Seminars Sorted by Series

We present a new approach to some of the universality limits that arise in orthogonal polynomials. For example, we show that if w is a fixed weight positive a.e. in [-1,1], and w satisfies a Dini condition in some subinterval of (-1,1), then there...

An outstanding problem in "Quantum Chaos" is to understand the complexity of highly excited quantum states, and in particular of eigenfunctions of the Laplacian on a compact manifold. One attribute that can be studied is complexity of the nodal set...

We want to describe the eigenmodes of the Laplace-Beltrami operator on a compact Riemannian manifold of negative curvature, in the high-energy limit. In this aim, we study the semiclassical measures (a.k.a. quantum limits) associated with (infinite)...

In this talk I will go over a new approach to the Beurling-Selberg extremal problem for even functions based on the solution for the Gaussian and tempered distribution arguments. This provides interesting applications to analytic number theory...

Loop-erased random walk (LERW) is a random self-avoiding curve obtained by erasing the loops of a random walk according to chronological order. Studying LERW on the two-dimensional integer lattice, Schramm introduced a model of one-parameter planar...

We consider a system of trapped spinless bosons interacting with a repulsive potential and subject to rotation. In the limit of rapid rotation and small scattering length, we show that the ground state energy converges to that of a simplified model...

Abstract - TristanRoy.pdf

Over the past decades, much attention has been devoted to the detection of small inhomogeneities in materials or tissues, using non-invasive techniques, primarily electromagnetic wave-fields. The characterization of the signature of small inclusions...

Concentration phenomena for Laplacian eigenfunctions can be studied by obtaining estimates for their $L^{p}$ growth. By considering eigenfunctions as quasimodes (approximate eigenfunctions) within the semiclassical framework we can extend such...

I will discuss the problem of determining the number of infinite-volume ground states in the Edwards-Anderson (nearest neighbor) spin glass model on Z^D for D \geq 2. There are no complete results for this problem even in D=2. I will focus on this...

This talk will be a progress report on an ongoing research project which is joint work with Ajay Chandra and Gianluca Guadagni and which concerns a p-adic analog of the Brydges-Mitter-Scoppola phi-4 model with fractional Laplacian in 3 dimensions...

Band matrices are a class of random operators with rows and columns indexed by elements of the d-dimensional lattice, and random entries H(u, v) which are small when the distance between u and v on the lattice is larger than W. We shall discuss the...

As originally proposed by Anderson (1958), a quantum system of many local degrees of freedom with short-range interactions and static disorder may fail to thermally equilibrate, even with strong interactions and high excitation energy density. This...

Deift--Simon and Poltoratskii--Remling proved upper bounds on the measure of the absolutely continuous spectrum of Jacobi matrices. Using methods of classical approximation theory, we give a new proof of their results, and generalize them in several...

We develop the droplet scaling theory for the low temperature critical behavior of two-dimensional Ising spin glasses. The models with integer bond energies vs. continuously-distributed bond energies are in the same universality class in a regime of...

Using the spectral multiplicities of the standard torus, we endow the Laplace eigenspace with Gaussian probability measure. This induces a notion of a random Gaussian Laplace eigenfunctions on the torus. We study the distribution of nodal length of...

Let (M,g) be a compact, closed manifold and g_u be a family of conformal metric deformations of g supported in a small ball B(\delta) of radius \delta>0. We show that for a class of such deformations, the corresponding Laplace eigenfunctions almost...

For many elegant mathematical examples, one can 1) find theories behind them, 2) understand why they exist in the first place, 3) explore the consequences in math and physics. If one takes the Euler number as such an example, then the theory behind...

The Gaussian central limit theorem says that for a wide class of stochastic systems, the bell curve (Gaussian distribution) describes the statistics for random fluctuations of important observables. In this talk I will look beyond this class of...

We explain an exact solution of the one-dimensional Kardar-Parisi-Zhang equation with sharp wedge initial data. Physically this solution describes the shape fluctuations of a thin film droplet formed by the stable phase expanding into the unstable...

