Seminars Sorted by Series

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...

The study of random matrix moments of moments has connections to number theory, combinatorics, and log-correlated fields. Our results give the leading order of these functions for integer moments parameters by exploiting connections with Gelfand...

I will discuss the extreme eigenvalue distributions of adjacency matrices of sparse random graphs, in particular the Erd{\H o}s-R{\'e}nyi graphs $G(N,p)$ and the random $d$-regular graphs. For Erd{\H o}s-R{\'e}nyi graphs, there is a crossover in the...

A random planar map is a canonical model for a discrete random surface which is studied in probability theory, combinatorics, mathematical physics, and geometry. Liouville quantum gravity is a canonical model for a random 2D Riemannian manifold with...

I will discuss an upcoming result proving the full finite-codimension non-linear asymptotic stability of the Schwarzschild family as solutions to the Einstein vacuum equations in the exterior of the black hole region. No symmetry is assumed. The...

I will discuss recent results on localization for the Anderson--Bernoulli model. This will include my work with Ding as well as work by Li--Zhang. Both develop new unique continuation results for the Laplacian on the integer lattice.

In this talk I will outline recent work in collaboration with Pierre Germain, Zaher Hani and Jalal Shatah regarding a rigorous derivation of the kinetic wave equation. The proof presented will rely of methods from PDE, statistical physics and number...

To solve the inverse spectral problem for the Schrödinger equation on a metric graph one needs to determine:
• the metric graph;
• the potential in the Schrödinger equation;
• the vertex conditions (connecting the edges together).
The inverse...

We will discuss a recent result with Tristan Buckmaster (Princeton) and Steve Shkoller (UC Davis) which establishes the finite-time shock formation for the 3d isentropic Euler equations with vorticity. We prove that for an open set of Sobolev-class...

I will illustrate a recent result obtained in collaboration with L. Spolaor and B. Velichkov concerning the regularity of the free boundaries in the two phase Bernoulli problems. The new main point is the analysis of the free boundary close to...

We shall discuss optimal conditions on the geometry of the domain responsible for solvability of the Dirichlet problem, or absolute continuity of the harmonic measure with respect to the Lebesgue measure. In rough terms, the question is: do Brownian...

The spin-S quantum spin chain, with a projection-based antiferromagnetic interaction, and the antiferromagnetic XXZ spin-1/2 chain exhibit different forms of translation symmetry breaking. Yet they are related to a common system of random loops...

I will discuss several results on mean-values of multiplicative functions, notably Halasz's method for estimating such averages. In the second lecture I will show how this circle of ideas can be used to derive "weak subconvexity" bounds for L...

I will discuss several results on mean-values of multiplicative functions, notably Halasz's method for estimating such averages. In the second lecture I will show how this circle of ideas can be used to derive "weak subconvexity" bounds for L...

In this mini-course we give a quick tour of sieve methods. The first lecture will deal with the rudiments of the basic theory and mention a few simple examples. The second talk will, for the most part, feature some more recent applications. The...

In this mini-course we give a quick tour of sieve methods. The first lecture will deal with the rudiments of the basic theory and mention a few simple examples. The second talk will, for the most part, feature some more recent applications. The...

We describe how various fundamental algebraic structures (involving, for example, number fields, class groups, and algebraic curves) can be parameterized via the orbits of appropriate group representations. By developing techniques to count such...

We describe how various fundamental algebraic structures (involving, for example, number fields, class groups, and algebraic curves) can be parameterized via the orbits of appropriate group representations. By developing techniques to count such...

This is an exposition of work on Artin's Conjecture on the zeros of p-adic forms.A variety of lines of attack are described ,going back to 1945.However there is particular emphasis on recent developments concerning quartic forms on the one hand ,and...

This is an exposition of work on Artin's Conjecture on the zeros of p-adic forms.A variety of lines of attack are described ,going back to 1945.However there is particular emphasis on recent developments concerning quartic forms on the one hand ,and...

The Schmidt subspace theorem is a deep generalization of Roth's theorem in Diophantine approximation to the setting of hyperplanes in projective space. Another well-known generalization of Roth's theorem is the theorem of Wirsing, which extends Roth...

We improve Kolyvagin's upper bound on the order of the p-primary part of the Shafarevich-Tate group of an elliptic curve of rank one over a quadratic imaginary field. In many cases, our bound is precisely the one predicted by the Birch and...

A sequence of primes $p_1, \dotsc , p_k$ is a prime chain if $p_{j+1} \equiv 1 \pmod{p_j}$ for each $j$. For example: 3, 7, 29, 59. We describe new estimates for counts of prime chains satisfying various properties, e.g. the number of chains with $p...

An overview of methods and results concerning small gaps between primes, between almost primes, and the consequences on values taken by certain arithmetical functions will be presented. The material is from the works with Goldston, Graham and Pintz...