# Geometric Analysis Seminar

## Geometric Analysis Seminar

This seminar will promote research in mathematical and physical topics which employ the use and creation of geometric analysis techniques. The goal of the presentations is to disseminate these topics and techniques to a mathematical research audience with expertise in analysis and partial differential equations (PDEs).

Some of the main themes the seminar will explore are

1. Eigenvalue problems for PDEs on manifolds,
2. Geometric flows.
3. Inverse problems for PDEs and geometric settings.
4. The introduction of microlocal techniques to a broader analysis audience.

The presentations will be scheduled for 45 minutes, with an additional 15 minutes for questions. The presentation structure will be about 10-15 minutes of introduction and literature review, 15 minutes for the exploration of the topic, and about 15-20 minutes for in-depth discussions or proofs.

## Multi-marginal optimal transport and graph theory

Multi-marginal optimal transport in the general mathematical problem of aligning several probability distributions with maximal efficiency, relative to a given cost function. While many applications for this problem have emerged over the past several years, in economics, physics, statistics and finance, among other areas, the structure of solutions is very delicate and depends on the cost function in ways that are still only partially understood. In this talk, I will introduce the problem and briefly outline the known theory of multi-marginal optimal transport, illustrating the theory with a few simple and intuitive examples. I will then go on to describe recent joint work with PhD student Adolfo Vargas-Jimenez on problems with cost functions coming from an underlying graph.

Brendan Pass is an associate professor in the Department of Mathematical and Statistical Sciences at the University of Alberta (Edmonton, Alberta, Canada). He works primarily on optimal transport, in particular multi-marginal problems, on which he is among the world’s leading experts. Pass is one of the founders of the Kantorovich Initiative, a nascent organization focused on interdisciplinary optimal transport research and supported by the Pacific Institute for the Mathematical Sciences (PIMS) and the US National Science Foundation (NSF). Pass’ 2011 PhD thesis at the University of Toronto garnered him the 2012 Cecil Graham Doctoral Dissertation Award from the Canadian Applied and Industrial Mathematics Society (CAIMS), and he has recently been awarded the 2021 CAIMS - PIMS Early Career Award, in recognition of his contributions to optimal transport theory.

## Inverse Problems for Nonlinear PDEs

In this talk, I will demonstrate the higher order linearization approach to solve several inverse boundary value problems for nonlinear PDEs modeling nonlinear electromagnetic optics including nonlinear time-harmonic Maxwell’s equations with Kerr-type and second harmonic generation nonlinearity. The problem will be reduced to solving for the coefficient functions from their integrals against multiple linear solutions. We will focus our discussion on different choices of linear solutions. A similar problem for nonlinear magnetic Schrodinger equation will be considered as a comparison.

Ting Zhou received her PhD in Mathematics at the University of Washington under the supervision of Gunther Uhlmann. Dr. Zhou was a CLE Moore instructor at MIT for three years (2011-2014), after which she held an associate professor position at Northeastern University until 2021. Dr. Zhou was awarded a Sloan Fellowship in 2015. Dr. Zhou is now a professor in the Department of Mathematics at Zhejiang University.

## Reconstruction of a manifold from travel time data

In this talk I will introduce several geometric data sets related to the distance function either on Riemannian or Finsler manifolds with boundary. I consider differences of these data sets, and for each of them I will provide geometric conditions that are sufficient to determine the isometry class of the manifold producing the data. This talk is based on joint works with Maarten V. de Hoop, Joonas Ilmavirta, Matti Lassas and Ella Pavlechko.

Teemu Saksala is an assistant professor at NC State University, Raleigh, North Carolina, USA who is working on geometric inverse problems arising from seismology. In this field, the general theme is to understand whether a geometric property in conjunction with some a priori information about the underlying geometric structure, is sufficient to determine the whole geometric structure. The isometry invariance is the natural gauge of these kinds of problems. Dr. Saksala got his PhD in 2017 at University of Helsinki, Finland. Prior to joining NC State he worked as a Simons postdoctoral fellow at Rice University from 2017-2020.

