theory of first order criticality for ropelength. We will discuss an extension of this work for the case of link conformations with rigid rotational symmetry. As an application we will prove that there is an infinite class of knots for which there are geometrically distinct ropelength critical conformations. This work is joint with Jason Cantarella, Jennifer Ellis, and Joe Fu.

INI 1  
10:00 to 10:20 
Regularity theory for knot energies Session: Knots in mathematics: Knot energies Chair: Jason Cantarella
In the past two decades, the introduction of several knotbased 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 selfavoiding functionals, socalled knot energies, that blow up on embedded curves converging to a curve with a selfintersection.
Due to the singularities which guarantee the selfrepulsion 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 EulerLagrange equation and derive higher regularity via a bootstrapping process.

INI 1  
10:20 to 10:40 
Critical Links and Unlinks Session: Knots in mathematics: Knot energies Chair: Jason Cantarella
The configuration spaces Hopf_k or McCool_k of kcomponent Hopflinks or unlinks can be understood using extensions of Hatcher's proof of the Smale Conjecture. We will describe lowenergy critical configurations for Möbius energy or Ropelength on Hopf_k or McCool_k, and sketch a picture of the bottom of the MorseSmale complex for each.
[This represents part of an ongoing project with Ryan Budney and John M. Sullivan]

INI 1  
10:40 to 11:00  Morning Coffee  
11:00 to 11:40 
Renormalized potential energies and their asymptotics Session: Knots in mathematics: Knot energies Chair: Jason Cantarella
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. 
INI 1  
11:40 to 12:00 
Higher order topological invariants from the ChernSimons action Session: Knots in mathematics: Knot energies Chair: Jason Cantarella 
INI 1  
12:00 to 12:20 
Helical Organization of Tropical Cyclones Session: Knots in mathematics: Knot energies Chair: Jason Cantarella
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 largescale helical vortex based on diagnosis of a set of integral helical and energetic characteristics. Using the data from a near cloudresolving 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 systemscale 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 systemscale 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. 
INI 1  
12:20 to 12:35 
Vortex knots in a BoseEinstein condensate Session: Knots in mathematics: Knot energies Chair: Jason Cantarella
I will present a method for numerically building a quantum vortex knot state in the single scalar field wave function of a BoseEinstein condensate. I will show how the two topologically simplest vortex knots wrapped over a torus evolve and may preserve their shapes by reporting results of the integration in time of the governing GrossPitaevskii equation.
In particular, I will focus on how the velocity of a vortex knot depends on the ratio of poloidal and toroidal radius: in a first approximation it is linear and, for smaller ratio, the knot travels faster. Finally, I will display mechanisms of vortex breaking by reconnections which produce simpler vortex rings whose number depends on initial knot topology.

INI 1  
12:35 to 13:30  Lunch at Wolfson Court  
13:30 to 13:50 
Solitons and Breathers on Quantized Superfluid Vortices Session: Knots in mathematics: Knot energies Chair: Clayton Shonkwiler
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 GrossPitaevskii 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 crossover regime of scales in superfluid turbulence.

INI 1  
13:50 to 14:10 
Branes, strings and boojums; topological defects in helium3 and the cosmos Session: Knots in mathematics: Knot energies Chair: Clayton Shonkwiler
The order parameter of the superfluid helium3 condensate exhibits broken symmetries that show analogs with those broken in the various transitions undergone by the Universe after the Big Bang. Fortunately for us, the helium3 order parameter is also sufficiently complex that the superfluid may exist in several phases, the two most stable being the A and B phases. At Lancaster we have developed techniques to investigate the properties of the interface between the A and B phases in the pure condensate limit, far below the superfluid transition temperature. The order parameter transforms continuously across the A–B boundary, making this interface the most coherent twodimensional structure to which we have experimental access. It has been argued that this ordered 2d surface in a 3d bulk matrix, separating the two phases, can provide a good analog of a cosmological brane separating two distinct quantum vacuum states. In superfluid helium3 the creation of such 2branes mu st lead to the formation of point and line defects in the texture of the 3d bulk, simply as a result of the constraints imposed by the interplay of the order parameter symmetries and the geometry of the container. Furthermore, our experiments have shown that removing the 2branes from the bulk, in a process analogous to brane annihilation, creates new line defects in large quantities. Such observations may provide insight into the formation of topological defects such as cosmic strings arising from brane interactions in the early Universe. Up to now our experimental techniques have only allowed us to infer the properties of the interface and defects by measuring how they impede the transport of quasiparticle excitations in the superfluid, which is essentially a remote measurement. Our new experiments allow us to directly probe the interface region.

