# Geometry & Analysis Seminar

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Fall 2021 G&A Zoominar
Wednesdays @ 4pm California Time
Meeting ID: 933 9188 4256
Passcode: seminar
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Professor Kasia Jankiewicz

Schedule (click dates for title and abstract)

 10/6 Joey Zou UC Santa Cruz 10/20 Xiaodong Wang Michigan State University 11/3 Bo Zhu University of Minnesota 11/17 Kyle Miller UC Santa Cruz 12/1 Yang Zhang University of Washington

Wednesday, Oct 6, at 4pm PST

Joey Zou, UC Santa Cruz

Microlocal Analysis with Applications to Seismic Inverse Problems

I will discuss the travel time tomography problem for elastic media, which asks to recover the composition of an elastic material given the data of the travel times of the material's elastic waves. This problem is particularly important in the transversely isotropic setting, as transverse isotropy forms a common model for the composition of the Earth's interior. The mathematical study of this problem relates to X-ray tomography and boundary rigidity problems studied by de Hoop, Stefanov, Uhlmann, Vasy, et al., which reduce the inverse problems to the microlocal analysis (i.e. analysis in phase space) of certain operators obtained from a pseudo-linearization argument; however in this setting the analysis is more subtle, as the operators obtained are somewhat degenerate (they resemble parabolic operators in a particular sense, rather than elliptic operators in previous works). In this talk, I will explain how to reduce the travel time tomography problem to a problem in microlocal analysis, as well as the analysis required to accommodate the operators involved.

Wednesday, Oct 20, at 4pm PST

Xiaodong Wang, Michigan State University

Uniqueness Results on a geometric PDE in Riemannian and CR Geometry Revisited

I will discuss some uniqueness results for a geometric nonlinear PDE related to the scalar curvature in Riemannian geometry and CR geometry. In the Riemannian case I will present a new proof of the uniqueness result assuming only a positive lower bound for Ricci curvature. I will explain how the same principle can be used in the CR case to establish a Jerison-Lee identity in a more general setting and a stronger uniqueness result.

Wednesday, Nov 3, at 4pm PST

Bo Zhu, University of Minnesota

Geometry of positive scalar curvature on complete 3-manifolds

In this talk, we first introduce some conjectures related to scalar curvature in the context of size geometry. Then, we will particularly discuss how the strictly positive scalar curvature affects the volume growth of the geodesic ball in complete three-dimensional manifolds with nonnegative scalar curvature. We will also cover related progress on this topic if time permits.

Wednesday, Nov 17, at 4pm PST

Kyle Miller, UC Santa Cruz

The homological arrow polynomial

The Kauffman bracket polynomial is an invariant of framed unoriented knots and links in the 3-sphere, giving the Jones polynomial via skein relations. Applying these skein relations to links in other 3-manifolds yields invariants in the Kauffman bracket skein module, which is a relatively complicated object. I will talk about a multivariable polynomial invariant for links in thickened surfaces that is obtained as a functional on the skein module, an evaluation of which coincides with the arrow polynomial of Kauffman and Dye. This polynomial is an invariant of virtual links (i.e., it is unchanged under destabilization along vertical annuli in the complement), and it has applications to checkerboard colorability and in proving nonexistence of essential vertical annuli.

Wednesday, Dec 1, at 4pm PST

Yang Zhang, UC Santa Cruz

Inverse boundary value problems for nonlinear hyperbolic equations on Lorentzian manifolds

Inverse problems of recovering the metric and nonlinear terms were originated in the work by Kurylev, Lassas, and Uhlmann for the semilinear wave equation $$square_g u(x) + a(x)u^2(x) = f(x)$$ in a manifold without boundary. The idea is to use the linearization and the nonlinear interactions of distorted plane waves to produce point source like singularities in an observable set. In this talk, I will discuss joint work with Gunther Uhlmann which considers the recovery of the metric and nonlinear terms for a quadratic derivative nonlinear wave equation on a Lorentzian manifold with boundary. The main difficulty that we need to handle here is caused by the presence of the boundary. I will first overview two related inverse boundary problems for semilinear wave equations considered by Hintz, Uhlmann, and Zhai. Our work builds on these previous results and then I will discuss our methods to overcome the difficulties