Seminars Sorted by Series

In this talk I will describe joint work with D. Alonso-Orán and A. Córdoba where we extend a result, proved independently by Kiselev-Nazarov-Volberg and Caffarelli-Vasseur, for the critical dissipative SQG equation on a two dimensional sphere. The...

"I will discuss some history and new results about the forced mean curvature flow in inhomogeneous media.  It is a model for interface propagation in quenched randomness in various physical settings, e.g. contact lines, phase interfaces in porous...

The path between two measures in the sense of optimal transport yields the notion of *displacement interpolation*. As observed by R. McCann, certain functionals that are not convex in the usual sense are nonetheless *displacement convex*. Following...

For the Obstacle Problem involving a convex fully nonlinear elliptic operator, we show that the singular set of the free boundary stratifies. The top stratum is locally covered by a $C^{1,\alpha}$-manifold, and the lower strata are covered by $C^{1...

I will prove the following surprising fact: on a given Kahler manifold (X, J, \omega), a Holder bound on the Kahler potential \phi implies a Holder bound on the distance function of the new Kahler metric \omega+dd^c \phi. Time permitting I will also...

Due to its importance in materials science where it models the slow relaxation of grain boundaries, multiphase mean curvature flow has received a lot of attention over the last decades.

In this talk, I want to present two theorems. The first one is...

In the talk we will focus on certain constructions of weak solutions to the Navier--Stokes inequality (NSI), \[ u \cdot \left( u_t - \nu \Delta + (u\cdot \nabla ) u+ \nabla p \right) \leq 0\] on $\mathbb R^3$. Such vector fields satisfy both the...

Flocking and swarming models which seek to explain pattern formation in mathematical biology often assume that organisms interact through a force which is attractive over large distances yet repulsive at short distances. Suppose this force is given...

Given a planar domain with sufficiently regular boundary, one can study periodic orbits of the associated billiard problem. Periodic orbits have a rich and quite intricate structure and it is natural to ask how much information about the domain is...

Some words in the title are between quotation marks because it is a matter of interpretation. For dynamical systems on finite dimensional spaces, one often equates observable events with positive Lebesgue measure sets, and invariant distributions...

We provide geometric sufficient conditions for Reifenberg flat sets of any integer dimension in Euclidean space to be parametrized by a Lipschitz map with Hölder derivatives. The conditions use a Jones type square function and all statements are...

Consider a (possibly time-dependent) vector field $v$ on the Euclidean space. The classical Cauchy-Lipschitz (also named Picard-Lindel\"of) Theorem states that, if the vector field $v$ is Lipschitz in space, for every initial datum $x$ there is a...

We present a purely variational approach to the regularity theory for the Monge-Ampère equation, or rather optimal transportation, introduced with M. Goldman. Following De Giorgi’s philosophy for the regularity theory of minimal surfaces, it is...

In 1965, M. Kac proved that discs were uniquely determined by their Dirichlet (or, Neumann) spectra. Until recently, disks were the only smooth plane domains known to be determined by their eigenvalues. Recently, H. Hezari and I proved that ellipses...

We show that a topologically mixing C^\infty Anosov flow on a 3 dimensional compact manifold is exponential mixing with respect to any equilibrium measure with Holder potential. This is a joint work with Masato Tsujii.

Given a Lipschitz map, it is often useful to chop the domain into pieces on which the map has simple behavior. For example, depending on the dimensions of source and target, one may ask for pieces on which the map behaves like a bi-Lipschitz...

We prove a sharp square function estimate for the cone in R^3 and consequently the local smoothing conjecture for the wave equation in 2+1 dimensions. The proof uses induction on scales and an incidence estimate for points and tubes. This is joint...

The construction of pure phases from ground states is performed for $ u > u_*(d)$ for all values of $d$ except for 39 special ones. For values $d$ with a single equivalence class all periodic ground states generate the corresponding pure phase which...

Wave maps are harmonic maps from a Lorentzian domain to a Riemannian target. Like solutions to many energy critical PDE, wave maps can develop singularities where the energy concentrates on arbitrary small scales but the norm stays bounded. Zooming...

Filling a curve with an oriented surface can sometimes be "cheaper by the dozen". For example, L. C. Young constructed a smooth curve drawn on a projective plane in $\mathbb R^n$ which is only about 1.5 times as hard to fill twice as it is to fill...

The study of nodal sets of Laplace eigenfunctions has intrigued many mathematicians over the years. The nodal count problem has its origins in the works of Strum (1936) and Courant (1923) which led to questions that remained open to this day. One...

Lévy matrices are symmetric random matrices whose entries are independent alpha-stable laws. Such distributions have infinite variance, and when alpha is less than 1, infinite mean. In the latter case these matrices are conjectured to exhibit a...