INI 1  
14:10 to 14:30 
Dynamics of Hopfions Session: Knots in mathematics: Knot energies Chair: Clayton Shonkwiler
Several materials, such as ferromagnets, spinor BoseEinstein condensates and some topological insulators, are now believed to support knotted structures. One of the most successful basemodels having stable knots is the FaddeevSkyrme 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.

INI 1  
14:30 to 17:00  Free Afternoon  
19:30 to 22:00  Conference Dinner at Trinity Hall 
09:00 to 09:20 
J Parsley (Wake Forest University) Cohomology reveals when helicity is a diffeomorphism invariant Session: Knots in mathematics: Tight Knots, etc. Chair: Rob Kusner
We consider the helicity of a vector field, which calculates the average linking number of the field’s flowlines. Helicity is invariant under certain diffeomorphisms of its domain – we seek to understand which ones.
Extending to differential (k+1)forms on domains R^{2k+1}, we express helicity as a cohomology class. This topological approach allows us to find a general formula for how much helicity changes when the form is pushed forward by a diffeomorphism of the domain. We classify the helicitypreserving diffeomorphisms on a given domain, finding new ones on the twoholed solid torus and proving that there are no new ones on the standard solid torus. This approach also leads us to define submanifold helicities: differential (k+1)forms on ndimensional subdomains of R^m.

INI 1  
09:20 to 09:40 
C Shonkwiler (University of Georgia) The geometry of random polygons Session: Knots in mathematics: Tight Knots, etc. Chair: Rob Kusner
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, fixedlength ngons 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 fixedlength ngons 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 2frames in complex nspace 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 ngon sampled according to this measure is exactly 1/(2n). I will also demonstrate a simple, lineartime 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).

INI 1  
09:40 to 10:00 
The Expected Total Curvature of Random Polygons Session: Knots in mathematics: Tight Knots, etc. Chair: Rob Kusner
We consider the expected value for the total curvature of a random closed polygon. Numerical experiments have suggested that as the number of edges becomes large, the difference between the expected total curvature of a random closed polygon and a random open polygon with the same number of turning angles approaches a positive constant. We show that this is true for a natural class of probability measures on polygons, and give a formula for the constant in terms of the moments of the edgelength distribution.
We then consider the symmetric measure on closed polygons of fixed total length constructed by Cantarella, Deguchi, and Shonkwiler. For this measure, the expected total curvature of a closed ngon is asymptotic to n pi/2 + pi/4 by our first result. With a more careful analysis, we are able to prove that the exact expected value of total curvature is n pi/2 + (2n/2n3) pi/4. As a consequence, we show that at least 1/3 of fixedlength hexagons and 1/11 of fixedlength heptagons in 3space are unknotted.

INI 1  
10:00 to 10:20 
On the energy spectrum of magnetic knots and links Session: Knots in mathematics: Tight Knots, etc. Chair: Rob Kusner
The groundstate energy spectrum of the first 250 zeroframed prime knots and links is studied by using an exact analytical expression derived by the constrained relaxation of standard magnetic flux tubes in ideal magnetohydrodynamics (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, 5790.

INI 1  
10:20 to 10:40 
Criticality theory for ropelength and related problems Session: Knots in mathematics: Tight Knots, etc. Chair: Rob Kusner
We consider certain new results related to the criticality theory for ropelength developed with Cantarella, Fu and Kusner.

INI 1  
10:40 to 11:10  Morning Coffee  
11:10 to 11:30 
Topological and Geometric Properties of tightly confined random polygons Session: Knots in mathematics: Tight Knots, etc. Chair: Rob Kusner
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.

INI 1  
11:30 to 11:45 
Equilibrium configurations of elastic torus knots (n,2) Session: Knots in mathematics: Tight Knots, etc. Chair: Rob Kusner
We study equilibria of braided structures made of two elastic rods with their centrelines remaining at constant distance from each other. The model is geometrically exact for large deformations. Each of the rods is modelled as thin, uniform, homogeneous, isotropic, inextensible, unshearable, intrinsically twisted, and to have circular crosssection. The governing equations are obtained by applying Hamilton's principle to the action which is a sum of the elastic strain energies and the constraints related to the inextensibility of the rods. Hamilton's principle is equivalent to the secondorder variational problem for the action expressed in reduced strainlike variables. The EulerLagrange equations are derived partly in EulerPoincare form and are a set of ODEs suitable for numerical solution.
We model torus knots (n,2) as closed configurations of the 2strand braid. We compute numerical solutions of this boundary value problem using path following. Closed 2braids buckle under increasing twist. We present a bifurcation diagram in the twistforce plane for torus knots (n,2). Each knot has a Vshaped nonbuckled branch with its vertex on the twist axis. There is a series of bifurcation points of buckling modes on both sides of each of the Vbranches. The 1st mode bifurcation points for n and n±4 are connected by transition curves that go through (unphysical) selfcrossing of the braid. Thus, all the knots turn out to be divided into two classes: one of them may be produced from the righthanded trefoil and the other from the lefthanded. Highermode postbuckled configurations lead to cable knots.
It is instructive to see how close our elastic knots can be tightened to the ideal shape. For the trefoil knot the tightest shape we could get has a ropelength of 32.85560666, which is remarkably close to the best current estimate. Careful examination reveals that the solution is free from selfintersections though the contact set remains a distorted circle.

