Berkeley Harmonic Analysis and Differential Equations Student Seminar
Fall
2018

Here is a list of links to material and conferences/workshops on PDEs and Harmonic Analysis.

 August 28th 3:40 - 5:00 PM 740 Evans Hall Oran Gannot Semiclassical diffraction by conormal potential singularities. I will describe joint work with Jared Wunsch on propagation of singularities for some semiclassical Schrodinger equations where the potential is conormal to a hypersurface, with applications to logarithmic resonance-free regions. Semiclassical singularities of a given strength propagate across the hypersurface up to a threshold depending on the regularity of the potential and the singularities along certain backwards branching bicharacteristics. September 4th 3:40 - 5:00 PM 740 Evans Hall Daniel Tataru Solitary waves in deep water. Solitary waves are waves on the surface of the water which keep a constant profile and which move with constant velocity. Two longstanding open problems have been whether such waves exist in deep water in the presence of either gravity or surface tension, but not both. This talk will provide the answers to both of these problems in two space dimensions. This is joint work with Mihaela Ifrim. September 11th 3:40 - 5:00 PM 740 Evans Hall Semyon Dyatlov An introduction to hyperbolic dynamics. I will show the Stable/Unstable Manifold Theorem for hyperbolic dynamical systems (maps and flows) which describes long time dynamics in a neighborhood of a given trajectory. Time permitting, I will also give examples of hyperbolic dynamical systems such as geodesic flows on manifolds of negative curvature and dispersive billiards. This talk will follow (a subset) of the notes https://arxiv.org/abs/1805.11660 September 18th 3:40 - 5:00 PM 740 Evans Hall Kevin O'Niell A sharpened inequality for twisted convolution. Consider the trilinear form for twisted convolution on R^{2d}: \mathcal{T}_t(\mathbf{f}) := \iint f_1(x)f_2(y)f_3(x+y)e^{it\sigma(x,y)}dxdy where sigma is a symplectic form and t is a real-valued parameter. It is known that in the case t != 0 the optimal constant for twisted convolution is the same as that for convolution, though no extremizers exist. Expanding about the manifold of triples of maximizers and t = 0 we prove a sharpened inequality for twisted convolution with an arbitrary antisymmetric form in place of sigma. September 25th 3:40 - 5:00 PM 740 Evans Hall Maciej Zworski Internal waves in stratified fluids and 0th order. Colin de Verdi\`ere and Saint-Raymond have recently found a fascinating connection between modeling of internal waves in stratified fluids and spectral theory of 0th order pseudodifferential operators on compact manifolds. The purpose of this talk is to motivate that connection and then explain challenges in spectral theory of 0th order operators, with and without viscosity. Some numerical simulations will illustrate how hyperbolic dynamics of certain classical flows results in concentration of velocities. A sketch of mathematics behind it will then be provided. October 2nd 3:40 - 5:00 PM 740 Evans Hall No HADES. October 9th 3:40 - 5:00 PM 740 Evans Hall No HADES. October 16th 3:40 - 5:00 PM 740 Evans Hall Jian Wang Meromorphic continuations of the resolvent of Laplacians on asymptotically hyperbolic manifolds: Vasy's method. I will present the method introduced by Andr\'as Vasy to prove meromorphic continuations of resolvents of Laplacians on asymptotically hyperbolic spaces in a simple model case. In particular, I will show the proof of Melrose's radial estimates indicating the idea behind the general case. October 23th 3:40 - 5:00 PM 740 Evans Hall James Rowan Almost conservation laws and low regularity well-posedness for nonlinear Schrodinger equations. For energy-subcritical nonlinear Schrodinger equations, the law of conservation of energy can be used to extend the local well-posedness theory for solutions with initial data in H_x^1 to a global well-posedness theory in H_x^1. If we want a global well-posedness theory in a Sobolev space H_x^s for 0 < s < 1, the energy may be infinite, and thus conservation of energy is unavailable. Colliander, Keel, Staffilani, Takaoka, and Tao, building off of earlier ideas of Bourgain, developed a method to prove global well-posedness at regularities below H_x^1 via "almost conserved" quantities. After applying a Fourier multiplier I which is the identity at low frequencies and is decaying at a rate like \xi^{s-1} at high frequencies, the energy of this modified function Iu can