The HADES seminar on Tuesday, September 22nd will be given by Jonathan Luk via Zoom from 3:40 to 5 pm.

Speaker: Jonathan Luk, Stanford

Abstract: I will discuss the construction of a class of low-regularity (merely $W^{1,2}$) solutions to the Einstein vacuum equations which have the property that the solutions are foliated by $2$-spheres so that the metric is more regular along the tangential directions of the $2$-spheres. I will first discuss a model semi-linear problem, then introduce the relevant geometric setup and give a sketch of the proof. This type of singular solutions is relevant to the problems of impulsive gravitational waves, high-frequency limits, null dust shells and the formation of trapped surfaces in general relativity (discussed in the Analysis and PDE seminar on 9/21).

The HADES seminar on Tuesday, September 15th will be given by Sung-Jin Oh via Zoom (please contact the organizer at “james_rowan at berkeley dot edu” for the Zoom ID) from 3:40 to 5 pm.

Speaker: Sung-Jin Oh

Abstract: Since the pioneering works of Krieger-Schlag-Tataru, Rodnianski-Sterbenz, Raphaël-Rodnianski and Merle-Raphaël-Rodnianski, the construction of finite-time blow-up solutions to geometric dispersive equations has been a topic of immense interest. In this expository talk, after a brief general review, I’ll try to describe in some detail the modulation-theoretic scheme of Raphaël-Rodnianski for constructing a finite time blow-up solution with smooth initial data. If time permits, I’ll present briefly a work-in-progress on an analogous blow-up construction for the self-dual Chern-Simons-Schrödinger equation (joint with Kihyun Kim and Soonsik Kwon).

The HADES seminar on Tuesday, September 29th will be given by Maciej Zworski via Zoomfrom 3:40 to 5 pm.

Speaker: Maciej Zworski

Abstract: As analysts we are used to smooth functions of compact support and after constructing one example of a bump function we are happy to apply it for many purposes. We also know that for any sequence of numbers we can construct a smooth function with that sequence as coefficients of its Taylor series. Can that map from sequences to functions be made linear? The answer is no for all sequences but yes for sequences satisfying certain growth conditions. I will prove the Denjoy–Carleman theorem which shows what growth is needed if you want to keep compact support, describe Carleson’s moment problem and talk about characterization of an important subclass of Gevrey functions. Those functions appear naturally in the theories of diffraction, of Landau diffusion for the Boltzmann equation, and of trace formulas for Anosov flows.

The HADES seminar on Tuesday, September 8will be given by James Rowan via Zoom (please contact the organizer at “james_rowan at berkeley dot edu” for the Zoom ID) from 3:40 to 5 pm.

Speaker:  James Rowan

Abstract:  I will present a recent paper by Merle, Raphael, Rodnianski, and Szeftel which constructs a new kind of blowup solution for certain supercritical nonlinear Schrodinger equations.  The mechanism is neither a rapid frequency cascade nor concentration of a [quasi]soliton, but rather a highly-oscillatory front blowup coming from a collection of special solutions to the self-similar spherically symmetric Euler equations.  The construction relies on studying the behavior of a wave equation in the phase and modulus variables and a fixed point argument to control the behavior of unstable modes.  Along the way I hope to showcase some common techniques in the study of nonlinear PDEs.

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.

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.

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. 

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 $x=\sum_{k=1}^\infty a_kx_k$. 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 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.