Analysis Seminar
20182019
Time  Location  Seminar Chair 

Tuesdays 9:45am (Thursdays 11am in 2019)  AP&M 7421  Andrej Zlatos 
Fall 2018
Date  Speaker  Title + Abstract 

October 16 
Benjamin Krause
Caltech 
Discrete analogues in Harmonic Analysis beyond the CalderonZygmund paradigm
Motivated by questions in pointwise ergodic theory, modern discrete harmonic analysis, as developed by Bourgain, has focused on understanding the oscillation of averaging operators  or related singular integral operators  along polynomial curves. In this talk we present the first example of a discrete analogue of polynomially modulated oscillatory singular integrals; this begins to unify the work of Bourgain, Stein, and SteinWainger. The argument combines a wide range of techniques from Euclidean harmonic analysis and analytic number theory. 
October 23 
Yuming Zhang
UCLA 
Porous Medium Equation with a Drift: Free Boundary Regularity
We study regularity properties of the free boundary for solutions of the porous medium equation with the presence of drift. We show that if the initial data has superquadratic growth at the free boundary, then the support strictly expands relative to the streamline, and that the movement is Holder continuous in time. Under additional information of directional monotonicity in space, we derive nondegeneracy of solutions and $C^{1,\alpha}$ regularity of free boundaries. Finally several examples of singularities are given that illustrate differences from the zero drift case. 
October 30 
Sameer Iyer
Princeton University 
Validity of Steady Prandtl Layer Expansions
Consider the vanishing viscosity limit for the 2D steady NavierStokes equations in the region $0\leq x \leq L$ and $0 \leq y<\infty$ with no slip boundary conditions at $y=0.$ For $L\ll 1,$ we justify the validity of the steady Prandtl layer expansion for scaled Prandtl layers, including the celebrated Blasius boundary layer. This is joint work with Yan Guo. 
November 6 
Yu Deng
USC 
Instability of the Couette flow in low regularity spaces
In an exciting paper, J. Bedrossian and N. Masmoudi established the stability of the 2D Couette flow in Gevrey spaces of index greater than 1/2. I will talk about recent joint work with N. Masmoudi, which proves, in the opposite direction, the instability of the Couette flow in Gevrey spaces of index smaller than 1/2. This confirms, to a large extent, that the transient growth predicted heuristically in earlier works does exist and has the expected strength. The proof is based on the framework of the stability result, with a few crucial new observations. I will also discuss related works regarding Landau damping, and possible extensions to infinite time. 
November 13 
Curtis Porter
NC State University 
Nondegeneracy in CR Geometry
CR geometry studies boundaries of domains in $\mathbb{C}^n$ and their generalizations. In characterizing CR structures, a central role is played by the Levi form $L$ of a CR manifold $M$, which measures the failure of the CR bundle to be integrable, so that when $L$ has a nontrivial kernel of constant rank, $M$ is foliated by complex manifolds. If the local transverse structure to this foliation still determines a CR manifold $N$, $M$ is called straightenable, and the TanakaChernMoser classification of CR hypersurfaces with nondegenerate Levi form can be applied to $N$. It remains to classify those $M$ for which $L$ is degenerate and no such straightening exists. This was accomplished in dimension 5 by Ebenfelt, IsaevZaitzev, MedoriSpiro, and Pocchiola. I will discuss their results, my progress on the problem in dimension 7, and my work (joint with Igor Zelenko) modifying Tanaka's prolongation procedure to treat the equivalence problem in arbitrary dimension. 
December 4 
Connor Mooney
UC Irvine 
TBA

Winter 2019
Date  Speaker  Title + Abstract 

January 10 
Tanya Christiansen
University of Missouri 
TBA

January 17 
Oran Gannot
Northwestern University 
TBA

January 24 
Joonhyun La
Princeton University 
TBA

January 31 
Jeffrey Galkowski
Northeastern University 
TBA

February 14 
Benjamin Harrop Griffiths
UCLA 
TBA

March 14 
Burak Erdogan
UIUC 
TBA

March 21 
Jared Speck
Vanderbilt University 
TBA

Spring 2019
Date  Speaker  Title + Abstract 

April 11 
Zaher Hani
University of Michigan 
TBA

May 2 
Katya Krupchik
UC Irvine 
TBA

May 9 
Siming He
Duke University 
TBA