be made to be finite, and the time derivative of E[Iu] can be estimated. The fact that the modified energy E[Iu] is "almost-conserved" is enough to extend local well-posedness to global well-posedness. I will present the proof of an almost-conservation law for the three-dimensional cubic NLS due to Colliander et al. and use it to show global well-posedness in H_x^s for s > 5/6. October 30th 3:40 - 5:00 PM 740 Evans Hall Maciej Zworski Introduction to the mathematics of graphene. We will consider the simplest model of graphene given by a hexagonal quantum graph and explain the appearance of the famous "Dirac points". All the relevant concepts, quantum graphs, density of states etc, will be explained from scratch. When the magnetic field is added interesting oscillations appear in physically observed quantities. Using semiclassical methods (with the strength of the magnetic field as the small parameter) we will give a geometric description of the density of states. This description will then be used to see magnetic oscillations such as the de Haas--van Alphen effect. Numerical example will also be presented. November 6th 3:40 - 5:00 PM 740 Evans Hall Colin Guillarmou Fried's conjecture in small dimensions. We explain how to use microlocal methods in order to show Fried's conjecture relating torsion and Ruelle zeta function in dimension 3 and some cases in dimension 5. In higher dimensions we show that the value of the Ruelle zeta function at 0 is a local invariant of the connection (thus independent of the Anosov flow) under certain spectral assumptions, providing new insights toward Fried's conjecture. Joint work with Nguyen Viet Dang, Gabriel Riviere, and Shu Shen. November 13th 3:40 - 5:00 PM 740 Evans Hall Nima Moini. On the problem of interacting bodies (Part I). The state of a system in the low density limit should be described (at the statistical level) by the kinetic density, i.e. by the probability of finding a particle with position x and velocity v at time t. This density is expected to evolve under both the effects of transport and binary elastic collisions, which are expressed in the Boltzmann equation. The Cauchy problem for this equation is still one of the most important open problems, a new concept appearing in the 1989 paper by DiPerna and Lions is the notions of renormalized solutions of transport equation, is the most recent breakthrough, they provide a proof of the global existence of weak solutions via compactness arguments without any a priori estimates on the derivatives. The regularity and uniqueness of these solutions are still open problems. On the other hand, in terms of connection with the physical problem of interacting bodies (liquid, gas, etc), it is necessary to study the qualitative behavior of system of particles with short range potentials, for example particles with short range binary interactions like hard spheres undergo elastic collisions or smooth, monotonic, compactly supported potentials. The point of interest is to show that as the number of the particles increases, behavior of the system will actually converge to the kind of evolution that is being described by the Boltzmann equation. In this first part of this talk, I will present a rigorous derivation of the Boltzmann equation as the low density limit of system of hard spheres based on works of Saint-Raymond, Cercignani, Gerasimenko and Petrina. As for using the DiPerna-Lions theory in this context, the first step would be to understand the counterpart of renormalization at the level of the microscopic dynamics. November 20th 3:40 - 5:00 PM 740 Evans Hall Thanksgiving November 27th 3:40 - 5:00 PM 740 Evans Hall No HADES (MSRI Workshop) December 4th 3:40 - 5:00 PM 740 Evans Hall Albert Ai Low regularity solutions for gravity water waves. We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.

## Upcoming Conferences and Summer Schools

Here is a link for those intersted in student support.

Past:

Chicago Summer School in Analysis: 6/17/2017 - 6/30/2017 at U Chicago.

Hausdorff School: Dispersive Equations, Solitons, and Blow-up: 9/4/2017 - 9/8/2017 at University of Bonn.

Prairie Analysis Seminar 2017: 9/8/17 - 9/9/17 at Kansas State University.

## Expository Articles and Info

Subcritical Scattering for Defocusing NLS, by Jason Murphy (UC Berkeley).

Generalizations of Fourier Analysis, and How to Apply Them, by W.T. Gowers.

A Study Guide for the l^2 Decoupling Theorem, by Jean Bourgain and Ciprian Demeter.

Notes on hyperbolic dynamics, by Semyon Dyatlov.