Video Lectures

We compute the subleading asymptotics of the ECH and elementary ECH capacities of toric domains and show that they recover the perimeter in the liminf, without any genericity required.  As an application, we give the first examples of the failure of...

Remarkable martensitic microstructures are observed in the alloy  Ti76Nb22Al2 , which undergoes a cubic to orthorhombic transformation with six martensitic variants Ui=UTi greater than 0 having middle eigenvalue λ2(Ui) very close to 1. Assuming that...

How and where do black hole mergers form in our Universe? After 10 years of GW science with LIGO/Virgo/Kagra, there still seems to be no direct indication of what the underlying astrophysical formation channels are. I will present ideas and ongoing...

Inspired by the Erdos--Renyi model for random graphs, Linial and Meshulam devised in 2006 a model for random 2-dimensional simplicial complexes. The goal of this talk (and the next) is to present some nice results about the behavior of these random...

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...

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...

Riemannian metrics are the simplest generalizations of Euclidean geometry to smooth manifolds. The Ricci curvature of a metric measures, in an averaged sense, how the geometry deviates from being flat. The tensor −2Ric can be viewed as a Laplacian...

Large language models (LLMs) sometimes generate statements that are plausible but factually incorrect—a phenomenon commonly called "hallucination." We argue that these errors are not mysterious failures of architecture or reasoning, but rather...

The Birkhoff attractor is a closed invariant subset associated with any dissipative twist map of the annulus (of dimension 2), which was introduced by Birkhoff in 1932. We will see that it can be generalized to higher dimensions using tools from...

The goal of these lectures is to present the fundamentals of Simpson’s correspondence, generalizing classical Hodge theory, between complex local systems and semistable Higgs bundles with vanishing Chern classes on smooth projective varieties.