When does the (Marked) Length Spectrum determine geometry of the billiard table?

The HADES seminar on Tuesday, April 21th will be given by Vadim Kaloshin via Zoom (please contact the organizer at “wangjian at berkeley dot edu” for the Zoom ID) from 3:40 to 5 pm.

Speaker: Vadim Kaloshin, University of Maryland, College Park

Abstract: We study the billiard on the plane ask: does the (Marked) Length Spectrum, i.e., the set of lengths of periodic orbits (together with their labeling), determine the geometry of the billiard table? This question is closely related to the well-known question: “Can you hear the shape of a drum?”

We report two results for planar domains having certain symmetry and analytic boundary. First, we consider billiards obtained by removing from the plane three strictly convex analytic obstacles satisfying the non-eclipse condition and a suitable symmetry. We show that under a non-degeneracy assumption, the Marked Length Spectrum determines the geometry of the billiard table. This is a joint work with J. De Simoi and M. Leguil.

Second, we consider billiards inside of a strictly convex planar domain having certain symmetry. We show that under a non-degeneracy assumptions, the Length Spectrum determines the geometry of the billiard table. This is a joint work with M. Leguil and K. Zhang. These results are analogous to results of Colin de Verdière,  Zelditch and Iantchenko-Sjöstrand-Zworski in terms of the (Marked) Length Spectrum.

Ruelle Zeta Function for Anosov Flows

The HADES seminar on Tuesday, April 14th will be given by Yonah Borns-Weil via Zoom (please contact the organizer at “wangjian at berkeley dot edu” for the Zoom ID) from 3:40 to 5 pm.

Speaker: Yonah Borns-Weil, Berkeley

Abstract: We discuss the meromorphic extension of the Ruelle zeta function for Anosov flows on a compact manifold. This was shown under an orientability condition by Giulietti, Liverani, and Policott in 2012, and then again by Dyatlov and Zworski in 2016 using microlocal analysis. We present the microlocal proof, and give a simple argument to remove the orientability assumption.

Spatially quasi-periodic traveling gravity-capillary waves

The HADES seminar on Tuesday, April 7th will be given by Xinyu Zhao via Zoom (please contact the organizer at “wangjian at berkeley dot edu” for the Zoom ID) from 3:40 to 5 pm.

Speaker: Xinyu Zhao, Berkeley

Abstract: For linearized gravity-capillary waves, it is possible that two periodic waves with different wave numbers travel at the same speed. If the ratio of their wave numbers is irrational, the motion of the superposition of the two waves is spatially quasi-periodic. I will present a numerical study of the spatially quasi-periodic gravity-capillary waves in deep water and introduce a conformal mapping formulation for the wave equations. This is a joint work with Jon Wilkening. 

Neural Networks for Mathematicians

The HADES seminar on Tuesday, February 25th will be given by Boris Hanin in Evans 740 from 3:40 to 5 pm.

Speaker: Boris Hanin, TAMU

Abstract: Neural networks are families of functions used in state-of-the-art approaches to practical problems coming from computer vision (self-driving cars), natural language processing (Google Translate), and reinforcement learning (AlphaGo). After defining what neural networks are and sketching how they are used, I will describe a number of practically important and mathematically interesting questions that arise in trying to understand why they perform so well. These problems touch on random matrix theory, combinatorics, stochastic processes, and ergodic theory.

Bases of non-negative functions in Hilbert spaces and free Banach lattices

The HADES seminar on Tuesday, February 18th will be given by Mitchell Taylor in Evans 740 from 3:40 to 5 pm.

Speaker: Mitchell Taylor, Berkeley

Abstract:  A basis of a Banach space $X$ is a sequence $(x_k)$ in $X$ such that for every $x\in X$ there is a unique sequence of scalars $(a_k)$ such that $x=\sum_{k=1}^\infty a_kx_k$. Examples of bases in $L_2([0,1])$ include the Haar, Walsh, and trigonometric bases. A question arising independently in Engineering and Stochastic PDE is whether $L_p([0,1])$ admits a basis with each of the $x_k$ being a non-negative function. It is a theorem of Bill Johnson and Gideon Schechtman that $L_1$ admits such a basis, and that any non-negative basis in $L_p$ must necessarily be conditional, i.e., it will fail to be a basis if the $(x_k)$ are permuted. In this talk I will give a construction of a non-negative basis in $L_2$, and at the end will discuss non-negative bases in general; in particular, the connection to free Banach lattices.

