Program

Mini courses

    • Thierry Giamarchi

      • Superconductivity in clean and disordered systems: challenges from the physical world

      • Superconductors are fantastic laboratories for remarkable phenomena.In particular, when put under magnetic field, Abrikosov showed, using a relatively simple equation (the so called Ginzburg-Landau equation) as a starting point, that the magnetic field is not homogeneous in such a system but penetrates as ``vortices'' that form a regular lattice. This discovery made in 1957 (for which he received the Physics Nobel prize in 2003) started a host of activity, not only on a practical sense but also on the theoretical and conceptual side to put the solution on a firm ground and explore its ramifications. This is especially important if one also takes into account the disorder inherent to any realistic material. In that case even understanding some basic points such as whether there is a lattice or not, becomes extremely challenging.

      • I will review this problem both for the clean and disordered case, with a special emphasis in trying to formulate the problem as cleanly as possible and trying to point out the main challenges and the points for which having a rigorous solution would be an invaluable asset.

    • Robert L. Jerrard

      • Binormal curvature flow.

      • The binormal curvature flow is a geometric evolution equation for curves in R^3, thought to describe vortex motion in certain ideal fluids. In particular, it is conjectured to govern dynamics, in certain limits, of vortex filaments in the solutions of the Gross-Pitaevskii equation, a nonlinear Schrodinger equation for which the conserved energy is the Ginzburg-Landau functional. This course will begin with a survey of some basic properties of the binormal curvature flow, after which it will focus on a notion of weak solutions of the binormal curvature flow. These weak solutions posses unexpected stability properties, and they provide the framework for the few rigorous results that are known to hold, linking the binormal curvature flow and the Gross-Pitaevskii equation.

    • Sylvia Serfaty

      • From Ginzburg-Landau to Coulomb gases : vortex lattices and crystallization questions

      • The starting point of the mini-course will be the analysis of vortex patterns in the Ginzburg-Landau model of superconductivity. When a superconducting sample is subjected to a large enough magnetic field, one observes the emergence of perfect triangular lattices of vortices, named Abrikosov lattices. I will describe the analysis developed with Etienne Sandier to try to explain this phenomenon. It also turns out that this is related to systems of particles with Coulomb and logarithmic interactions and with temperature, also called Coulomb gases and log gases, and which are of interest in statistical mechanics and random matrix theory. I will explain how in the limit of low temperature such systems are expected to "crystallize" to the same Abrikosov lattice patterns.

      • This is based mainly on joint works with Etienne Sandier, Nicolas Rougerie, and Thomas Leblé.

    • Didier Smets

      • Vortex motion in inhomogeneous Gross-Pitaevskii equation

      • We will outline the analysis of vortex motion in the Gross-Pitaevskii equation in the case where the background density and/or the metric tensor are inhomogeneous. This has application in particular in the case of perturbation of ground states in 2D BEC by vortices, or for vortex rings in the 3D homogeneous equation where interaction leads to the leapfrogging phenomenon.

Talks

    • Amandine Aftalion

      • Two component Bose Einstein condensates: coexistence, segregation and vortex patterns

      • Two component condensates are described by 2 wave functions minimizing a Gross Pitaevskii type energy. Though the basic coupling is only through the modulus, it can produce effects on the vortex patterns. We will describe the main features in the case of coexistence and segregation of the components, and also in the case of extra spin orbit or Rabi coupling.

      • We will explain how some segregation cases can be analyzed through a Gamma limit leading to a phase separation problem of de Giorgi type, and how we hope to explain the appearance of vortex sheets. In the coexistence cases, we obtain an asymptotic expansion of the energy taking into account the various types of defects. The first term in the expansion is related to the Thomas Fermi limit of the profile and relies on singular perturbation techniques, while the next ones require a more precise analysis of the defects cores described by a vortex/spike problem.

