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.

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 $x=\sum _{k=1}^{\mathrm{\infty }}{a}_k{x}_k$. 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 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.

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.

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.

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.

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.

The HADES seminar on Tuesday, November 26th will be given by Jiaxi Huang in Evans 740 from 3:40 to 5 pm.

Speaker: Jiaxi Huang, USTC

Abstract: In this talk, we will consider the Ericksen-Leslie’s hyperbolic system for compressible liquid crystal model in three spatial dimensions. Global regularity for small and smooth initial data near equilibrium is proved for the case that the system is a nonlinear coupling of compressible Navier-Stokes equations with wave map to ${\mathbb{S}}^2$. Our argument is a combination of vector field method and Fourier analysis. The main strategy to prove global regularity relies on an interplay between the control of high order energies and decay estimates, which is based on the idea inspired by the method of space-time resonances. In particular, the different behaviors of the decay properties of the density and velocity field for compressible fluids at different frequencies play a key role. Joint work with Ning Jiang, Yi-Long Luo, and Lifeng Zhao.

The HADES seminar on Tuesday, November 19th will be given by Benjamin Pineau in Evans 740 from 3:40 to 5 pm.

Speaker: Benjamin Pineau, Berkeley

Abstract: It is a classical result of Leray from the 1930s, that for appropriate initial data and domain, there exists a global weak solution (now known as a Leray-Hopf solution) to the n-dimensional, incompressible Navier-Stokes equations. For n ≥ 3, the question of uniqueness, and regularity of Leray-Hopf solutions remains open. On the other hand, by imposing certain “integrability” conditions on a weak solution, one can often establish global regularity using energy-type arguments. These types of conditions are often referred to as Prodi-Serrin type criteria. In this talk, I will present a relatively simple method for establishing global regularity of a weak solution, provided a certain quantity (e.g. velocity, pressure, etc.) satisfies a particular weak-Lebesgue “integrability” condition. This allows one to generalize several regularity criteria in the literature.

The HADES seminar on Tuesday, November 12th will be given by Thibault Lefebvre in Evans 740 from 3:40 to 5 pm.

Speaker: Thibault Lefebvre, Paris XI

Abstract: In 1985, Burns and Katok conjectured that the marked length spectrum of a negatively-curved Riemannian manifold (namely the collection of lengths of closed geodesics marked by the free homotopy of the manifold) should determine the metric up to isometries. This conjecture was independently proved for surfaces in 1990 by Croke and Otal but since then little progress has been accomplished in higher dimensions until our recent proof of the local version of the conjecture, obtained in collaboration with C. Guillarmou. Considering a geometric point of view in the moduli space of isometry classes, I will explain a new proof of this local version of the conjecture which relies on the notion of geodesic stretch. If time permits, I will show that this fits into a more general framework which generalizes Thurston’s distance and the pressure metric (initially defined on Teichmuller space) to the setting of variable curvature and higher dimensions. Joint work with C. Guillarmou, G. Knieper