INI 1  
11:45 to 12:05 
S Blatt (University of Warwick) The gradient flow of O'Hara's knot energies Session: Knots in mathematics: Tight Knots, etc. Chair: Rob Kusner
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 19911994.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.

INI 1  
12:05 to 12:25 
S Tanda (Hokkaido University) Topological Crystals and the quantum effects Session: Knots in mathematics: Tight Knots, etc. Chair: Rob Kusner
We report the discovery of Mobius, Ring, Figure8, Hopflink Crystals in NbSe3, conventionally grown as ribbons and whiskers.
We also reveal their formation mechanisms of which two crucial components are the spherical selenium (Se) droplet, which a NbSe3 ber wraps around due to surface tension, and the monoclinic (P2(1)/m) crystal symmetry inherent in NbSe3, which induces a twist in the strip when bent. Our crystals provide a nonfictious topological Mobius world governed by a nontrivial realspace topology. We classified these topological crystals as an intermediary between condensed matter physics and mathematics. Moreover, we observed AharonovBohm effect of chargedensity wave and Frolich type superconductor as electronic properties using topological crystals.

INI 1  
12:25 to 13:30  Lunch at Wolfson Court  
14:00 to 14:20 
Quantum vortex reconnections Session: Quantized flux tubes and vortices Chair: Robert Kerr 
INI 1  
14:20 to 14:40 
Knots in light and fluids Session: Quantized flux tubes and vortices Chair: Robert Kerr
To tie a shoelace into a knot is a relatively simple affair. Tying a knot in a field is a different story, because the whole of space must be filled in a way that matches the knot being tied at the core. The possibility of such localized knottedness in a spacefilling field has fascinated physicists and mathematicians ever since Kelvin’s 'vortex atom' hypothesis, in which the atoms of the periodic table were hypothesized to correspond to closed vortex loops of different knot types. An intriguing physical manifestation of the interplay between knots and fields is the possibility of having knotted dynamical excitations. I will discuss some remarkably intricate and stable topological structures that can exist in light fields whose evolution is governed entirely by the geometric structure of the field. A special solution based on a structure known as a Robinson Congruence that was rediscovered in different contexts will serve as a basis for the discussion. I will th en turn to hydrodynamics and discuss topologically nontrivial vortex configurations in fluids. My lab's website can be found at http://irvinelab.uchicago.edu 
INI 1  
14:40 to 14:55 
N Proukakis (Newcastle University) Vortex Dynamics and Turbulence in Confined Quantum Gases Session: Quantized flux tubes and vortices Chair: Robert Kerr
Quantised vortices are known to arise in ultralow 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 twodimensional regime. As realistic cold atom experiments are conducted at nonzero temperatures, where the condensate coexists 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.

INI 1  
14:55 to 15:10 
Interpretation of quasiparticle scattering measurements in $^3$HeB: a three dimensional numerical analysis Session: Quantized flux tubes and vortices Chair: Robert Kerr
Present research is concerned with numerical modelling of Andreev scattering technique used for detection of quantized vortices in $^3$HeB. The results of numerical analysis of Andreev reflection by threedimensional turbulent structures are reported. We analyse the Andreev scattering by a dense vortex tangle and calculate the spectral characteristics of the retroreflected beam of thermal excitations. The obtained results are in agreement with experimental observations.

INI 1  
15:10 to 15:30  Afternoon Tea  
15:30 to 15:45 
Macroscopic bundles of vortex rings in superfluid helium Session: Quantized flux tubes and vortices Chair: Robert Kerr
It is well known that two coaxial vortex rings can leapfrog 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 nonzero temperatures, and show how in this case the presence of normal fluid rotation is necessary for the stability of the bundle.