This is joint work with Govind Menon, Sheehan Olver and Thomas Trogdon. The speaker will present evidence for universality in numerical computations with random data. Given a (possibly stochastic) numerical algorithm with random input data, the time...

This is joint work with Tom Trogdon. Here the author shows how to prove universality rigorously for certain numerical algorithms of the type described in the first lecture. The proofs rely on recent state of the art results from random matrix theory...

I will discuss some results and open questions related to transport properties of 1D quasiperiodic operators with absolutely continuous spectra, and their relations to integrable many-body systems. Most of the results will be based on https://arxiv...

The talk is about the dynamics of a tracer particle coupled strongly to a dense non-interacting electron gas in two dimensions. I will present a recent result that shows that for high densities the tracer particle moves freely for very long times.

In this talk we will quantify the coupling asymptotics for the Lyapunov exponent (LE) of a one-frequency quasi-periodic Schrödinger operator with analytic potential sampling function. By proving an asymptotic formula for the LE valid for all...

Given $d, k\in\mathbb N$, let $P_j$ be an integer-valued polynomial of $k$ variables for every $1\le j \le d$. Suppose that $(X, \mathcal{B}, \mu)$ is a $\sigma$-finite measure space with a family of invertible commuting and measure preserving...

We present joint work with Jan Maas showing that Quantum Markov semigroups satisfying a detailed balance condition are gradient flow for quantum relative entropy, and use this prove some conjectured inequalities arising in quantum information theory...

We discuss the problem of singularity formation for some of the basic equations of incompressible fluid mechanics such as the incompressible Euler equation and the surface quasi-geostrophic (SQG) equation. We begin by going over some of the...

Hyperuniform many-particle systems have density fluctuations that are anomalously suppressed at long wavelengths compared to the fluctuations in typical disordered point configurations, such as ideal gases, liquids and glasses. All perfect crystals...

Recently we proved with Durcik, Kovac, Skreb variational estimates providing sharp quantitative norm convergence results for bilinear ergodic averages with respect to two commuting transformations. The proof uses so called twisted technology...

The intertwining wave operators are basic objects in the scattering theory of a Hamiltonian given as the sum of a Laplacian with a potential. These Hamiltonians are the classical Schroedinger operators of quantum mechanics. For the three dimensional...

It is widely believed that the generic dynamics of nonlinear dispersive equations in the whole space is described by solitary waves and linear dispersions. More precisely, over large times, solutions tend to de-couple into solitary waves plus...

I'll first introduce the Markoff-Hurwitz equation and explain how it plays a fundamental role in different areas of mathematics. The main result I'll discuss is a true asymptotic formula for the number of real points in a fixed orbit of the...

A polygon with rational angles can be unfolded and glued into a finite genus Riemann surface equipped with a flat metric and some singularities. The moduli space of all such structures carries an action of the group $\mathrm{PSL}(2,\mathbb R)$ and...

I will review recent progress on some dispersive type PDEs including the nonlinear magnetic and fractional Schroedinger equations with singular and unbounded potentials. This study is motivated by related open questions in the area in oder to...

We consider Laplace eigenfunctions of a metric graph satisfying Neumann-Kirchhoff conditions on every vertex. The nodal count of a given eigenfunction is the number of points at which it vanishes. The nodal count of the n-th eigenfunction was shown...

In this talk, I'm going to explain my recent work with Steve Zelditch, where we prove that, when $M$ is a principle $S^1$-bundle equipped with a generic Kaluza-Klein metric, the nodal counting of eigenfunctions is typically $2$, independent of the...

Courant's theorem states that the k-th eigenfunction of the Laplace operator on a closed Riemannian manifold has at most k nodal domains. Given a ball of radius r, we will discuss how many of nodal domains can intersect a ball (depending on r and k)...

Every area-minimizing hypercone having only an isolated singularity fits into a foliation by smooth, area-minimizing hypersurfaces asymptotic to the cone itself. In this talk I will present the following epsilon-regularity result: every minimal...