Andreas Seeger
TITLE: On Spherical Maximal Functions
ABSTRACT: The lectures will be on various types of spherical maximal functions, both in the Euclidean setting and in the setting of two step nilpotent Lie groups. Emphasis will be put on open problems in this subject.
Hong Wang
TITLE: Kakeya sets and union of tubes
ABSTRACT: We discuss techniques (Cordoba argument, projection theory) to prove volume bounds on union of tubes in R^2 and R^3, with applications to the Kakeya problem. This is joint work with Josh Zahl.
David Beltran
TITLE: L^p-L^q bounds for helical maximal averages
ABSTRACT: In 1997 Schlag and Sogge used Fourier analytic tools to prove the sharp range of L^p- L^q bounds, with q>p, for the circular maximal function (which had been previously established by Schlag via geometric arguments). In a recent joint work with Duncan and Hickman we have proved, using also Fourier analysis, a 3-dimensional version of the Schlag--Sogge result in which the circular averages are replaced by helical averages. A key ingredient is a localized variant of a trilinear Fourier restriction estimate for cones.
Matt Blair
TITLE: L^p and nonconcentration estimates for eigenfunctions
ABSTRACT: We will begin by surveying Sogge’s celebrated L^p estimates for eigenfunctions of the Laplacian. On manifolds of nonpositive curvature, it is now known that these estimates admit a logarithmic improvement. We then describe a work in progress in which similar gains are achieved, but the nonpositive curvature condition is replaced by a semi-hyperbolic dynamical assumption. Since the results will concern values of p beneath the critical (Stein-Tomas) exponent, the crucial matter is to show a nonconcentration estimate for L^2 mass when microlocalized in small, frequency-dependent, phase space tubes about a geodesic segment.
Ciprian Demeter
TITLE: Moment inequalities for exponential sums supported on thin sets
ABSTRACT: I will describe a few recent results related to the behavior of both random and deterministic exponential sums.
Larry Guth
TITLE: Bounds on the density of zeroes of the Riemann zeta function in strips
ABSTRACT: We survey bounds about the density of zeroes of the Riemann zeta function and how such bounds relate to the distribution of primes. We discuss recent new work on the density of zeroes by Maynard and me.
Oana Ivanovici
TITLE: Dispersion for 1D Schrödinger on the half line : quantum ping-pong and exponential sums
ABSTRACT: We investigate the one-dimensional semi-classical Schrödinger equation on the half-line, subject to a linear potential and Dirichlet boundary conditions. Our focus is on establishing dispersive and Strichartz estimates for this setting. Analogous to the higher-dimensional Friedlander model, the dispersive estimates exhibit a time-dependent 1/4 -loss, which is both intermittent and sharp.
Regarding Strichartz estimates, we conjecture that the optimal loss is slightly greater than 1/6 (i.e., 1/6 +), a bound that would follow from the generalized Lindelöf hypothesis. Nevertheless, even without this assumption, we prove that Van der Corput’s j-derivative tests enable an improvement over the 1/4 -loss typically derived from the dispersive bounds via the TT* argument. We further expect that analogous Strichartz bounds should hold within the domain of the Friedlander model in higher dimensions.
Matie Machedon
TITLE: Estimates for a system of Hartree-Fock-Bogoliubov type.
ABSTRACT: The HFB equations are a coupled system of non-linear Schr\"odinger equations approximating solutions to the many-body Schr\"odinger equation. I will describe the derivation of these equations (following a paper by M. Grillakis and me, based on earlier work with D. Margetis) and the estimates satisfied by the solution, obtained in recent papers with Jacky Chong , Xin Dong, Manoussos Grillakis and Zehua Zhao. These results were extended to the sharpest available form by Xiaoqi Huang.
Bill Minicozzi
TITLE: Gradient estimates for scalar curvature
ABSTRACT: I will talk about some work with Toby Colding on monotonicity formulas for harmonic functions and an application to a sharp gradient estimate for three manifolds with nonnegative scalar curvature.
Makoto Nakamura
TITLE: Remarks on the Cauchy problem for nonlinear Klein-Gordon equations in the de Sitter spacetime.
ABSTRACT: The Cauchy problem for nonlinear Klein-Gordon equations is considered in the de Sitter spacetime. The local and global well-posedness of the Cauchy problem is considered in Sobolev spaces. Blowing-up solutions are also considered.
Fabrice Planchon
TITLE: Space-time pointwise bounds for the wave green function on convex domains
ABSTRACT: We review how parametrices allow for refinements of fixed time decay estimates and in particular decay away from the (complicated) front set. We then illustrate how this seems required to improve Strichartz estimates on general convex domains as well as making (some) progress on spectral projector estimates, and speculate on the relevance of decoupling in this context.
Wilhelm Schlag
TITLE: Stability analysis of topological solitons and applications of the distorted Fourier transform
ABSTRACT: We will review some orbital and asymptotic stability results in Hamiltonian equations. A common tool in asymptotic stability proofs is given by the distorted Fourier transform. We will briefly review how this tool is derived and describe some of its applications. A particular challenge is to develop this tool for non-selfadjoint matrix operators which commonly arise when linearizing a nonlinear Schrödinger equation around a soliton. Much of the talk will deal with a particular instance of this problem in the setting of Ginzburg-Landau vortices.
Hart Smith
TITLE: Uniform regularity estimates for a family of degenerating subelliptic equations
ABSTRACT: We consider a class of sum-of-squares operators that model small stochastic perturbations of a vector flow. The equations exhibit both subelliptic and semiclassical features. The diffusion terms we consider act only in a subspace of directions, which leads to a subelliptic parabolic equation. The diffusion comes with a small parameter, and the goal is to establish uniform regularity estimates as the parameter approaches 0.
Examples of natural interest include kinetic Brownian motion on a compact manifold, and Fokker-Planck equations with diffusion in the momentum coordinates. I will focus on the latter in this talk, and show that a parametrix can be obtained to arbitrary order using non-isotropic heat kernel methods, with uniform control as the diffusion parameter tends to 0.
John Toth
TITLE: $L^2$ restriction bounds for analytic continuations of quantum ergodic Laplace eigenfunctions.
ABSTRACT: We prove a quantum ergodic restriction (QER) theorem for real hypersurfaces $\Sigma \subet X,$ where $X$ is the Grauert tube associated with a real-analytic, compact Riemannian manifold. As an application, we obtain $h$ independent upper and lower bounds for the $L^2$ - restrictions of the FBI transform of quantum ergodic Laplace eigenfunctions restricted to $\Sigma$ satisfying certain generic geometric conditions. This is joint work with X. Xiao.
Maciej Zworski
TITLE: Classically forbidden regions in twisted bilayer graphene
ABSTRACT: Twisted bilayer graphene (TBG) is a setting of a remarkable theoretical (Bistritzer--MacDonald, 2011) and experimental (Cao et al 2018) discovery, that two sheets of graphene twisted by certain (magic) angles display unusual electronic properties such as superconductivity. The key in the mathematical treatment of this is the existence of protected states in the so-called chiral model (Tarnopolsky--Kruchkov--Vishwanath, 2019).
In this talk I will explain exponential decay (as the angle of twisting goes to zero) of these states near the hexagon spanned by the stacking points (points of high symmetry). Near interior points of the edges it follows from general results (joint with M Hitrik) and based on the geometry of Poisson brackets (with a different proof recently provided by J Sjostrand). Near the stacking points (vertices of the hexagon) it follows from an analytic hypoellipticity argument based on the specific structure of the operator (joint work with Z Tao).