## Hypoellipticity via sums of squares

Many results on hypoellipticity of second order operators rely on the assumption that the operator can be written as a sum of squares of vector fields (e.g. Hormander's bracket condition, and Christ's hypoellipticity theorem for infinitely degenerate operators). For operators that are not sub elliptic and not sums of squares, hypoellipticity has been only proved in some very special cases, for example, when L=L_1+g(x)L_2 and L_1 and L_2 are subelliptic.
In this talk I will address the question of hypoellipticity for a general divergence form operator, whose matrix is comparable, but not necessarily equal, to a diagonal matrix of a special form. The idea is to find sharp sufficient conditions which guarantee that a smooth positive matrix can be written as a sum of squares of positive dyads with sufficient degree of smoothness. Interestingly, this question has not been completely resolved even for scalar positive functions.

I work in the area of Real Analysis, more precisely, degenerate elliptic PDEs, studying properties of solutions to such equations and associated metric measure spaces. I received my PhD from the University of Calgary in 2013, and I have been an Assistant Professor at Reed College since 2017 after completing postdoc appointments at the University of Pennsylvania, and at McMaster University.

## Lorentzian Calderón problem under curvature bounds

We introduce a method of solving inverse boundary value problems for wave equations on Lorentzian manifolds and show that zeroth order coefficients can be recovered under certain curvature bounds. The set of Lorentzian metrics satisfying the curvature bounds has a non-empty interior in the sense of smooth, compactly supported perturbations of the metric, whereas all previous results on this problem impose conditions on the metric that force it to be real analytic with respect to a suitably defined time variable.  The analogous problem on Riemannian manifolds is called the Calderón problem, and in this case the known results require the metric to be independent of one of the variables. Our approach is based on a new unique continuation result in the exterior of the double null cone emanating from a point. The approach shares features with the classical Boundary Control method and can be viewed as a generalization of this method to cases where no real analyticity is assumed. The talk is based on joint work with Spyros Alexakis and Lauri Oksanen.

Ali Feizmohammadi received his Ph.D in June 2018 under the supervision of Prof. Spyros Alexakis and Prof. Adrian Nachman. From 2018 to 2021 Dr. Feizmohammadi was a Research Associate at University College London working with Prof. Lauri Oksanen. Since 2021, he is a Simons postdoctoral fellow at the Fields institute for research in mathematical sciences. His research interests lie in partial differential equations and geometric analysis, especially the rapidly expanding field of inverse problems. Dr. Feizmohammadi has also worked on finite element methods associated to data assimilation and control problems subject to wave equations.

## Inverse boundary problems for biharmonic operators and nonlinear PDEs on Riemannian manifolds

In an inverse boundary problem, one seeks to determine the coefficients of a PDE inside a domain, describing internal properties, from the knowledge of boundary values of solutions of the PDE, encoding boundary measurements. Applications of such problems range from medical imaging to non-destructive testing. In this talk, starting with the fundamental Calderon inverse conductivity problem, we shall first discuss inverse boundary problems for first order perturbations of biharmonic operators in the setting of compact Riemannian manifolds with boundary. Specifically, we shall present a global uniqueness result as well as a reconstruction procedure for the latter inverse boundary problem on conformally transversally anisotropic Riemannian manifolds of dimensions three and higher. Finally, we shall also discuss briefly inverse boundary problems for nonlinear magnetic Schroedinger operators on a compact complex manifold, illustrating the recent insight that the presence of nonlinearity may help when solving inverse problems.

Lili Yan is currently a PhD candidate at the University of California, Irvine. Her PhD supervisor is Katya Krupchyk. Lili's research interests are in the ﬁeld of partial diﬀerential equations, inverse problems, and semiclassical analysis.

