The Ginzburg - Landau theory was first developed to explain and predict properties of superconductors, but had a profound influence on physics well beyond its original area. It had the first demonstration of the Higgs mechanism and it became a fundamental part of the standard model in the elementary particle physics. The theory is based on a pair of coupled nonlinear equations for a complex function (called order parameter or Higgs field) and a vector field (magnetic potential or gauge field). They are the simplest representatives of a large family of equations appearing in physics and mathematics. (The latest variant of these equations is the Seiberg - Witten equations.) Geometrically, these are equations for the section of a principal bundle and the connection on this bundle. Besides of importance in physics, they contain beautiful mathematics (some of the mathematics was discovered independently by A. Turing in his explanation of patterns of animal coats). In this talk I will review recent results involving key solutions of these equations - the magnetic vortices and vortex lattices, their existence, stability and dynamics, and how they relate to various theta functions appearing in number theory.

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)

Using a Ginzburg-Landau model, we study the vortex behavior of a rectangular thin film superconductor subjected to an applied current fed into a portion of the sides and an applied magnetic field directed orthogonal to the film. Through a center manifold reduction we develop a rigorous bifurcation theory for the appearance of periodic solutions in certain parameter regimes near the normal state. The leading order dynamics yield in particular a motion law for kinematic vortices moving up and down the center line of the sample. We also present computations that reveal the co-existence and periodic evolution of kinematic and magnetic vortices. This is joint work with Lydia Peres Hari and Jacob Rubinstein.

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)

We investigate the structure of nematic liquid crystal thin films described by the Landau-de Gennes tensor-valued order parameter model with Dirichlet boundary conditions on the sides of nonzero degree. We prove that as the elasticity constant goes to zero in the energy, a limiting uniaxial nematic texture forms with a finite number of defects, all of degree 1/2 or -1/2, corresponding to vertical disclination lines at those locations.

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)

We will complete the results presented in the February workshop. In particular we will prove that the dispersive blow-up property holds for the NLS (both "elliptic" and "nonelliptic") in any dimensions and also for the Davey-Stewartson systems. The talk is based on a joint work with Jerry Bona, Gustavo Ponce and Christof Sparber.

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)

We consider nonlinear equations of Schrödinger type including nonlinear damping terms. This class of equations is purely dispersive but no longer Hamiltonian. We shall prove several results ensuring global existence of solutions on the energy space and also discuss the influence of the damping term on the long time behavior of solutions (and their possible extinction).

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)

We consider a homogenization problem for magnetic GL functional in domains with a large number of small holes. For sufficiently strong magnetic field, a large number of vortices are pinned by the holes. We establish a scaling relation between sizes of holes and the magnitude of the external magnetic field when pinned vortices form a hierarchy of nested subdomains with different multiplicity that manifests a physical phenomenon of vortex phase separation. This is a joint work with V. Rybalko, V. Vinokur and O. Iarioshenko.

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)

Abstract: Smectic C* liquid crystal films are modeled by a relaxed Frank energy, where the elasticity splay and bend constants are positive but may differ. Our film is modeled by a two dimensional vector field on a planar domain where the field has fixed boundary data with degree d>0. We study the limiting pattern for a sequence of minimizers of the energy and prove that the pattern contains d degree one defects and that it has a either a radial or circular asymptotic form near each defect depending on the relative values of the elasticity constants. We further characterize a renormalized energy for the problem and show that it is minimized by the limit. This is joint work with Sean Colbert-Kelly.

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)

Motivated by seeking stationary solutions to the Euler equations with prescribed vortex topology, H.K. Moffatt has described in the 80s a diffusion process, called "magnetic relaxation", for 3D divergence-free vector fields that (formally) preserves the knot structure of their integral lines. (See also the book by V.I. Arnold and B. Khesin.) The magnetic relaxation equation is a highly degenerate parabolic PDE which admits as equilibrium points all stationary solutions of the Euler equations. Combining ideas from P.-L. Lions for the Euler equations and Ambrosio-Gigli-Savar'e for the scalar heat equation, we provide a concept of "dissipative solutions" that enforces first the "weak-strong" uniqueness principle in any space dimensions and, second, the existence of global solutions at least in two space dimensions.

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)

We discuss some old and some not-so-old results on the phase diagrams for quantized vortex states in type-II superconductors. These results are based on both rigorous analysis and numerical simulations of the time-dependent Ginzburg-Landau models near the critical transition temperature. They incorporate the effects of both an external magnetic field and an applied electric current as well as the sample geometry and topology.

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)

The aim of this talk is to present a result concerning the existence of classical solution for the waterbag model with a continuum of waterbag, which can been viewed as an infinite dimensional system of first-order conservation laws. The waterbag model, which constitutes a special class of exact weak solution of the Vlasov equation, is at the cross road of different problems in mathematical physics such as semi-classical approximation in quantum mechanics, long-wave approximation in fluid mechanics, gyrokinetic models and acoustic waves in plasma. The proof of the existence of a continuum of regular waterbag relies on a generalized definition of hyperbolicity for an integrodifferential hyperbolic system of equations, some results in singular integral operators theory and harmonic analysis, Riemann-Hilbert boundary value problem and energy estimates.

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)

We present different methods to compute vortex states of a rotating Bose-Einstein condensate by direct minimization of the Gross-Pitaevskii energy functional. We extensively compare imaginary time integration methods with improved steepest descent methods based on Sobolev gradients and Newton methods. In particular, we show that a careful choice of the gradient could considerably improve convergence properties. A rich variety of vortex arrangements (single-line vortex, Abrikosov lattice, giant vortex) is obtained using different trapping potentials, corresponding to real laboratory experiments performed at ENS Paris in the group of J. Dalibard. Configurations with arrays of condensates in 1D rotating optical lattices are also presented.

