Analysis Seminar
20172018
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
USC 
Postponed to November 21 
October 31 
InJee Jeong
Princeton University 
Finite time blowup for strong solutions to the 3D Euler equations
We show finite time blowup 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 scaleinvariant (radially homogeneous) solutions, whose dynamics is governed by a 1D system. Then we make a cutoff argument to ensure finiteness of energy. 
November 7 
Hantaek Bae
UNIST 
Regularity and decay properties of the incompressible NavierStokes equations
In this talk, I will consider the incompressible NavierStokesequations 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 NavierStokes 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 LogLipschitz regularity of the velocity field. 
November 14 
Hung Tran
UWMadison 
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
USC 
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 $(n1)$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. NavierStokes equations) and equations with nonanalytic 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 wellposedness
A major open problem in the periodic homogenization theory of HamiltonJacobi 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 nonconvex 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 Einsteinscalar 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 Einsteinnull dust system, allowing for both ingoing and outgoing dust. This system is a singular reduction of the spherically symmetric Einsteinmassless 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 Einsteinmassless Vlasov system. 
Winter 2018
Date  Speaker  Title + Abstract 

January 9 
Gautam Iyer
CMU 
TBA

January 16 
Peter Hintz
UC Berkeley 
TBA

January 23 
Michele CotiZelati
Imperial College 
TBA

January 30 
Huy Nguyen
Princeton University 
TBA

February 6 
Ben Dodson
JHU 
TBA

February 13 
Bingyuan Liu
UC Riverside 
TBA

February 13 at 3pm 
Laurent Stolovitch
University of Nice 
TBA

February 20 
Klaus Widmayer
Brown University 
TBA

February 27 
Hao Jia
University of Minnesota 
TBA

March 6 
Hanlong Fang
Rutgers University 
TBA

March 13 
Stefan Steinerberger
Yale University 
TBA

Spring 2018
Date  Speaker  Title + Abstract 

April 10 
Katya Krupchyk
UC Irvine 
TBA

April 17 
Anna Mazzucato
Penn State 
TBA
