# Analysis Seminar

## 2018-2019

Time Location Seminar Chair
Thursdays 11am AP&M 7321 Andrej Zlatoš
Fall 2018             Winter 2019 Spring 2019

### 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
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

Cancelled

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
UCLA
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
UIUC
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
A new formulation of multidimensional compressible Euler flow with vorticity and entropy: miraculous geo-analytic structures and applications to shocks
I will describe my recent works, some joint with M. Disconzi and J. Luk, on the compressible Euler equations and their relativistic analog. The starting point is new formulations of the equations exhibiting miraculous geo-analytic structures, including i) a sharp decomposition of the flow into geometric wave and transport-div- curl parts, ii) null form source terms, and iii) structures that allow one to propagate one additional degree of differentiability (compared to standard estimates) for the entropy and vorticity. I will then describe a main application: the study of stable shock formation, without symmetry assumptions, in more than one spatial dimension. I will emphasize the role that nonlinear geometric optics plays in the analysis and highlight how the new formulations allow for its implementation. Finally, I will describe some important open problems, and I will connect the results to the broader goal of obtaining a rigorous mathematical theory that models the long-time behavior of solutions in the presence of shock singularities.

### Spring 2019

Date Speaker Title + Abstract
April 2
(Tue) at 3pm
Franc Forstnerič
University of Ljubljana
Minimal surfaces by way of complex analysis
After a brief historical introduction, I will present some recent developments in the theory of minimal surfaces in Euclidean spaces which have been obtained by complex analytic methods. The emphasis will be on results pertaining to the global theory of minimal surfaces including Runge and Mergelyan approximation, the conformal Calabi-Yau problem, properly immersed and embedded minimal surfaces, and a new result on the Gauss map of minimal surfaces.
April 4 Yuming Zhang
UCLA
An obstacle problem in parallel search in marketing
We study an obstacle problem coming from consumer search in a product market. I will discuss several properties concerning the geometry of the free boundary. The difficulty is to determine how the geometry depends on the dimension d. This is a joint work with T. Tony Ke, Wenpin Tang and J. Miguel Villas-Boas.
April 11 Zaher Hani
University of Michigan
On the kinetic description of the long-time behavior of dispersive PDE
Wave turbulence theory claims that at very long timescales, and in appropriate limiting regimes, the effective behavior of a nonlinear dispersive PDE on a large domain can be described by a kinetic equation called the "wave kinetic equation". This is the wave-analog of Boltzmann's equation for particle collisions. We shall consider the nonlinear Schrodinger equation on a large box with periodic boundary conditions, and explore some of its effective long-time behaviors at time scales that are shorter than the conjectured kinetic time scale, but still long enough to exhibit the onset of the kinetic behavior. (This is joint work with Tristan Buckmaster, Pierre Germain, and Jalal Shatah).
April 18 Hart Smith
University of Washington
Dispersive estimates for the wave equation on manifolds of bounded curvature
I will discuss some recent work that establishes local-in-time dispersive estimates for the wave equation on compact Riemannian manifolds, assuming just uniform bounds on the sectional curvatures and some mild a priori regularity for the metric tensor. The estimates we obtain include $L^2$-$L^p$ bounds on unit width spectral projection operators, as well as the full range of Strichartz estimates. The proof combines elements of paradifferential and Fourier integral operator theory. This work is joint with Y. Chen.
April 25 In-Jee Jeong
Korea IAS
On the Cauchy problem for the Hall-MHD system without resistivity
The Hall-magnetohydrodynamics (MHD) system is obtained from the ideal MHD system by incorporating a quadratic second-order correction, called the Hall current term, that takes into account the motion of electrons relative to positive ions. In recent work with Sung-Jin Oh, we investigated the Cauchy problem in the irresistive case. We first study the linearized systems around a special class of stationary magnetic fields with certain symmetries, and obtain ill- and well-posedness results, depending on the profile of the magnetic field. We then pass from linear to nonlinear results: near a non-zero constant magnetic field, the system is well-posed but it is ill-posed (in the strongest sense of Hadamard) near the trivial magnetic field. We are mainly guided by the behavior of bicharacteristics for the principal symbol. The key tools are: dispersive smoothing in the well-posedness case and construction of degenerating wave packets together with a systematic use of a generalization of the energy identity in the ill-posedness case.
May 2 Katya Krupchyk
UC Irvine
Inverse boundary problems for elliptic PDE in low regularity setting
