Seminars Sorted by Series

This talk will be about the polynomial method and its applications to questions that have traditionally been tackled by Fourier analysis, with emphasis on the Kakeya conjecture, the cap set problem, arithmetic progressions in dense sets, and the...

We discuss an application of the SUSY approach to the analysis of spectral characteristics of hermitian and non hermitian random band matrices. In 1D case the obtained integral representations for correlation functions of characteristic polynomials...

At the base of spontaneous pattern formation is universally believed to be the competition between short range attractive and long range repulsive forces. Though such a phenomenon is observed in experiments and simulations, a rigorous understanding...

There has been much recent progress on the local solution theory for geometric singular SPDEs. However, the global theory is still largely open. In my talk, I will discuss the global well-posedness of the stochastic Abelian-Higgs model in two...

An inertial manifold is a positively invariant smooth finite-dimensional manifold which contains the global attractor and which attracts the trajectories at a uniform exponential rate. It follows that the infinite-dimensional dynamical system is...

Fourier Quasicrystals (FQ) are defined as crystalline measures $$ \mu = \sum_{\lambda \in \Lambda} a_\lambda \delta_\lambda, \quad \hat{\mu} = \sum_{s \in S} b_s \delta_s, $$ so that not only $ \mu $ (and hence $ \hat{\mu} $) are tempered...

The classical Serrin’s overdetermined theorem states that a C^2 bounded domain, which admits a function with constant Laplacian that satisfies both constant Dirichlet and Neumann boundary conditions, must necessarily be a ball. While extensions of...

In this talk, we will explore recent developments in the study of coherent structures evolving by incompressible flows. Our focus will be on the behavior of fluid interfaces and vortex filaments. We include the dynamics of gravity Stokes interfaces...

I will describe the duality of incompressible Navier-Stokes fluid dynamics in three dimensions, leading to its reformulation in terms of a one-dimensional momentum loop equation.
The decaying turbulence is a solution of this equation equivalent to a...

The question of asbolute continuity, with respect to the reference measure, of the harmonic measure on a domain with rough boundary has been the object of many important results. Here we ask about the similar question, but where the Dirichlet...

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In this talk, I will present several a priori interior and boundary trace estimates for the 3D incompressible Navier–Stokes equation, which recover and extend the current picture of higher derivative estimates in the mixed norm. Then we discuss the...

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In this overview talk we will explore a variational approach to the problem of Spectral Minimal Partitions (SMPs).  The problem is to partition a domain or a manifold into k subdomains so that the first Dirichlet eigenvalue on each subdomain is as...

This talk is about the study of the Boltzmann equation in the diffusive limit in a channel domain $\mathbb{T}^2\times (-1,1)$ nearby the 3D kinetic Couette flow.  We will begin the talk with a substantial introduction for non-experts.  Our result...

Suppose f is a function with Fourier transform supported on the unit sphere in $R^d$. Elias Stein conjectured in the 1960s that the $L^p$ norm of f is bounded by the $L^p$ norm of its Fourier transform, for any $p> 2d/(d-1)$.  We propose to study...

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More than 120 years ago, Minkowski published a seminal paper that laid the foundation for the field of convex geometry (as well as several other areas of mathematics). Despite numerous advances in the intervening years, there are fundamental...

A minimal partition is a decomposition of a manifold into disjoint sets that minimizes a spectral energy functional. In the bipartite case minimal partitions are closely related to eigenfunctions of the Laplacian, but in the non-bipartite case they...

In this talk, I will discuss some results concerning the geometry and topology of manifolds on which the first eigenvalue of the operator -γΔ + Ric is bounded below. Here, γ is a positive number, Δ is the Laplacian, and Ric denotes the pointwise...

Parallels between elliptic and parabolic theory of partial differential equations have long been explored. In particular, since elliptic theory can be seen as a steady-state version of parabolic theory, if a parabolic estimate holds, then by...

I will present recent work with Hairer, Rosati and Yi establishing quantitative lower bounds for the top Lyapunov exponent of linear PDEs driven by two-dimensional stochastic Navier-Stokes equations on the torus. For both the advection-diffusion...

In the early 80s Hatcher proved the Smale Conjecture, asserting that the diffeomorphism group of the three-sphere retracts onto its isometry group.  The corresponding problem for $RP^3$ was open nearly 40 years, and resolved only in 2019 by a...

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We use the min-max construction to find closed hypersurfaces which are stationary with respect to anisotropic elliptic integrands in any closed n-dimensional manifold . These surfaces are regular outside a closed set of zero n-3 dimension. The...

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Several recent groundbreaking results in geometric measure theory, homogeneous dynamics and number theory ultimately rely on a key result of Bourgain known as Bourgain's Projection Theorem (of course, each of these results require many other tools...

The mathematical core of deep learning is function approximation by neural networks trained on data using stochastic gradient descent. I will present a collection of sharp results on training dynamics for the deep linear network (DLN), a...

Let Ω be an open set in a Euclidean space X of dimension (n+1) and ϕ be a uniformly convex smooth norm on X. Consider an n-dimensional unit-density varifold V in Ω, whose generalised mean curvature vector, computed with respect to ϕ, is bounded...

The Hopf-Tsuji-Sullivan theorem states that the geodesic flow on (an infinite) Riemann surface is ergodic iff the Poincare series is divergent iff the Brownian motion is recurrent. Infinite Riemann surfaces can be built by gluing infinitely many...

In the Calculus of Variations, convexity plays a seemingly irreplaceable role.
For vectorial problems, however, for good reasons a whole zoo of its generalizations was introduced. It ranges  from notions based on very classical
observation over...

Remarkable martensitic microstructures are observed in the alloy  $Ti_{76}Nb_{22}Al_{2}$ , which undergoes a cubic to orthorhombic transformation with six martensitic variants $\mathbf U_i=\mathbf U_i^T>0$ having middle eigenvalue $\lambda_2(\mathbf...

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Ricci solitons are the self-similar solutions to the Ricci flow, which is the heat equation for Riemannian metrics, and they model singularity formation. We survey various estimates for Ricci solitons in dimension 4. This is mainly the work of...

A translation surface is a closed surface that is obtained by gluing edges of a polygon in parallel. The group $GL_2(R)$ acts on the collection translation surfaces of a fixed genus g. For a fixed translation surface S and t>0, we obtain a...

In 2003, Bressan proposed a conjecture on the mixing efficiency of incompressible flows, which remains open. This talk surveys progress toward resolving Bressan’s mixing conjecture and presents a new result confirming its asymptotic validity for...

We exhibit a new class of self-similar implosion solutions for the full compressible Euler equations. For any value of the adiabatic exponent, we construct a sequence of implosion profiles that are smooth before collapse and have an explicit...

Ricci solitons, introduced by R. Hamilton in the mid-80s, are self-similar solutions to the Ricci flow and natural generalizations of Einstein manifolds. Shrinking Ricci solitons, in particular,  model Type I singularities of the Ricci flow and...

I will prove Gromov's conjecture that every 3-manifold of positive scalar curvature contains a short closed geodesic. The proof uses Min-Max theory of minimal surfaces and a combinatorial version of mean curvature flow. Time permitting, I will...

Berger and Coburn conjectured that a Toeplitz operator on Bargmann--Fock space is bounded if and only if the heat transform of its symbol at time $t=1/4$ is bounded, the borderline time singled out by the Weyl calculus under the Bargmann transform...

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