Georgia Institute of Technology College of Sciences

We consider the mean field Bose gas on the unit torus at temperatures proportional to the critical temperature of the Bose—Einstein condensation phase transition. We discuss trace norm convergence of the Gibbs state to a state given by a convex combination of quasi-free states. Two consequences of this relation are precise asymptotic formulas for the two-point function and the distribution of the number of particles in the condensate. A crucial ingredient of the proof is a novel abstract correlation inequality.

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

As highly tunable platforms with exotic rich phase diagrams, moiré materials have captured the hearts and minds of physicists. Moiré materials arise when 2D crystal layers are stacked at relative twists. Their almost periodicity and multiscale behavior make these materials particularly mathematically appealing. We will describe the challenges in establishing a framework to study (phonon/vibrational) wave propagation in these materials, and explain how to overcome them.

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.

Understanding the route to thermalization of a physical system is a fundamental problem in statistical mechanics. When a system is initialized far from thermodynamical equilibrium, many interesting phenomena may arise. Among them, a lot of interest is attained by systems subjected to periodic driving (Floquet systems), which under certain circumstances can undergo a two-stage long dynamics referred to as "prethermalization", showing nontrivial physical features.

The anomaly cancellation is a basic property of the Standard Model, crucial for its consistence. We consider a lattice chiral gauge theory of massless Wilson fermions interacting with a non-compact massiveU(1) field coupled with left- and right-handed fermions in four dimensions. We prove in the infinite volume limit, for weak coupling and inverse lattice step of the order of boson mass, that the anomaly vanishes up to subleading corrections and under the same condition as in the continuum.

The mathematically rigorous derivation of nonlinear Boltzmann equations from first principles in interacting physical systems is an extremely active research area in Analysis, Mathematical Physics, and Applied Mathematics. In classical physical systems, rigorous results of this type have been obtained for some models. In the quantum case on the other hand, the problem has essentially remained open.

The mathematical theory of the recovery of quantum states stored in a quantum memory, is intimately related to the subadditivity property of the entropy function, and the class of states known as quantum Markov chains. In this talk we will introduce some of the basic ideas of this area of quantum information theory. We discuss a theorem regarding recovery of a widely studied class of quantum states, the matrix product states, and its implication for the mutual information stored over separated regions of a one dimensional quantum memory.