Georgia Institute of Technology College of Sciences

The Ginzburg-Landau model is a phenomenological description of superconductivity. A key feature of type-II superconductors is the emergence of singularities, known as vortices, which occur when the external magnetic field exceeds the first critical field. Determining the location and number of these vortices is crucial. Furthermore, the presence of impurities in the material can influence the configuration of these singularities.

In this talk, I will present an estimation of the first critical field for inhomogeneous type-II superconductors and show that the model admits stable local minimizers without vortices, corresponding to Meissner type solutions, even when the external magnetic field intensity significantly exceeds the first critical field, approaching the so-called superheating field. This work is in collaboration with Matías Díaz-Vera.

As is well known, many materials freeze at low temperatures. Microscopically,
  this means that their molecules form a phase where there is long range order
  in their positions. Despite their ubiquity, proving that these freezing
  transitions occur in realistic microscopic models has been a significant
  challenge, and it remains an open problem in continuum models at positive
  temperatures. In this talk, I will focus on lattice particle models, in which
  the positions of particles are discrete, and discuss a general criterion
  under which crystallization can be proved to occur. The class of models that
  the criterion applies to are those in which there is *no sliding*, that is,
  particles are largely locked in place when the density is large. The tool
  used in the proof is Pirogov-Sinai theory and cluster expansions. I will
  present the criterion in its general formulation, and discuss some concrete
  examples. This is joint work with Qidong He and Joel L. Lebowitz.

We will discuss the quantum dynamics associated with ergodic Schroedinger operators. Anderson localization (pure point spectrum with exponentially decaying eigenfunctions) has been obtained for a variety of ergodic operator families, but it is well known that Anderson localization is highly unstable and can also be destroyed by generic rank one perturbations. For quasiperiodic operators, it also sensitively depends on the arithmetic properties of the phase (a seemingly irrelevant parameter from the point of view of the physics of the problem) and doesn’t hold generically. These instabilities are also present for the physically relevant notion of dynamical localization. In this talk, we will discuss the notion of discrepancy and present current and ongoing work establishing novel upper bounds of the discrepancy for skew-shift sequences. As an application of our bounds, we improve the quantum dynamical bounds in Liu [2023] and Jitomirskaya-Powell [2022].

This presentation is dedicated to extending both defocusing and focusing Calogero–Moser–Sutherland derivative nonlinear Schrödinger equations (CMSdNLS), which are introduced in Abanov–Bettelheim–Wiegmann [arXiv:0810.5327], Gérard-Lenzmann [arXiv:2208.04105] and R. Badreddine [arXiv:2303.01087, arXiv:2307.01592], to a system of two matrix-valued variables. This new system is an integrable extension and perturbation of the original CMSdNLS equations. Thanks to the conjugation acting method, I can establish the explicit expression for general solutions on the torus and on the real line in my work [hal-04227081].