Lévy matrices are symmetric random matrices whose entry distributions have power law tails and infinite variance. They are predicted to exhibit an Anderson-type phase transition separating a region of delocalized eigenvectors from one with localized eigenvectors. We will discuss the context for this conjecture, and describe a result…

The Phi^4, and generally Phi^p measures, which are extensively studied in quantum field theory, also occur naturally as invariant Gibbs measures for certain (dispersive) Hamiltonian PDEs and parabolic SPDEs. A fundamental question is to rigorously justify the invariance of such measures under said dynamics, which leads to deep questions in the…

In recent years, experiments have shown that twisted bilayer graphene and other so-called ``moiré materials'' realize a variety of important strongly-correlated electronic phases, such as superconductivity and fractional quantum anomalous Hall states. I will present a rigorous multiple-scales analysis justifying the (single-particle) Bistritzer…

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 problems in…

Entanglement is a fundamental property of quantum mechanics, and plays an increasing role in our understanding of many-body systems, in and out of equilibrium. In multipartite systems, different forms of entanglement can exist between various sets of particles, and its detection, even theoretical, remains an outstanding challenge. In this talk,…

Bulk-edge correspondence is an important part of the theory of topological insulators which relates topological invariants in the bulk of an insulator to those of its topologically protected edge modes. Despite numerous rigorous proofs, this paradigm has recently been shown to fail in continuum models for topological phases where the…

I will give an overview of a perspective on Polchinski's continuous formulation of the renormalization group, developed over the last few years with T. Bodineau and B. Dagallier, as well as some applications to functional inequalities and sample path regularity of Euclidean field theories.

Time permitting, I will also…

I will discuss some of the present conceptual and theoretical (but not mathematical) understanding of the many-body localized (MBL) phase and its instabilities. In most cases, for the MBL phase to remain stable in the limit of an infinite system this limit needs to be taken differently from the standard thermodynamic limit (Gopalakrishnan…

I will discuss how symmetries of quantum spin systems can be realized. For a given realization of a symmetry group G of a 1d spin system, I will define the anomalous index that takes values in the cohomology H^4(BG) of the classifying space of the group. I will show that a G-invariant system with a non-trivial anomalous index can not have a…

A non-uniform deformation of a honeycomb medium induces effective-magnetic and effective-electric fields. One may choose a deformation which gives rise to a constant perpendicular effective-magnetic field with Landau-level spectrum (flat bands). In the setting of photonic crystals, the tight binding model is generally not applicable. I’ll…

The concept of "quasi-periodic" sets, functions, and measures is

prevalent in many fields including Mathematical Physics,

Fourier Analysis, and Number Theory. The Poisson summation formula provides a “Fourier characterization” for discrete periodic sets, saying that the Fourier transform of the counting measure of a discrete…

We consider a complex Ginibre ensemble of random matrices with a deformation $H=H_0+A$, where $H_0$ is a Gaussian complex Ginibre matrix and $A$ is a rather general deformation matrix. The analysis of such ensemble is motivated by many problems of random matrix theory and its applications. We use the Grassmann integration methods to obtain…

We consider various configuration exponents of Hamiltonian paths drawn on bipartite random planar maps. Estimates from exact enumerations are compared with predictions based on the KPZ relations, as applied to exponents on the regular hexagonal lattice. Surprisingly, a naive use of KPZ does not reproduce all the measured exponents, but an…

While the analysis of mean-field spin glass models has seen tremendous progress in the last twenty years, lattice spin glasses have remained largely intractable. I will talk about recent progress on this topic, giving the first proof of glassy behavior in the Edwards-Anderson model of lattice spin glasses.

We consider two models of random loops where we prove breaking of translational symmetry. The first is a mirror model, where the loops are formed by light rays bouncing in a labyrinth of randomly oriented mirrors. The second is a probabilistic representation of a quantum spin chain, and can be obtained as a limit of the first, for inhomogeneous…

Localization/delocalization transition in random Schrödinger operators cannot in general be seen from the behavior of the corresponding Integrated Density of States (IDS). Here we consider a random Schrödinger operator appearing in the study of certain reinforced random processes in connection with a supersymmetric sigma-model, and show that…

In electrostatic terminology, an electrical field of a stationary point process is a vector field whose distributional divergence is equal to the counting measure of the point process minus the Lebesgue measure. In the talk we will give a simple answer to the following question: when does a planar stationary point process generate a stationary…

Much attention has been given to systems of interacting Bosons in the dilute regime, where powerful theoretical tools such as Bogolyubov theory give detailed and accurate predictions. In this talk, I will discuss a different approach to studying the ground state of Boson systems, which Carlen, Lieb and I have recently found to be accurate at…

The resolvents of finite volume restricted Hamiltonians, GxxΛ(⍵), have long been used to describe the localization of quantum systems. More recently, projected Green's functions (pGfs) -- finite volume restrictions of the resolvent -- have been applied to translation invariant free fermion systems, and the pGf zero eigenvalues have been shown…

A minimal surface in a random environment (MSRE) is a surface which minimizes the sum of its elastic energy and its environment potential energy, subject to prescribed boundary values. Apart from their intrinsic interest, such surfaces are further motivated by connections with disordered spin systems and first-passage percolation models. We…

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