  • Lia Bronsard

    • Structure des vortex pour des supraconducteurs non conventionnel avec symétrie p

    • We study vortices in p-wave superconductors in a Ginzburg-Landau setting. The state of the superconductor is described by a pair of complex wave functions, and the p-wave symmetric energy functional couples these in both the kinetic (gradient) and potential energy terms, giving rise to systems of partial differential equations which are nonlinear and coupled in their second derivative terms. We prove the existence of energy minimizing solutions in bounded domains in the plane, and consider the existence and qualitative properties (such as the asymptotic behavior) of equivariant solutions defined in the whole plane. The coupling of the equations at highest order changes the nature of the solutions, and many of the usual properties of classical Ginzburg-Landau vortices either do not hold for the p-wave solutions or are not immediately evident.

  • Mickaël Dos Santos

    • Minimization of the simplified Ginzburg-Landau energy in a planar domain with prescribed degrees

    • In this talk we focus on the simplified Ginzburg-Landau energy [without magnetic field] defined in a planar domain. We impose degree type conditions on the boundary. We will present some "old" and new results about existence/non existence of minimizers of the simplified Ginzburg-Landau energy with prescribed degrees.

  • Yuxin Ge

    • Generalized Ginzburg-Landau Equations in high dimensions

    • In this talk, we present some results on the critical points to the generalized Ginzburg-Landau equations in dimensions n≥ 3 which satisfy a suitable energy bound, but are not necessarily energy-minimizers. When the parameter in the equations tend to zero, such solutions are shown to converge to singular n-harmonic maps into spheres which are conformally invariant, and the convergence is strong away from a finite set consisting 1) of the infinite energy singularities of the limiting map, and 2) of points where bubbling off of finite energy n-harmonic maps takes place. The latter case is specific to dimensions greater than 2. This is a joint work with E. Sandier et P. Zhang.

  • Rejeb Hadiji

    • Minimization of a Quasi-linear Ginzburg-Landau type energy

    • Mathieu Lewin

      • Superfluidity and Bogoliubov theory: rigorous derivation for mean-field many-body systems

      • One of the most famous problems is to understand the occurence of superfluidity in cold Bose gases, starting from many-body quantum mechanics. A physical explanation was provided by Landau and Bogoliubov in terms of the excitation spectrum, but a mathematical proof of this effect is still lacking. I will review recent advances in this direction for the simpler mean-field model, where the interaction has an intensity of order $1/N$, with $N$ the number of particles in the system.

      • Based on joint works with Phan Thanh Nam, Nicolas Rougerie, Benjamin Schlein, Sylvia Serfaty and Jan Philip Solovej.

  • Evelyne Miot

    • Collisions of vortex filaments

    • We study the dynamics of vortex filaments according to a model introduced by Klein, Majda and Damodaran. More precisely, we focus on the issue of collisions between the filaments. In the cases of counter-rotating pairs of filaments and of an arbitrary number of filaments with polygonial symmetry, we prove the existence of a solution exhibiting a self-similar collision in finite time. This is a joint work with Valeria Banica and Erwan Faou.

    • Nicolas Rougerie

      • The surface superconductivity regime in Ginzburg-Landau theory

      • I will present new results about the ground state of the Ginzburg-Landau functional for type II superconductors in magnetic fields varying between the second and third critical fields. In this regime, superconductivity is a surface phenomenon, restricted to a thin layer along the boundary of the sample. We prove that in this regime, the Ginzburg-Landau energy is to subleading order entirely determined by the minimization of simplified 1D functionals. The leading order of the energy is given by a universal, sample-independent, problem, whereas corrections depend on the curvature of the sample. Refined estimates on the Ginzburg-Landau minimizer follow from these energy estimates. In particular we settle in the affirmative a conjecture of X. B. Pan about the uniform distribution of superconductivity along the boundary. joint work with Michele Correggi.

    • Michael Sigal

      • Stability of Vortex Lattices

      • In this talk I will review some recent results on existence and stability of the magnetic vortex lattice solutions of the Ginzburg - Landau equations of superconductivity (and particle physics). Certain automorphic functions play a key role in the theory described.