The Travel Time Tomography Inverse Problem for Transversely Isotropic Elastic Media

The HADES seminar on Tuesday, February 4th will be given by Joey Zou in Evans 740 from 3:40 to 5 pm.

Speaker: Joey Zou, Stanford

Abstract:  I will discuss the travel time tomography problem for the elastic wave equation, where the aim is to recover elastic coefficients in the interior of an elastic medium given the travel times of the corresponding elastic waves. I will consider in particular the transversely isotropic case, which provides a reasonable seismological model for the interior of the Earth or other planets. By applying techniques from boundary rigidity problems, our problem is reduced to the microlocal analysis of certain operators obtained from a pseudo-linearization argument. These operators are not quite elliptic, but they strongly resemble parabolic operators, for which a symbol calculus first constructed by Boutet de Monvel can be applied. I will describe how to use this calculus to solve the problem given certain global assumptions, and if time permits I will discuss current work to modify this calculus in order to solve the problem more locally.

Gaussian Regularization of the Pseudospectrum and Davies’ Conjecture

The HADES seminar on Tuesday, February 11th will be given by Nikhil Srivastava in Evans 740 from 3:40 to 5 pm.

Speaker: Nikhil Srivastava, Berkeley

Abstract: A diagonalizable matrix has linearly independent eigenvectors. Since the set of nondiagonalizable matrices has measure zero, every matrix is a limit of diagonalizable matrices. We prove a quantitative version of this fact: every n x n complex matrix is within distance delta in the operator norm of a matrix whose eigenvectors have condition number poly(n)/delta, confirming a conjecture of E. B. Davies. The proof is based on regularizing the pseudospectrum of an arbitrary matrix with a complex Gaussian perturbation.

Joint work with J. Banks, A. Kulkarni, S. Mukherjee.


On the Cauchy problem for degenerate dispersive equations

The HADES seminar on Tuesday, January 21st will be given by Sung-Jin Oh in Evans 732 from 3:40 to 5 pm.

Speaker: Sung-Jin Oh, Berkeley

Abstract: In plasma physics or fluid dynamics, one sometimes encounters a degenerate dispersive equation, i.e., a nonlinear dispersive equation whose dispersion relation is degenerate (i.e., vanishes at some points). A satisfactory understanding of the Cauchy problem for such equations is still missing, largely due to the appearance of challenging (and interesting!) phenomena from degenerate dispersion, such as the strong focusing of bicharacteristics near the degeneracy.

The purpose of this talk is to provide an introduction to this topic, by focusing on simple examples. In the first part of my talk, I’ll work with simple linear models, namely linear degenerate Schrödinger equations on the line, to demonstrate some key phenomena related to degenerate dispersion. Then in the second part of my talk, I’ll describe some nonlinear illposedness results for a quasilinear degenerate Schrödinger equation on the line, whose proof builds off of the understanding of the linear models. This talk is based on joint work with In-Jee Jeong.

Sub-Riemannian limit of the differential form heat kernels of contact manifolds

The HADES seminar on Tuesday, December 10th will be given by Hadrian Quan in Evans 740 from 3:40 to 5 pm.

Speaker: Hadrian Quan, UIUC

Abstract: In this talk I will report on joint work with Pierre Albin in which we study heat kernel of the Hodge Laplacian on a contact manifold endowed with a family of Riemannian metrics that blow-up the directions transverse to the contact distribution. We apply this to analyze the behavior of global spectral invariants such as the eta-invariant and the determinant of the Laplacian. Time permitting, I will discuss connections with the Rumin complex, and the associated limit of Analytic Torsion.

Propagation of singularities for gravity-capillary waves

The HADES seminar on Tuesday, December 3rd will be given by Hui Zhu in Evans 740 from 3:40 to 5 pm.

Speaker: Hui Zhu, MSRI

Abstract: The surface tension makes free surfaces of fluids instantaneously smooth. For 2D gravity-capillary waves, this phenomenon has been justified by Christianson–Hur–Staffilani and Alazard–Burq–Zuily as local smoothing effects. In this talk, I will present a microlocal justification of this phenomenon for gravity-capillary waves in arbitrary dimensions. My main results are two propagation theorems for some quasi-homogeneous wavefront sets of gravity-capillary waves.