Analysis Seminar


Time Location Seminar Chair
Thursdays 11am (Tuesdays 9:45am in Fall 2018) AP&M 7321 Andrej ZlatoŇ°

Fall 2018

Date Speaker Title + Abstract
October 16 Benjamin Krause
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
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
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
Singular Solutions to Parabolic Systems
Regularity results for linear elliptic and parabolic systems with measurable coefficients play an important role in the calculus of variations. Morrey showed that in two dimensions, solutions to linear elliptic systems are continuous. We will discuss some surprising recent examples of discontinuity formation in the plane for the parabolic problem.

Winter 2019

Date Speaker Title + Abstract
January 10 Tanya Christiansen
University of Missouri


January 17 Oran Gannot
Northwestern University
Semiclassical diffraction by conormal potential singularities
I will describe joint work with Jared Wunsch on propagation of singularities for some semiclassical Schrodinger equations where the potential has singularities normal to an interface. Semiclassical singularities of a given strength propagate across the interface, but only up to a threshold. This is due to diffracted singularities which are weaker than the incident singularity by a factor depending on the regularity of the potential. Time permitting, I will give applications to logarithmic resonance-free regions in scattering theory.
January 24 Joonhyun La
Princeton University
On a kinetic model of polymeric fluids
In this talk, we prove global well-posedness of a system describing behavior of dilute flexible polymeric fluids. This model is based on kinetic theory, and a main difficulty for this system is its multi-scale nature. A new function space, based on moments, is introduced to address this issue, and this function space allows us to deal with larger initial data.
January 31 Jeffrey Galkowski
Northeastern University
Concentration and Growth of Laplace Eigenfunctions
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of $L^2$ mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration including high $L^p$ norms and Weyl laws; in each case obtaining quantitative improvements over the known bounds.
February 14 Benjamin Harrop Griffiths
Vortex filament solutions of the Navier-Stokes equations
From Helmholtz to vaping hipsters, the dynamics of vortex filaments, i.e. fluids with vorticity concentrated along a smooth curve, has been a topic of significant interest in fluid dynamics. The global well-posedness of vortex filaments with small circulation follows from the theory of mild solutions of the 3d Navier-Stokes equations at critical regularity. However, for filaments with large circulation these results no longer apply. In this talk we discuss a proof of well-posedness (in a suitable sense) for vortex filaments of arbitrary circulation. Besides their physical interest, these results are the first to give well-posedness in a neighborhood of large self-similar solutions of the 3d Navier-Stokes without additional symmetry assumptions. This is joint work with Jacob Bedrossian and Pierre Germain.
March 7
at 2pm in AP&M 5829
Dimitri Zaitsev
Trinity College Dublin
Geometry of real hypersurfaces meets Subelliptic PDEs
In his seminal work from 1979, Joseph J. Kohn invented his theory of multiplier ideal sheaves connecting a priori estimates for the d-bar problem with local boundary invariants constructed in purely algebraic way. I will explain the origin and motivation of the problem, and how Kohn's algorithm reduces it to a problem in local geometry of the boundary of a domain. I then present my work with Sung Yeon Kim based on the technique of jet vanishing orders, and show how it can be used to control the effectivity of multipliers in Kohn's algorithm, subsequently leading to precise a priori estimates.
March 14 Burak Erdogan
Fractal solutions of dispersive PDE on the torus
In this talk we discuss qualitative behavior of certain solutions to linear and nonlinear dispersive partial differential equations such as Schrodinger and Korteweg-de Vries equations. In particular, we will present results on the fractal dimension of the solution graph and the dependence of solution profile on the algebraic properties of time.
March 19
(Tue) at 3pm
Nicoletta Tardini
Universita di Firenze
Cohomological properties of complex manifolds
The $\partial\overline\partial$-lemma is an important obstruction to Kahlerianity on compact complex manifolds. In this talk we will describe the relations between this property and the cohomology groups that one can define on complex manifolds. These are joint works with Daniele Angella, Tatsuo Suwa and Adriano Tomassini.
March 21 Jared Speck
Vanderbilt University

Spring 2019

Date Speaker Title + Abstract
April 4 Yuming Zhang
April 11 Zaher Hani
University of Michigan
April 18 Hart Smith
University of Washington
May 2 Katya Krupchik
UC Irvine
May 9 Siming He
Duke University
May 16 Yao Yao
Georgia Tech