Analysis Seminar


Time Location Seminar Chair
Tuesdays 9:45am AP&M 7421 Andrej Zlatos

Fall 2017

Date Speaker Title + Abstract
September 27
(Wed) at 3pm
Yuan Yuan
Syracuse University
Diameter rigidity for Kahler manifolds with positive bisectional curvature
I will discuss the recent work with Gang Liu on the diameter rigidity for Kahler manifolds with positive bisectional curvature.
October 24 Igor Kukavica

Postponed to November 21

October 31 In-Jee Jeong
Princeton University
Finite time blow-up for strong solutions to the 3D Euler equations
We show finite time blow-up for strong solutions to the 3D Euler equations on the exterior of a cone. The solutions we construct have finite energy, and velocity is axisymmetric and are Lipschitz continuous before the blow up time. We achieve this by first analyzing scale-invariant (radially homogeneous) solutions, whose dynamics is governed by a 1D system. Then we make a cut-off argument to ensure finiteness of energy.
November 7 Hantaek Bae
Regularity and decay properties of the incompressible Navier-Stokes equations
In this talk, I will consider the incompressible Navier-Stokesequations in the mild solution setting. Using this setting, I will show how to obtain analyticity of mild solutions using the Gevrey regularity technique. This regularity enables to get decay rates of weak solutions of the Navier-Stokes equations. This idea can be applied to other dissipative equations with analytic nonlinearities. I will finally consider the regularity of the flow map of mild solutions using the Log-Lipschitz regularity of the velocity field.
November 14 Hung Tran
On some selection problems for fully nonlinear, degenerate elliptic PDEs
I will describe some interesting selection problems for fully nonlinear, degenerate elliptic PDEs. In particular, I will focus on the vanishing discount procedure and show the convergence result via a new variational technique.
November 14
at 3pm
Nikolay Shcherbina
University of Wuppertal
Squeezing functions and Cantor sets
We construct "large" Cantor sets whose complements resemble the unit disk arbitrarily well from the point of view of the squeezing function, and we construct "large" Cantor sets whose complements do not resemble the unit disk from the point of view of the squeezing function.
November 21 Igor Kukavica
On the size of the nodal sets of solutions of elliptic and parabolic PDEs
We present several results on the size of the nodal (zero) set for solutions of partial differential equations of elliptic and parabolic type. In particular, we show a sharp upper bound for the $(n-1)$-dimensional Hausdorff measure of the nodal sets of the eigenfunctions of regular analytic elliptic problems in ${\mathbb R}^n$. We also show certain more recent results concerning semilinear equations (e.g. Navier-Stokes equations) and equations with non-analytic coefficients. The results on the size of nodal sets are connected to quantitative unique continuation, i.e., on the estimate of the order of vanishing of solutions of PDEs at a point. The results on unique continuation are joint with Ignatova and Camliyurt.
November 28 Yifeng Yu
UC Irvine
Some properties of the mysterious effective Hamiltonian: a journey beyond well-posedness
A major open problem in the periodic homogenization theory of Hamilton-Jacobi equations is to understand "deep" properties of the effective equation, in particular, how the effective Hamiltonian depends on the original Hamiltonian. In this talk, I will present some recent progress in both the convex and non-convex settings.
December 5 Georgios Moschidis
Princeton University
A proof of the instability of AdS spacetime for the Einstein's null dust system
The AdS instability conjecture is a conjecture about the initial value problem for Einstein vacuum equations with a negative cosmological constant. Proposed by Dafermos and Holzegel in 2006, the conjecture states that generic, arbitrarily small perturbations to the initial data of the AdS spacetime, under evolution by the vacuum Einstein equations with reflecting boundary conditions on conformal infinity, lead to the formation of black holes. Following the work of Bizon and Rostworowski in 2011, a vast amount of numerical and heuristic works have been dedicated to the study of this conjecture, focusing mainly on the simpler setting of the spherically symmetric Einstein--scalar field system. In this talk, we will provide the first rigorous proof of the AdS instability conjecture in the simplest possible setting, namely for the Einstein--null dust system, allowing for both ingoing and outgoing dust. This system is a singular reduction of the spherically symmetric Einstein--massless Vlasov system, in the case when the Vlasov field is supported only on radial geodesics. In order to overcome the "trivial" break down occurring once the null dust reaches the center $r=0$, we will study the evolution of the system in the exterior of an inner mirror with positive radius $r_{0}$ and prove the conjecture in this setting. After presenting our proof, we will briefly explain how the main ideas can be extended to more general matter fields, including the regular Einstein--massless Vlasov system.

Winter 2018

Date Speaker Title + Abstract
January 9 Gautam Iyer
January 16 Peter Hintz
UC Berkeley
January 23 Michele Coti-Zelati
Imperial College
January 30 Huy Nguyen
Princeton University
February 6 Ben Dodson
February 13 Bingyuan Liu
UC Riverside
February 13
at 3pm
Laurent Stolovitch
University of Nice
February 20 Klaus Widmayer
Brown University
February 27 Hao Jia
University of Minnesota
March 6 Hanlong Fang
Rutgers University
March 13 Stefan Steinerberger
Yale University

Spring 2018

Date Speaker Title + Abstract
April 10 Katya Krupchyk
UC Irvine
April 17 Anna Mazzucato
Penn State