INI 1  
15:45 to 16:05 
Knots and links of disclination lines in chiral nematic colloids Session: Quantized flux tubes and vortices Chair: Robert Kerr
Nematic braids formed by disclination lines entangling colloidal particles in nematic liquid crystal are geometrically stabilized and restricted by topology. Experiments with nematic braids show rich variety of knotted and linked disclinations loops that can be manipulated and rewired by laser light. We describe a simple rewiring formalism and demonstrate how the selflinking number of nematic ribbons enables a classification of entangled structures. Controlled formation of arbitrary microscopic links and knots in nematic colloids provides a new route to the fabrication of soft matter with special topological features.
References:
[1] M. Ravnik, M. Škarabot, S. Žumer, U. Tkalec, I. Poberaj, D. Babič, N. Osterman, I. Muševič, Phys. Rev. Lett. 99, 247801 (2007). [2] U. Tkalec, M. Ravnik, S. Žumer, I. Muševič, Phys. Rev. Lett. 103, 127801 (2009). [3] S. Čopar, S. Žumer, Phys. Rev. Lett. 106, 177801 (2011). [4] U. Tkalec, M. Ravnik, S. Čopar, S. Žumer, I. Muševič, Science 333, 62 (2011). [5] S. Čopar, T. Porenta, S. Žumer, Phys. Rev. E 84, 051702 (2011). [6] G. P. Alexander, B. G. Chen, E. A. Matsumoto, R. D. Kamien, Rev. Mod. Phys. 84, 497 (2012).

INI 1  
16:05 to 16:25 
Experiments with tangles of quantized vortex lines in superfluid 4He in the T=0 limit Session: Quantized flux tubes and vortices Chair: Robert Kerr
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 largescale 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 largescale polarization), homogeneous quasiclassical turbulence (tangles in which the dominant energy is concentrated in largescale bundleseddies), 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.

INI 1  
16:25 to 16:45 
J Bohr ([DTU]) Torus Knots and Links, Twist Neutrality and Biological Applications Session: Quantized flux tubes and vortices Chair: Robert Kerr
We present mathematical restrictions for torus knots and links, and for bent helices. The
concept of twist neutrality is developed for bent and coiled structures and biological
applications are reviewed [1,2].
[1] K. Olsen and J. Bohr, Geometry of the toroidal Nhelix: optimalpacking and zerotwist.
New Journal of Physics 14, 023063 (2012).
[2] J. Bohr and K. Olsen, Twist neutrality and the diameter of the nucleosome core particle.
Phys. Rev. Lett. 108, 098101 (2012).

INI 1 
09:00 to 09:40 
Magnetic Fields in the Early Universe  Chiral Effects and Topology Session: Linking in magnetic fields and other physical systems Chair: Levon Pogosian
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 byproduct, the discussion will hint at processes that might lead to the production of magnetic monopoles.

INI 1  
09:40 to 10:20 
P Akhmet'ev ([IZMIRAN]) Asymptotic higher ergodic invariants of magnetic lines Session: Linking in magnetic fields and other physical systems Chair: Levon Pogosian
V.I.Arnol'd in 1984 formulated the following problem: "To transform asymptotic ergodic definition of Hopf invariant of a divergencefree vector field to Novikov's theory, which generalizes Withehead product in homotopy groups"'.
We shall call divergencefree fields by magnetic fields. Asymptotic invariants of magnetic fields, in particular, the theorem by V.I.Arnol'd about asymptotic Gaussian linking number, is a bridge, which relates differential equitations and topology. We consider 3D case, the most important for applications.
Asymptotic invariants are derived from a finitetype invariant of links, which has to be satisfied corresponding limit relations. Ergodicity of such an invariant means that this invariant is welldefined as the mean value of an integrable function, which is defined on the finitetype configuration space $K$, associated with magnetic lines.
At the previous step of the construction we introduce a simplest infinite family of invariants: asymptotic linking coefficients. The definition of the invariants is simple: the helicity density is a welldefined function on the space $K$, the coefficients are welldefined as the corresponding integral momentum of this function. Using this general construction, a higher asymptotic ergodic invariant is welldefined. Assuming the the magnetic field is represented by a $\delta$support with contains 3 closed magnetic lines equipped with unite magnetic flows, this higher invariant is equal to the corresponding Vassiliev's invariant of classical links of the order 7, and this invariant is not a function of the pairwise linking numbers of components. When the length of generic magnetic lines tends to $\infty$, the asymptotic of the invariant is equal to 12, this is less then twice order $14$ of the invariant.
Preliminary results arXiv:1105.5876 was presented at the Conference "`Entanglement and Linking"' (Pisa) 1819 May (2011).

