Seminars Sorted by Series

I will talk about joint work with my student Reuben Drogin.  We prove delocalization for the homogeneous Anderson model on an infinite regular tree (or Caley graph or Bethe lattice) with small bounded disorder. This extends earlier results of Klein...

Capillary surfaces model the geometry of liquid interfaces meeting a container at an angle, and arise naturally as (constrained) minimizers of the Gauss free energy. We will discuss recent progress in understanding the size of the singular set of...

The compressible Euler equation can lead to the emergence of shock discontinuities in finite time, notably observed behind supersonic planes. A very natural way to justify these singularities involves studying solutions as inviscid limits of Navier...

This talk is concerned with solutions of the 3D incompressible Navier-Stokes equations that are bounded in a critical space. From small initial data, these solutions are known to be globally well-posed due to classical work of Fujita-Kato and others...

We will describe a recent result on the stability of twisting in 2d Hamiltonian flows and its application to various questions concerning the long-time behavior of 2d Euler solutions. We will look at stability properties of stationary solutions...

I will discuss the problem of understanding the long-time behavior of Ricci flow on a compact Kähler manifold, assuming that a solution exists for all positive time. Inspired by an analogy with the minimal model program in algebraic geometry, Song...

A fractal uncertainty principle (FUP) roughly says that a 
function and its Fourier transform cannot both be concentrated on a 
fractal set. These were introduced to harmonic analysis in order to 
prove new results in quantum chaos: if eigenfunctions...

I will describe some new "coarse-graining" methods in quantitative homogenization and how they can be used to give rigorous versions of certain heuristic "renormalization group" arguments in physics, with a focus on several examples.

I will present joint work with Jacek Jendrej. We consider classical scalar fields in dimension 1+1 with a symmetric double-well self-interaction potential. Examples of such equations are the phi-4 model and the sine-Gordon equation. These nonlinear...

Hilbert’s sixth problem asks for a mathematically rigorous justification of the macroscopic laws of statistical physics from the microscopic laws of dynamics. The classical setting of this problem is the justification of Boltzmann’s kinetic equation...

We study multiply warped product geometries MN:=Bn×Fn1×···×FnA 

g = g_B + \sum_{a=1}^A v_a^2 g_{F^{n_a}} and show that for an open set of initial data within multiply warped product geometries the Ricci flow starting at any of those develops...

We consider an energy model for N ordered elastic membranes subject to forcing and boundary conditions. The heights of the membranes are described by real functions u_1, u_2,...,u_N, which minimize an energy functional involving the Dirichlet...

Extremal black holes are special solutions of Einstein’s equations which have absolute zero temperature in the thermodynamic analogy of black hole mechanics. In this talk, I will present a proof that extremal black holes arise on the critical...

Gamow introduced the liquid drop model in 1928 as a model for the nucleus. I will discuss some recent work (with Ian Ruohoniemi) concerning roundness of minimizers. 

An evolving surface is a mean curvature flow if the normal component of its velocity field is given by the mean curvature. First introduced in the physics literature in the 1950s, the mean curvature flow equation has been studied intensely by...

New results on quantum tunneling between deep potential wells, in the presence of a strong constant magnetic field are presented. This includes a family of double well potentials containing examples for which the low-energy eigenvalue splitting...

The lateral periodic condition has been imposed canonically and investigated extensively in the study of a channel flow, which unfortunately is not compatible with either the celebrated Reynolds' experiment or the self-similar Blasius boundary layer...

Semiclassical measures are a standard object studied in quantum chaos, capturing macroscopic behavior of sequences of eigenfunctions in the high energy limit. They have a long history of study going back to the Quantum Ergodicity theorem and the...

In 1962, Yudovich established the well-posedness of the two-dimensional incompressible Euler equations for solutions with bounded vorticity. However, uniqueness within the broader class of solutions with L^p vorticity remains a key unresolved...

Let $M(n,m)$ denote the real $m\times n$ matrices. A continuous function $f\colon M(n,m)\to \R$ is called {\it Morrey quasi-convex} if  $$\int_{\R^n}(f(A+\nabla\vf(x))-f(A))\,dx\ge 0$$ for each smooth, compactly supported $\vf\colon\R^n\to\R^m$ and...

I will describe a new bipartite Euler system-type construction system for GSp_4 and its inner forms, based on the special cycles appearing in the Kudla program (for instance, Shimura curves on Siegel threefolds). This leads to new results towards...

Sen's theorem on the ramification of a p-adic analytic Galois extension of p-adic local fields shows that its perfectoidness is equivalent to the non-vanishing of its arithmetic Sen operator. By developing p-adic Hodge theory for general valuation...

By Deligne's Hodge theory, the integral cohomology groups H^n(X^h, Z) of the C-analytification of a separated scheme X of finite type over C are provided with a mixed Hodge structure, functorial in X. Given a non-Archimedean field K isomorphic to...

Eisenstein proved, in 1852, that if a function f(z) is algebraic, then its Taylor expansion at a point has coefficients lying in some finitely-generated Z-algebra. I will explain ongoing joint work with Josh Lam which studies the extent to which the...

Local Langlands correspondence (LLC) is a conjectural finite-to-one map from representations of a p-adic reductive group G to the set of L-parameters for G. Recently there have been two major advances in this area: Kaletha's characterization of the...

Topology of the Hitchin system has been studied for decades, and interesting connections were found to orbital integrals, non-abelian Hodge theory, mirror symmetry etc. I will explain that a large part of the symmetries in these geometries above are...

Studying the \’etale cohomology of Shimura varieties with Hecke and Galois actions provides an avenue toward understanding the Langlands correspondence. 
While the structure of the rational cohomology groups is predicted conjectures of Kottwitz and...

The analytic de Rham stack is a new construction in Analytic Geometry whose theory of quasi-coherent sheaves encodes a notion of p-adic D-modules. It has the virtue that can be defined even under lack of differentials (eg. for perfectoid spaces or...

Kazhdan-Lusztig theory provides a pattern of applying tools of algebraic geometry,
such as the theory of Frobenius or Hodge weights, to numerical problems of representation theory.
These techniques have been used in representation theory over a field...

Ohta described the ordinary part of the 'etale cohomology of towers of modular curves in terms of Hida families. Ohta's approach crucially depended on the one-dimensional nature of modular curves. In this talk, I will present joint work with Chris...

We show how the language of motivic non-archimedean homotopy theory can be used to define p-adic cohomology theories and prove new results about them. For example, we show how to define solid relative rigid cohomology and deduce a version of

I will discuss a theorem obtained in joint work with Wyss and Ziegler, which is devoted to describing the Frobenius traces for the IC sheaf on moduli space of objects in symmetric abelian categories linear over a finite field. The formula is...

Pretannakian categories, that is k-linear abelian categories with finite dimensional Hom groups, given with a commutative and associative tensor product, with a unit object (such that End(1)=k) and duals, can be viewed as generalizations of linear...