Seminars Sorted by Series

Lower scalar curvature bounds on spin Riemannian manifolds exhibit remarkable rigidity properties determined by spectral properties of Dirac operators. For instance, a fundamental result of Llarull states that there is no smooth Riemannian metric on...

I will discuss how to use mean curvature flow to give nearly optimal lower bounds on the density of topologically non-trivial minimal hypercones in low dimensions. This compliments work of Ilmanen-White who gave analogous bounds for topologically...

In this talk I will present a general framework to establish the stability of  inequalities of the form >= F(u); where L is a positive linear operator and F is a 2-homogeneous nonlinear functional.

We will then see how this framework can be...

There is a celebrated connection between minimal (or constant mean curvature) hypersurfaces and Ricci curvature in Riemannian Geometry, often boiling down to the presence of a Ricci term in the second variation formula for the area. The first goal...

The presence of branch points and so-called density gaps heavily complicates the regularity theory for minimal submanifolds or, more precisely, stationary integral varifolds. Density gaps illustrate a mismatched symmetry between nearby varifolds...

The Teukolsky equation is one of the fundamental equations governing linear gravitational perturbations of the Kerr black hole family as solutions to the vacuum Einstein equations. We show that solutions arising from suitably regular initial data...

I will talk about three interesting ingredients that goes into the results on H\"{o}rmander type operators I presented at Princeton (joint with Shaoming Guo and Hong Wang). They are all related to algebraic or geometric properties of multivariate...

We obtain new bounds on the additive energy of (Ahlfors-David type) regular measures in both one and higher dimensions, which implies expansion results for sums and products of the associated regular sets, as well as more general nonlinear functions...

Schedule: https://www.ias.edu/math/bourgain16/schedule

Schedule: https://www.ias.edu/math/bourgain16/schedule

Schedule: https://www.ias.edu/math/bourgain16/schedule

The loop equation for circulation PDF as functional of the loop shape is derived from the Hopf equation. This equation is $\textbf{exactly}$ equivalent to the Schrödinger equation in loop space, with viscosity playing the role of Planck's constant...

I will review some recent progress on regularity properties of vortex and SQG patches. In particular, I will present an example of a vortex patch with continuous initial curvature that immediately becomes infinite but returns to $(C^2)$ class at all...

Solitons are particle-like solutions to dispersive evolution equations whose shapes persist as time evolves. In some situations, these solitons appear due to the balance between nonlinear effects and dispersion, in other situations their existence...

Given a linear equation whose principal term is given by a degenerate dispersive pseudo-differential operator, we provide a framework for the construction of degenerating wave packet solutions. As an application, we prove strong ill-posedness for...

This talk will be about a ferromagnetic spin system called the Blume-Capel model. It was introduced in the '60s to model an exotic multi-critical phase transition observed in the magnetisation of uranium oxide. Mathematically speaking, the model can...

The Brascamp-Lieb inequality is a fundamental inequality in analysis, generalizing more classical inequalities such as Holder's inequality, the Loomis-Whitney inequality, and Young's convolution inequality: it controls the size of a product of...

We consider general two-dimensional autonomous velocity fields and prove that their mixing and dissipation features are limited to algebraic rates. As an application, we consider a standard cellular flow on a periodic box, and explore potential...

We construct solutions with prescribed radiation fields for wave equations with polynomially decaying sources close to the lightcone. In this setting, which is motivated by semi-linear wave equations satisfying the weak null condition, solutions to...

This lecture is devoted to a survey on explicit stability results in Gagliardo-Nirenberg-Sobolev and logarithmic Sobolev inequalities. Generalized entropy methods based on carré du champ computations and nonlinear diffusion flows can be used for...

Spacetimes formed from the cartesian product of Minkowski space and a flat torus play an important role as toy models for theories of supergravity and string theory. In this talk, I will discuss an upcoming work, joint with Huneau and Stingo...

We discuss a one-phase degenerate free boundary problem which arises from the minimization of the so-called Alt-Phillips functional $$\int_\Omega (|\nabla u|^2 + u^\gamma \chi_{\{u>0\}}) dx$$ for $\gamma \in (-2,0).$ We establish partial regularity...

A large toolbox of numerical schemes for dispersive equations has been established, based on different discretization techniques such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the full equation...

Submanifolds with intrinsic Lipschitz regularity in Carnot groups (i.e.,
stratified groups endowed with a sub-Riemannian structure) can be
introduced using the theory of intrinsic Lipschitz graphs started years
ago by B. Franchi, R. Serapioni and F...

The question of producing a foliation of the n-dimensional Euclidean space with k-dimensional submanifolds which are tangent to a prescribed k-dimensional simple vectorfield is part of the celebrated Frobenius theorem: a decomposition in smooth...

This talk is a discussion about the extremal points of the unit ball with respect to the Hessian-Schatten variation seminorm, i.e. the total variation of the second distributional differential with respect to the Schatten matrix norm. The main...

In this talk, we will start by describing how classical tools from probability offer a robust framework to understand the dynamics of waves via appropriate ensembles on phase space rather than particular microscopic dynamical trajectories. We will...

We study the probability that a geodesic passes through a prescribed edge in first-passage percolation on Z^d. Benjamini, Kalai and Schramm conjectured that this probability tends to zero as the length of the geodesic tends to infinity, as long as...

We consider 2D quantum materials (non-magnetic and constant magnetic field cases), modeled by a continuum Schroedinger operator, whose potential is a sum of translates of an atomic well, centered on the vertices of a discrete subset of the plane...

An interesting feature of General Relativity is the presence of singularities which can occur in even the simplest examples such as the Schwarzschild spacetime. However, in this case the singularity is cloaked behind the event horizon of the black...

I will describe the construction of a harmonic measure that reproduces a harmonic function from its Robin boundary data, which is a combination of the value of the function and its normal derivative. I shall discuss the surprising fact that this...

In 1982, S. T. Yau conjectured that there exists at least four embedded minimal 2-spheres in the 3-sphere with an arbitrary metric. In this talk, we will show that this conjecture holds true for bumpy metrics and metrics with positive Ricci...

In the search for possible blow-up of the incompressible Navier-Stokes equations, there has been much recent attention on the class of axisymmetric solutions with swirl. Several interesting structures of this system have led to regularity criteria...

In spite of tremendous progress in the mean-field theory of spin glasses in the last forty years, culminating in Giorgio Parisi’s Nobel Prize in 2021, the more “realistic” short-range spin glass models have remained almost completely intractable. In...

Interaction models discussed here are the (asymmetric) quantum Rabi model (QRM), which describes the interaction between a photon and two-level atoms, and the non-commutative harmonic oscillator (NCHO). The latter can be considered as a covering...

What happens to an Lp function when one truncates its Fourier transform to a domain? This question is now rather well understood, thanks to famous results by Marcel Riesz and Charles Fefferman, and the answer depends on the domain: if it is a...

Let $G$ be a compact Lie group acting on a closed manifold $M$. Partially motivated by work of Uhlenbeck (1976), we explore the generic properties of Laplace eigenfunctions associated to $G$-invariant metrics on $M$. We find that, in the case where...

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...