Rivière-Fabes Symposium on Analysis and PDE

Weighted Fourier extension estimates

If we want the solution to the free Schrodinger equation to converge to its initial data pointwise, what's the minimal regularity condition for the initial data should be? I will present recent progress on this classic question of Carleson. This pointwise convergence problem is closely related to other problems in PDE and geometric measure theory, including spherical average Fourier decay rates of fractal measures, Falconer distance set conjecture, etc. All these problems can be approached by Weighted Fourier extension estimates

Falconer's distance set problem

A classical question in geometric measure theory, introduced by Falconer in the 80s is, how large does the Hausdorff dimension of a compact subset in Euclidean space need to be to ensure that the Lebesgue measure of its set of pairwise Euclidean distances is positive. In this talk, I'll report some recent progress on this problem, which combines several ingredients including radial projection estimates, and the refined decoupling theory. 

A constructive proof of nearly self-similar blowup of 2D Boussinesq and 3D Euler equations with smooth data

Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this talk, I will present a new result with Dr. Jiajie Chen in which we prove finite time blowup of the 2D Boussinesq and 3D Euler equations with smooth initial data. There are several essential difficulties in establishing such blowup result. We use the dynamic rescaling formulation and turn the problem of proving finite time singularity into a problem of proving stability of an approximate self-similar profile. A crucial step is to establish linear stability and control a number of nonlocal terms. We decompose the solution operator into a leading order operator that enjoys sharp stability estimates plus a finite rank perturbation operator that can be estimated by constructing space-time numerical solutions with rigorous error control. This enables us to establish nonlinear stability of the approximate self-similar profile and prove stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth initial data and boundary. This provides the first rigorous justification of the Hou-Luo blowup scenario.

Potentially singular behavior of 3D incompressible Navier-Stokes equations

Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this talk, I will present some new numerical evidence that the 3D Navier-Stokes equations develop nearly self-similar singular scaling properties with maximum vorticity increased by a factor of $10^7$. This potentially singular behavior is induced by a potential finite time singularity of the 3D Euler equations. Unlike the Hou-Luo blowup scenario, the potential singularity of the 3D Euler and Navier-Stokes equations occurs at the origin. We have applied several blowup criteria to study the potentially singular behavior of the Navier-Stokes equations. The Beale-Kato-Majda blow-up criterion, the blowup criteria based on the growth of enstrophy and negative pressure, the Ladyzhenskaya-Prodi-Serrin regularity criteria all seem to imply that the Navier-Stokes equations develop nearly singular behavior.  Finally, we present some new numerical evidence that a class of generalized axisymmetric Euler and Navier-Stokes equations with time dependent fractional dimension seem to develop asymptotically self-similar blowup.

From energy minimization to Fourier interpolation

I will talk about the recent results in the sphere packing and energy minimization problems, more precisely, the solutions
of 8- and 24-dimensional cases of the Cohn-Elkies and the Cohn-Kumar conjectures about linear programming bounds for these problems. I will then explain how these conjectures naturally lead one to consider an unusual interpolation formula involving values of a radial Schwartz function and its Fourier transform, and explain the ideas behind its proof.

Fourier uniqueness pairs, interpolation formulas, and modular forms

I will continue the discussion of the Fourier interpolation formulas used in the proof of the 8- and 24-dimensional Cohn-Kumar conjecture and put them it into a more general context of Fourier uniqueness pairs. I will then discuss number-theoretic constructions of tight Fourier uniqueness pairs and associated interpolation formulas using modular forms, and I will also talk about general results on Fourier uniqueness pairs recently obtained by Kulikov-Nazarov-Sodin.

Nonlinear stability of Kerr for small angular momentum I. Introduction to the Kerr stability conjecture

In the first lecture, I will introduce the Einstein equations and the corresponding evolution problem. I will then review some of the techniques used in the stability of Minkowski. I will end the lecture by presenting Kerr black holes and the Kerr stability conjecture.

Nonlinear stability of Kerr for small angular momentum II. History, statement and ideas of the proof

In the second lecture, I will recall the history of the Kerr stability conjecture. I will then focus on a recent work on the resolution of the black hole stability conjecture for small angular momentum.  

3:00 – 3:25 p.m.: Coffee and Check-in, 120 Vincent Hall

3:30 – 4:30 p.m.: Talk 1: Thomas Hou, 16 Vincent Hall

4:30 – 5:00 p.m.: Coffee Break, 120 Vincent Hall 

5:00 – 6:00 p.m.: Talk 2: Xiumin Du, 16 Vincent Hall

9:00 – 9:30 a.m.: Coffee and light breakfast, 120 Vincent Hall

9:30 – 10:30 a.m.: Talk 3: Jérémie Szeftel, 16 Vincent Hall

10:30 – 11:00 a.m.: Coffee, 120 Vincent Hall 

11:00 – Noon: Talk 4: Danylo Radchenko, 16 Vincent Hall

Noon  – 2:00 p.m.: Lunch Break

2:00 – 3:00 p.m.: Talk 5: Xiumin Du, 16 Vincent Hall

3:00 – 3:30 p.m.: Coffee, 120 Vincent Hall

3:30 – 4:30 p.m.: Talk 6: Thomas Hou, 16 Vincent Hall

5:30 p.m.:  Reception and Symposium Banquet, Campus Club (on 4th floor of the Coffman Memorial Union building)

8:30 – 9:00 a.m.: Coffee and light breakfast, 120 Vincent Hall

9:00 – 10:00 a.m.: Talk 7: Danylo Radchenko, 16 Vincent Hall

10:00 – 10:30 a.m.: Coffee, 120 Vincent Hall

10:30 – 11:30 a.m.: Talk 8: Jérémie Szeftel, 16 Vincent Hall