INI 1  
10:20 to 10:40  Morning Coffee  
10:40 to 11:20 
Topological Solitons from Geometry Session: Linking in magnetic fields and other physical systems Chair: Levon Pogosian
Solitons are localised nonsingular 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 spacetime 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.

INI 1  
11:20 to 11:40 
M Nitta (Keio University) Creating Vortons and Knot Solitons via Domain Wall Pair Annihilation in BEC and Field Theory Session: Topological Solitons Chair: Konrad Bajer
We show that when a vortexstring is stretched between a pair of a domain wall and an antidomain wall in two component BoseEinstein condensates, there remains a vorton after the pair annihilation. We also show that the same configuration in the mass deformed FaddeevSkyrme model results in a knot soliton (Hopfion) after the pair annihilation.

INI 1  
11:40 to 12:00 
T Tchrakian ([DIAS]) Abelian and nonAbelian Hopfions in all odd dimensions Session: Topological Solitons Chair: Konrad Bajer
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 ChernPontryagin (CP) densities or their descendants, are total divergence, the corresponding quantities for Hopfions are the ChernSimons (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 SkyrmeFadde'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 nonAbelian theories, namely empolying projectivecomplex and Grassmannian models, respectively. It is in this sense that we have used the terminology of Abelian and nonAbelian 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.

INI 1  
12:00 to 12:20 
E Babaev ([UMass Amherst and KTH Stockholm]) Skyrmions and Hopfions in exotic superconductors Session: Topological Solitons Chair: Konrad Bajer 
INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
13:30 to 13:50 
D Harland (Loughborough University) Modelling Hopf solitons with elastic rods Session: Topological Solitons continued Chair: Tom Kephart
I will review recent progress in modelling knotted solitons in the SkyrmeFaddeev 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.

INI 1  
13:50 to 14:10 
Fermionic quantization of knot solitons Session: Topological Solitons continued Chair: Tom Kephart
Knot solitons arise as global energy minimizers in field theories such as the FaddeevSkyrme model. Such field theories, when quantized, are inherently bosonic because the fundamental fields represent scalar bosons. Nonetheless, the solitons they support can be given fermionic exchange statistics, provided the classical field configuration space has the right algebraic topology. In this talk I will review a computation of the fundamental group of the FaddeevSkyrme configuration space, and show how this allows a consistent fermionic quantization of knot solitons. This is based on (separate) collaborations with Dave Auckly and Steffen Krusch.

INI 1  
14:10 to 14:30  Afternoon Tea  
14:30 to 14:50 
Folding and collapse in stringlike structures Session: Topological Solitons continued Chair: Tom Kephart
We argue that the the physics of folding and collapse of stringlike 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 picometers rootmeansquare distance form the experimentally constructed structure. Our result proposes that there are at least some 10^20 topological solitons in each human body.

INI 1  
14:50 to 15:10 
S Krusch (University of Kent) Fermions coupled to Hopf Solitons Session: Topological Solitons continued Chair: Tom Kephart
Solitons in the Skyrme and the FaddeevSkyrme model share many similarities. While no topologically nontrivial exact solutions are known in flat space there is a minimal energy charge one soliton on the 3sphere of sufficiently small radius in both models. The charge one Skyrmion is given by the identity map, whereas the charge one Hopf soliton is given by the Hopf map. Also, the solitons in both models can be semiclassically quantized as fermions, by defining the wave function of the covering space of configuration space and imposing FinkelsteinRubinstein constraints. When fermions are chirally coupled to Skyrmions the resulting Dirac equation can be solved explicitly on the 3sphere in the background of a charge one Skyrmion. In this talk, I describe how to couple fermions to Hopf solitons.

INI 1  
15:10 to 15:25 
J Garaud ([UMass Amherst and KTH Stockholm]) Topological solitons in multicomponent superconductors : from babySkyrmions to vortex loops Session: Topological Solitons continued Chair: Tom Kephart
The crucial importance of topological excitations in the physics of superconductivity made GinzburgLandau vortices one of the most studied example of topological defects. Multiband/multicomponent superconductors extends usual GinzburgLandau theory by considering more than one scalar field (several superconducting order parameters).
Family materials, where superconductivity is multiband/multicomponent, has recently been growing. Because of additional fields and new broken symmetries, the zoo of topological defects is much richer in multicomponent systems (e.g. Linelike 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 multicomponent/multiband superconductors.

INI 1  
15:25 to 15:40  Closing Remarks  INI 1 