In this talk, we shall discuss recent progress in the global uniqueness issues for inverse boundary problems for second order elliptic equations, such as the conductivity and magnetic Schrodinger equations, with low regularity coefficients. Generally speaking, in an inverse boundary problem, one wishes to determine the coefficients of a PDE inside a domain from the knowledge of its solutions along the boundary of the domain. While ubiquitous in practice, the mathematical analysis of such problems is quite challenging, and the consideration of the low regularity setting, motivated by applications, brings additional substantial difficulties. In this talk, we shall discuss the case of full, as well as partial, measurements, both for domains in the Euclidean space, as well as in the more general setting of transversally anisotropic compact Riemannian manifolds with boundary. Some of the important ingredients in our approach are semiclassical Carleman estimates with limiting Carleman weights with an optimal gain of derivatives, precise smoothing estimates, as well as a construction of Gaussian beam quasimodes in a low regularity setting. This is joint work with Gunther Uhlmann.
May 7
(Tue) at 2pm in APM 7218
Keegan Flood
University of Auckland
C-projective metrizability and CR submanifolds
The c-projective metrizability equation is an invariant overdetermined linear geometric PDE on an almost c-projective manifold governing the existence of quasi-Kahler metrics compatible with the c-projective structure. I will show that the degeneracy locus of a solution to the c-projective metrizability equation satisfying a generic condition on its prolonged system is a smoothly embedded submanifold of codimension 1 which inherits a partially-integrable nondegenerate almost CR structure. Phrased differently, this result explicitly links the Levi-form of the boundary CR structure of a c-projectively compact quasi-Kahler manifold satisfying a non-vanishing "generalized scalar curvature" condition to the interior metric.
May 9 Siming He
Duke University
Suppression of Chemotactic collapse through fluid-mixing and fast-splitting
The Patlak-Keller-Segel equations (PKS) are widely applied to model the chemotaxis phenomena in biology. It is well-known that if the total mass of the initial cell density is large enough, the PKS equations exhibit finite time blow-up. In this talk, I present some recent results on applying additional fluid flows to suppress chemotactic blow-up in the PKS equations. These are joint works with Jacob Bedrossian and Eitan Tadmor.
May 9
at 2pm in APM 7218
Jeffrey Case
Penn State
Sharp Sobolev trace inequalities via conformal geometry
Escobar proved a sharp Sobolev inequality for the embedding of $W^{1,2}(X^{n+1})$ into $L^{2n/(n-1)}(\partial X)$ by exploiting the conformal properties of the Laplacian in X and the normal derivative along the boundary. More recently, an alternative proof was given by using a Dirichlet-to-Neumann operator along the boundary and its close relationship to the 1/2-power of the Laplacian. In this talk, I describe a new relationship between the conformally covariant fractional powers of the Laplacian due to Graham--Zworski and higher-order Dirichlet-to-Neumann operators in the interior, and use it to prove sharp Sobolev inequalities for embeddings of $W^{k,2}$. Other consequences of this relationship, such as a surprising maximum principle for the conformal 3/2-power of the Laplacian, will also be discussed.
May 16 Yao Yao
Georgia Tech
Radial symmetry of stationary and uniformly-rotating solutions in 2D incompressible fluid equations
In this talk, I will discuss some recent work on radial symmetry property for stationary or uniformly-rotating solutions for 2D Euler and SQG equation, where we aim to answer the question whether every stationary/uniformly-rotating solution must be radially symmetric, if the vorticity is compactly supported. This is a joint work with Javier Gómez-Serrano, Jaemin Park and Jia Shi.
May 30 Gautam Iyer
Carnegie Mellon
Anomalous diffusion in one and two dimensional combs
We study the effective behavior of a Brownian motion in both one and two dimensional comb like domains. This problem arises in a variety of physical situations such as transport in tissues, and linear porous media. We show convergence to a limiting process when when both the spacing between the teeth, and the probability of entering a tooth vanish at the same rate. This limiting process exhibits an anomalous diffusive behavior, and can be described as a Brownian motion time-changed by the local time of an independent sticky Brownian motion. At the PDE level, this leads to equations that have fractional time derivatives and are similar to the Bassett differential equation.
May 30
at 3pm in APM 7218
Hang Xu
Johns Hopkins
On the asymptotic properties of the Bergman kernel
Consider the Bergman kernel associated to the tensor power of a positive line bundle on a compact Kähler manifold. We will present our work on its near-diagonal asymptotic and off-diagonal decay properties. This is joint work with H. Hezari and Z. Lu.
June 6 Steve Shkoller
UC Davis
Water waves with time-dependent and deformable angled crests (or corners)
I will describe a new set of estimates for the 2d water waves problem, in which the free surface has an angled crest (or corner) with a time-dependent angle that changes with the evolution of the water wave, and with a corner vertex that can move in all directions. There are no symmetry constraints on the crest, and the fluid can have bulk vorticity. This is joint work with D. Coutand.