We will briefly review Kolmogorov’s (41) theory of homogeneous turbulence and Onsager’s (49) conjecture that in 3-dimensional turbulent flows energy dissipation might exist even in the limit of vanishing viscosity. Although over the past 60 years...

Given a set $E$ of Hausdorff dimension $s > d/2$ in $\mathbb{R}^d$ , Falconer conjectured that its distance set $\Delta(E)=\{ |x-y|: x, y \in E\}$ should have positive Lebesgue measure. When $d$ is even, we show that $\dim_H E>d/2+1/4$ implies $|...

Motivated by some work of Thurston on defining a Teichmuller theory based on best Lipschitz maps between surfaces, we study infinity-harmonic maps from a manifold to a circle. The best Lipschitz constant is taken on on a geodesic lamination...

In the theory of turbulence, a famous conjecture of Onsager asserts that the threshold Hölder regularity for the total kinetic energy conservation of (spatially periodic) Euler flows is 1/3. In particular, there are Hölder continuous Euler flows...

An extension of Gromov compactness theorem ensures that any family of manifolds with convex boundaries, uniform bound on the dimension and uniform lower bound on the Ricci curvature is precompact in the Gromov-Hausdorff topology. In this talk, we...

For the incompressible Navier-Stokes equations, classical results state that weak solutions are unique in the so-called Ladyzhenskaya-Prodi-Serrin regime. A scaling analysis suggests that classical uniqueness results are sharp, but current...

We will discuss recent developments of the theory of a-contraction with shifts to study the stability of discontinuous solutions of systems of equations modeling inviscid compressible flows, like the compressible Euler equation.

The Stefan problem, dating back to the XIX century, aims to describe the evolution of a solid-liquid interface, typically a block of ice melting in water. A celebrated work of Luis Caffarelli from the 1970's established that the ice-water interface...

The rigorous calculation of the ground state energy of dilute Bose gases has been a challenging problem since the 1950s. In particular, it is of interest to understand the extent to which the Bogoliubov pairing theory correctly describes the ground...

We consider systems of $N$ particles interacting through a repulsive potential in the Gross-Pitaevskii regime. We prove complete Bose-Einstein condensation and we determine the form of the low-energy spectrum, in the limit of large $N$. Our results...

Our study is motivated by earlier results about nodal count of Laplacian eigenfunctions on manifolds and graphs that share the same flavor: a normalized nodal count is equal to the Morse index of a certain energy functional at the critical point...

We will discuss the mean curvature flow of $n$-dimensional submanifolds in $\mathbb{R}^{n+k}$ satisfying a pinching condition $|A|^2 c|H|^2$ introduced by Andrews and Baker (2010). For suitable constants $c$, these flows resemble flows of convex...

We consider a class of interacting particle systems with two types, A and B which perform independent random walks at different speeds. Type A particles turn into type B when they meet another type B particle. This class of systems includes models...

I will survey results related to graph comparison; graph comparison is a certain type of restriction on a metric spaces which is encoded by a given graph.

In 2017 Lucio Galeati understood that a suitable scaling limit of certain hyperbolic PDEs with noise may lead to deterministic parabolic equations. Since then, in collaboration with Lucio and Dejun Luo, we have understood the phenomenon from several...

Given a polynomial $P$ of constant degree in $d$ variables and consider the oscillatory integral \[I_P = \int_{[0,1]^d} e(P(\xi)) \, \mathrm{d}\xi.\] Assuming the number $d$ of variables is also fixed, what is a good upper bound of $|I_P|$? In this...

We consider a system of $N$ particles evolving according to the gradient flow of their Coulomb or Riesz interaction, or a similar conservative flow, and possible added random diffusion. By Riesz interaction, we mean inverse power $s$ of the distance...

The study of hyperkaehler manifolds of lowest dimension (and of gauge theory on them) leads to a chain of generalizations of the notion of a quiver: quivers, bows, slings, and monowalls. This talk focuses on bows, their representations, and...

In this talk, we will discuss some joint work with Alexandru Ionescu on the nonlinear inviscid damping near point vortex and monotone shear flows in a finite channel. We will put these results in the context of long time behavior of 2d Euler...

Consider a point particle flying freely on the torus and elastically bouncing back from the boundary of fixed smooth convex obstacles. This is the celebrated Sinai billiard, a rare example of a deterministic dynamical system where rigorous results...

Mean curvature flow (MCF) is a geometric heat equation where a submanifold evolves to minimize its area. A central problem is to understand the singularities that form and what these imply for the flow. I will talk about joint work with Toby Colding...

Korevaar and Schoen introduced, in a seminal paper in 1993, the notion of `Dirichlet energy’ for a map from a smooth Riemannian manifold to a metric space. They used such concept to extend to metric-valued maps the regularity theory by Eells-Sampson...