Analysis Seminar

2018-2019

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 Calderon-Zygmund 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 Stein-Wainger. 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 super-quadratic 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 Navier-Stokes 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 Tanaka-Chern-Moser 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, Isaev-Zaitzev, Medori-Spiro, 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