Schedule 

In this talk, I will prove Anderson localization for potentials generated by hyperbolic transformations via positivity and large deviations of the Lypapunov exponent. This is a joint work with Artur Avila and David Damanik.

We extend methods of Ding and Smart from their breakthrough paper in 2020 which showed Anderson localization for certain random Schrödinger operators on $\ell^2(\mathbb{Z}^2)$ via a quantitative unique continuation principle and Wegner estimate. We replace the requirement of identical distribution with the requirement of a uniform bound on the essential range of potential and a uniform positive lower bound on the variance of the variables giving the potential. Under those assumptions, we recover the unique continuation and Wegner lemma results, using Bernoulli decompositions and modifications of the arguments therein. This leads to a localization result at the bottom of the spectrum.

BREAK

We shall 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].

As a classical object, periodic Jacobi operators still attracts attention as its connection with quasi-periodic counterparts and other models from mathematical physics. In this talk after reviewing estimates on the sizes of the spectral bands and gaps of periodic Jacobi matrices, I will discuss some recent improvements and how logarithmic potential theory helps us to get them.

 LUNCH BREAK

The last decade has witnessed tremendous experimental progress in the study of "Floquet media", crystalline materials whose properties are altered by time-periodic parametric forcing. Understanding these materials from their underlying, first-principle PDE models, however, remains an open problem. A key question is whether Floquet Edge Modes - waves that propagate only along the edge of the material - exist in PDE models.

We address this problem by considering a single electron in a 1D material with a domain wall. We first derive a homogenized (low-energy) Dirac equation for this setup. In the context of the Driven Dirac model, we find that edge modes are only meta-stable, decaying due to a resonance phenomenon known as the Fermi Golden Rule. Key to our analysis are the dispersive decay estimates of the Dirac model, which are non-standard due to the presence of the domain wall.  

Topological insulators are central objects in condensed matter physics. They refer to insulating phases of matter (i.e. the evolution is described by a Hamiltonian with a spectral gap) to which one can associate a non-trivial topological invariant. When two insulators with distinct topological invariants are glued together, protected gapless currents emerge along the interface: the material becomes a conductor along its edge. Furthermore, the edge conductance is quantized and it equals the difference of the bulk topological invariants for straight interfaces. This fundamental result is called the bulk-edge correspondence. In this talk, we discuss bulk-edge correspondence for topological insulators with curved interface. This is based on a joint work with Alexis Drouot.

BREAK

I will present a framework for studying randomized open quantum dynamics by means of the repeated interactions formalism. I will present some basic notions and ergodic theorems. This may serve as a precursor to my AMS special session talk.

We study the spectral ζ -function associated with a Schrödinger equation endowed with a Pöschl--Teller potential. In particular, we show that for particular self-adjoint extensions, the spectral ζ -function possesses a very unusual and remarkable structure consisting of a series of branch points located at every nonpositive integer value of s. By comparing the Pöschl--Teller potential to the classic Bessel potential, we further illustrate that perturbing a given potential by even a smooth potential on a finite interval can greatly affect the meromorphic structure and branch points of the spectral $\zeta$-function in surprising ways.  Based on joint work with Guglielmo Fucci and Mateusz Piorkowski.

We first present the global setup of the semiclassical Sturm-Lioville.

The deterministic inviscid primitive equations (also called the hydrostatic Euler equations) are known to be ill-posed in Sobolev spaces and in Gevrey classes of order strictly greater than 1, and some of their analytic solutions exist only locally in time and exhibit finite-time blowup. In this talk, I will discuss that suitable random noise can restore the local well-posedness and prevent finite-time blowups. Specifically, random diffusion addresses the ill-posedness in certain Gevrey classes, allowing us to establish the local well-posedness almost surely and the global existence of solutions with high probability. In the case of random damping (linear multiplicative noise), the noise prevents analytic solutions from forming singularities in finite time, resulting in globally existing solutions with high probability.

Understanding fluids with small viscosity is one of the most fundamental problems in mathematical fluid dynamics. This issue remains unresolved in general for domains with curved boundaries due to the presence of boundary layers and large vorticity in the inviscid limit. We present a framework that precisely captures the pointwise behavior of the vorticity for the Navier-Stokes equations on domains with boundaries, under no-slip boundary conditions. Through an in-depth study of the linear problem with a nonlocal boundary condition for vorticity on the half-space, we demonstrate that the inviscid limit holds for the fully nonlinear Navier-Stokes equations if the initial data is locally analytic near the boundary, whether on a general bounded domain or the exterior of a disk.

Graphene is an exciting new two-dimensional material. Though it was considered theoretical for a long time, it was isolated about 20 years ago. Since then, it has drawn much attention due to its numerous exciting properties. More recently, it was discovered that when twisting two layers of graphene with respect to each other, at certain angles called ``magic angles", exotic transport properties emerge. The work of Bistritzer and MacDonald first predicted this, and they derived an effective approximate model for this phenomenon. 

 In this work, we take the first steps towards establishing these results without relying on the Bistritzer-MacDonald approximate model. We study twisted bilayer graphene in AA stacking when the twisting angle is commensurate, where the resulting potential is still periodic, and will focus on the continuum setting. In this setting, we show the existence of Dirac cones under some technical conditions, in this new setting, as well as give a full description of the commensurate angles.  

 In this talk, I will introduce the main phenomena of twisted bilayer graphene, state our main results in detail, and finally sketch the proof. 

BREAK

We study Floquet isospectrality of the zero potential for the discrete periodic Schr¨odinger operator acting on functions on the n-dimensional square lattice. The classical Ambarzumyan Theorem states for the square lattice, the zero potential has no trivial real Floquet isospectral potentials. In this talk we will provide explicit complex potentials Floquet isospectral to the zero potential using combinatorial methods.

We apply the Gordon lemma to investigate the spectral type of long range operators over circle homeomorphisms. Our result shows that when the potential term is a measurable function over $C^{1+BV}$-circle diffeomorphism, the operators have no point spectrum.

LUNCH + DRIVING TO SAN ANTONIO