# Baoping Liu (Peking)

The APDE seminar on Monday, 3/6, will be given by Baoping Liu (Peking University) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (sjoh@math.berkeley.edu).

Title: Large time asymptotics for nonlinear Schrödinger equation

Abstract:  We consider the Schrödinger equation with a general interaction term, which is localized in space. Under the assumption of radial symmetry, and uniformly boundedness of the solution in $H^1(\mathbb{R}^3)$ norm, we prove it is asymptotic to a free wave and a weakly localized part.  We derive further properties of the localized part such as smoothness and boundedness of the dilation operator.  This is joint work with A. Soffer.

# Sebastian Herr (Bielefeld)

The APDE seminar on Monday, 2/28, will be given by Sebastian Herr (Bielefeld University) online via Zoom from 9:10am to 10:00am PST (NOTE THE SPECIAL TIME). To participate, email Sung-Jin Oh (sjoh@math.berkeley.edu).

Title: Global wellposedness for the energy-critical Zakharov system below the ground state

Abstract: The Zakharov system is a quadratically coupled system of a Schroedinger and a wave equation, which is related to the focussing cubic Schroedinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that it is globally well-posed in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schr\”odinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate.

# Pierre Germain (NYU)

The APDE seminar on Monday, 2/14, will be given by Pierre Germain (NYU) both in-person (891 Evans) and online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (sjoh@math.berkeley.edu).

Title: Boundedness of spectral projectors on Riemannian manifolds

Abstract: Given the Laplace(-Beltrami) operator on a Riemannian manifold, consider the spectral projector on (generalized) eigenfunctions whose eigenvalue lies in an interval of size $\delta$ around a central value $L$. We ask the question of optimal $L^2$ to $L^p$ bounds for this operator. Some cases are classical: for the Euclidean space, this is equivalent to the Stein-Tomas theorem; and for general manifolds, bounds due to Sogge are optimal for $\delta > 1$. The case $\delta < 1$ is particularly interesting since it is connected with the global geometry of the manifold. I will present new results for the hyperbolic space (joint with Tristan Leger), and the Euclidean torus (joint with Simon Myerson).

# SPECIAL SEMINAR: Svetlana Jitomirskaya (UCI)

In lieu of the regular APDE seminar on Monday, 2/7, I would like to advertise the talk of Svetlana Jitomirskaya (UCI), which will be given in-person (740 Evans) and online via Zoom from 4:10pm to 5:00pm PST. The Zoom meeting room link is:

https://berkeley.zoom.us/j/95575804119

Title: Treating Small Denominators without KAM

Abstract:  Small denominator problems appear in various areas of analysis, PDE, and dynamical systems, including spectral theory of quasiperiodic Schrödinger operators, non-linear Schrödinger equations, and non-linear wave equations. These problems have traditionally been approached by KAM-type constructions. We will discuss the new methods, originally developed in the spectral theory of quasiperiodic Schrödinger operators, that are both considerably simpler and lead to results completely unattainable through KAM techniques. For quasiperiodic operators, these methods have enabled precise treatment of various types of resonances and their combinations, leading to .proofs of sharp (arithmetic) spectral transitions, the ten martini problem, and the discovery of universal hierarchical structures of eigenfunctions.