## The Neumann problem in graph Lipschitz domains in the plane

New aspects of the solvability of the classical Neumann boundary value problem in a graph Lipschitz domain in the plane will be presented. When the domain is the upper half-plane and the boundary data is assumed to belong to weighted  Lebesgue or weighted Lorentz spaces, it will be shown that the solvability of the Neumann problem in these settings may be characterized in terms of Muckenhoupt weights and related weights, respectively. For a general graph Lipschitz domain $\Omega$, as proved in an unpublished work by E. Fabes and C. Kenig, there exists $\varepsilon_\Omega>0$ such that the Neumann problem is solvable with data in $L^p(\partial\Omega)$ for $1<p<2+\varepsilon_\Omega;$ it will be shown that the Neumann problem is solvable at the endpoint $2+\varepsilon_\Omega$ with data in the  Lorentz space $L^{2+\varepsilon_\Omega,1}(\partial\Omega).$ Examples of the results in Schwarz-Christoffel Lipschitz domains and related domains will be given. This is joint work with María Jesús Carro (Universidad Complutense de Madrid) and Carmen Ortiz-Caraballo (Universidad de Extremadura).

Virginia Naibo earned her undergraduate degree, or Licenciatura, in mathematics from Universidad Nacional de Rosario and her doctorate in mathematics from Universidad Nacional del Litoral, Argentina. She held a three-year postdoctoral position at the University of Kansas and was a tenure-track assistant professor at Rose-Hulman Institute of Technology for a year before joining the faculty of the mathematics department at Kansas State University, where she is professor and associate department head.
Naibo’s research interests are in the area of Fourier analysis; in a broad sense, this branch of mathematics allows the study of signals, such as sounds and images, by breaking them down into fundamental pieces that are less complex and, therefore, easier to examine. Her more recent work concerns the study of different aspects of linear and bilinear pseudodifferential operators and singular integrals, Leibniz-type rules, commutator estimates and function spaces, among other topics. Applications of her work to analysis and partial differential equations include pointwise multiplication properties of function spaces, well-posedness results for Euler, Navier-Stokes and Korteweg-de Vries equations as well as for the Ideal Magneto Hydrodynamic equations, smoothing properties of Schrödinger semigroups, and scattering properties of solutions to systems of partial differential equations associated to local and nonlocal operators.

## A Harnack inequality for certain degenerate/singular elliptic PDEs shaped by convex functions

Certain convex functions on $R^n$ produce quasi-distances and Borel measures on $R^n$. Furthermore, their Hessians can shape degenerate/singular elliptic operators. We will describe geometric and measure-theoretic conditions on convex functions that allow to prove a Harnack inequality for nonnegative solutions to their associated elliptic operators.

Diego Maldonado is a Professor in the analysis group in the Department of Mathematics at Kansas State University. He researches singular elliptic PDEs, with a focus on regularity estimates for the Monge-Ampère equation and singular Calderón-Zygmond integrals.

## Forty Years of Calderón's Problem

Calderón's problem (also called electrical impedance tomography) asks the question of whether one can determine the electrical conductivity of a medium by making voltage and current measurements at the boundary. I will give a survey of some of the progress made on this problem, including the more recent progress on solving similar problems for nonlinear equations and nonlocal operators.

Gunther Uhlmann is a Walker Family Endowed Professor of Mathematics at the University of Washington. He is also currently a Si Yuan Professor in the Institute for Advanced Study at the Hong Kong University of Science and Technology and a Fellow of the American Mathematical Society (AMS). His research has made seminal contributions to the fields of inverse problems, microlocal analysis, scattering theory, and image analysis. For his work in inverse problems, which includes his work on Calderón’s Problem, he has been awarded numerous distinguished positions, honours and prizes, which include: the AMS Bochner Memorial Prize (2011), the SIAM Kleinman Prize (2011), the Soloman Lefschetz Medal (2017), and the AMS-SIAM Birkhoff Prize in Applied Mathematics (2021).