The APDE seminar on Monday, 2/2, will be given by Patrick Flynn (UCLA) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Linear decay of the beta-plane equation near Couette flow on the plane

Abstract: We prove time decay for the linearized beta-plane equation near shear flow on the plane. Specifically, we show that the profiles of the velocity field components decay polynomially on any compact set, and identify specific rates of decay. Our proof entails the analysis of oscillatory integrals with homogeneous phase and multipliers that diverge in the infinite time limit. To handle this singular limit, we prove a Van der Corput type estimate, followed by a multi-scale asymptotic analysis of the phase and multipliers. This is joint work with Jacob Bedrossian and Sameer Iyer.

The APDE seminar on Monday, 1/26, will be given by our own Sung-Jin Oh (UC Berkeley) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: A physical-space approach to global asymptotics for variable-coefficient Schrödinger equations

Abstract: In this talk, I will discuss a new physical-space approach to establish the time decay and global asymptotics of solutions to variable-coefficient Schrödinger equations in (3+1)-dimensions. A key innovation in our methodology is the concept of a “good commutator,” which extends the classical commuting vector field method, and which combines well with Ifrim-Tataru’s testing by wave packets. As an immediate nonlinear application, we obtain new small data global existence and asymptotics results for quasilinear Schrödinger equations with cubic, Hamiltonian nonlinearity, variable coefficients in their linear part, and possibly outside obstacles. This talk is based on an upcoming work with F. Pasqualotto (UCSD) and N. Tang (UC Berkeley).

The APDE seminar on Monday, 12/1, will be given by Mingfeng Chen (UW Madison) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PDT. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Rational points near space curves

Abstract: How many rational points with a bounded denominator lie in a small neighborhood of a given manifold? This fundamental question in Diophantine approximation connects to dynamics, number theory, and harmonic analysis, with applications to problems like Khinchin’s theorem for manifolds and the dimension growth conjecture.
In this talk, I will present new results that establish the main conjecture for the case of space curves. This is joint work with A. Seeger, R. Srivastava and N. Technau.

The APDE seminar on Monday, 11/24, will be given by Shukun Wu  (IU Bloomington) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PDT. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Weighted L^2 estimates and applications to L^p problems.

Abstract: We will discuss some weighted L^2 estimates in the plane and their applications to a couple of L^p problems. These include the almost everywhere convergence of the planar Bochner-Riesz means, decay of circular L^p-means of Fourier transform of fractal measures, estimates for the maximal Schrödinger operator and the maximal extension operator, and an L^p analogue of the Mizohata–Takeuchi conjecture.

The APDE seminar on Monday, 11/17, will be given by Serban Cicortas (Princeton) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PDT. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Critical collapse in 2+1 gravity

Abstract: Starting with the work of Choptuik ’92, numerical relativity predicts that naked singularity spacetimes arise on the threshold of dispersion and black hole formation, a phenomenon referred to as critical collapse. In this talk, I will present for 2+1 gravity the first rigorous construction of threshold naked singularities in general relativity. Joint work with Igor Rodnianski (Princeton University).

The APDE seminar on Monday, 11/10, will be given by Katya Krupchyk (UC Irvine) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PDT. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Fractional Anisotropic Calderón Problem

Abstract: The classical anisotropic Calderón problem, in its geometric formulation, asks whether a Riemannian metric, or more generally a compact Riemannian manifold with boundary, can be recovered from the Dirichlet-to-Neumann map for the Laplace–Beltrami operator, given on the boundary of the manifold. The problem remains open in general for smooth metrics in dimensions three and higher.
In this talk, we will present uniqueness results for the fractional anisotropic Calderón problem, a nonlocal analogue of the classical anisotropic Calderón problem, in dimensions two and higher, in two settings: on smooth closed Riemannian manifolds with source-to-solution data, and on domains in Euclidean space with external measurements. Specifically, we will show that the source-to-solution map for the fractional Laplace–Beltrami operator, known on an arbitrary open subset of a smooth closed Riemannian manifold, determines the manifold up to isometry. In the Euclidean case, for smooth Riemannian metrics that coincide with the Euclidean metric outside a compact set, we will demonstrate that the partial exterior Dirichlet-to-Neumann map for the fractional Laplace–Beltrami operator, known on arbitrary open subsets in the exterior of the domain, determines the Riemannian metric up to diffeomorphism fixing the exterior. The talk is based on joint works with Ali Feizmohammadi, Tuhin Ghosh, Angkana Rüland, Johannes Sjöstrand, and Gunther Uhlmann.

The APDE seminar on Monday, 11/03, will be given by Robert Jerrard (Toronto) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PDT. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Vortex reconnection in 3d near-critical Abelian Higgs models

Abstract: The critical Abelian Higgs model (AHM) is a system of nonlinear wave equations arising in particle physics. We construct solutions of this system in 3+1 dimensions that exhibit a number of slowly-moving nearly parallel vortex filaments. The leading-order dynamics of this ensemble of filaments are described by a wave map into the modulo space, a manifold carrying a natural Riemannian structure that parametrizes stationary 2D solutions of the AHM. These results allow for the study of the poorly-understood phenomenon of vortex reconnection in this setting. In particular, it is shown that in the regime studied, reconnection is the generic outcome of collisions of pairs of vortex filaments. Extremely similar results are also proved for the critical Abelian Higgs heat flow, modeling certain superconductors, near-critical versions of these equations, and in higher dimensions. This work is joint with Masoud Geevechi.

The APDE seminar on Monday, 10/27, will be given by Beomjong Kwak (KAIST) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PDT. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Global well-posedness of the cubic nonlinear Schrödinger equation on T^2

Abstract: In this talk, we present the global well-posedness for the cubic nonlinear Schrödinger equation for periodic initial data in the mass-critical dimension d=2 for large initial data in H^s, s>0. The result is based on a new inverse Strichartz inequality, which is proved by using incidence geometry and additive combinatorics, such as the inverse theorem for the Gowers uniformity norms by Green-Tao-Ziegler. This is based on joint works with Sebastian Herr.

The APDE seminar on Monday, 10/20, will be given by Allison Byars (UW Madison) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PDT. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Global dynamics for the derivative nonlinear Schrödinger equation

Abstract: We will discuss the long-time dynamics of the derivative nonlinear Schrödinger equation. For small, localized initial data, where no solitons arise, we prove dispersive estimates globally in time. Under the same assumptions, we further prove modified scattering and asymptotic completeness. To the best of our knowledge, this is the first result to achieve an asymptotic completeness theory in a quasilinear setting. Our approach combines the method of testing by wave packets of Ifrim and Tataru, a bootstrap argument, and the Klainerman–Sobolev vector field method.