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)

The appearance of quantized vortices is regarded as the key signature of superfluidity and superconductivity, and their phenomenological properties have been well captured by the Ginzburg-Landau-Schrodinger (GLSE) equation and the Gross-Pitaevskii equation (GPE). In this talk, we will propose accurate and efficient numerical methods for simulating GLSE and GPE. Then we apply them to study various issues about the quantized vortex phenomena, including vortex dynamics, sound-vortex interaction, radiation, pinning effect and the validity of the reduced dynamical law (RDL) which govern the motion of the vortex centers in GLSE as well as the dynamics and interaction of quantized vortex lattices in GPE with rotational term.

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)

Many scientific, engineering and financial applications require solving high-dimensional PDEs. However, traditional tensor product based algorithms suffer from the so called "curse of dimensionality". We shall construct a new sparse spectral method for high-dimensional problems, and present, in particular, rigorous error estimates as well as efficient numerical algorithms for elliptic equations in both bounded and unbounded domains. As an application, we shall use the proposed sparse spectral method to solve the N-particle electronic Schrodinger equation.

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)

In this talk, we will describe efficient and high-order algorithms for solving the Poisson and fractional Poisson equations in free space in both two and three dimensions. The problem is closely related to the dipolar interactions in Bose-Einstein condensates. The performance of the algorithm is illustrated via several numerical examples.

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)

A convergence analysis of time-splitting pseudo-spectral methods adapted for time-dependent Gross-Pitaevskii equations with additional rotation term is given. For the time integration high-order exponential operator splitting methods are studied, and the space discretization relies on the generalized-Laguerre-Fourier spectral method with respect to the (x,y)-variables as well as the Hermite spectral method in the z-direction. Essential ingredients in the stability and error analysis are a general functional analytic framework of abstract nonlinear evolution equations, fractional power spaces defined by the principal linear part, Sobolev-type inequalities in curved rectangles, and results on the asymptotical distribution of the nodes and weights associated with Gauss-Laguerre quadrature. The obtained global error estimate ensures that the nonstiff convergence order of the time integrator and the spectral accuracy of the spatial discretization are retained, provided that the problem data satisfy suitable regularity requirements. A numerical example confirms the theoretical convergence estimate.

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)

We present a series of experiments in the context of 1D physics with ultracold atoms, combining optical lattice potentials with the capability to tune the strength of the onsite particle interaction U. For an array of tilted 1D chains with site-to-site tilt E and initial unity occupation we record the dynamics after a quench to the paramagnetic-to-anti-ferromagnetic phase transition point U≈E by monitoring the number of doublons created as a function of time after the quench. We observe characteristic oscillations from which we deduce a shift of the resonance condition as time progresses. For U/2≈E and U/3≈E we observe coupling to next-nearest neighbors and beyond. We find evidence of higher-order super-exchange interaction scaling as J^3/U^2.

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)

I will present the new derivation of a new PDE, starting from NLS on the 2-torus, in the limit of small data, and high frequency. As I will explain, this is closely connected to the theory of weak turbulence. Furthermore, the limiting equation has striking properties, which I will describe. This is joint work with Erwan Faou and Zaher Hani.

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)

Dislocations are line defects and the primary carriers of plastic deformation in crystalline materials. Dislocations have the property that the increment of the displacement vector around a dislocation is the Burgers vector, which is similar to the vortices in fluid dynamics or superconductivity. The study of plasticity based on dislocations is very challenging due to the multiscale nature of dislocation modeling: on one hand, the interaction of dislocations is long-range; and on the other hand, there are many short-range interactions that play important roles in the evolution of dislocation microstructures. I will present some of our recent work on modeling and simulation of dislocations at multiple length scales.

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)

Many problems arising from applied sciences can be abstractly formulated as a system that navigates over a complex energy landscape of high or infinite dimensions. Well known examples include nucleation events during phase transitions, conformational changes of bio-molecules, chemical reactions, some extreme events that lead to materials failure, etc. The system spends most of time in metastable states and jumps from one metastable state to another infrequently. In this talk, I will introduce the mathematical theory and computational techniques for modeling rare events.

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)

The main objective of this talk is to drive a unified field model coupling four interactions, based on the principle of interaction dynamics (PID) and the principle of representation invariance (PID). Intuitively, PID takes the variation of the action functional under energy-momentum conservation constraint. PRI requires that physical laws be independent of representations of the gauge groups. One important outcome of this unified field model is a natural duality between the interacting fields $(g, A, W^a, S^k)$, corresponding to graviton, photon, intermediate vector bosons $W^pm$ and $Z$ and gluons, and the adjoint bosonic fields $(Phi_mu, phi^0, phi^a_w, phi^k_s)$. This duality predicts two Higgs particles of similar mass with one due to weak interaction and the other due to strong interaction. The unified field model can be naturally decoupled to study individual interactions, leading to 1) modified Einstein equations, giving rise to a unified theory for dark matter and dark energy, 2) three levels of strong interaction potentials for quark, nucleon/hadron, and atom respectively, and 3) two weak interaction potentials. These potential/force formulas offer a clear mechanism for both quark confinement and asymptotic freedom---a longstanding problem in particle physics. Also, with this unified model, we derive a weakton model of elementary particles, leading to an explanation of all known sub-atomic decays and the creation/annihilation of matter/antimatter particles, as well as the baryon asymmetry problem.This is joint with Tian Ma.

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)

• Event: Workshop on "Quantized vortices in superfluidity and superconductivity and related problems" (2013)