The HADES seminar on Tuesday, October 14th, will be at 3:30pm in Room 740.

Speaker: Zachary Lee

Abstract: The Nonlinear Schrödinger Equation (NLS) arises in various physical contexts, notably in models of Bose–Einstein condensation and nonlinear optics. I will begin by outlining these motivations and by presenting several heuristics—scaling, dispersion, and symmetry—that shed light on the qualitative behavior of its solutions. I will then turn to a rigorous analysis based on the Duhamel formulation of the equation, together with Strichartz estimates, conservation laws, and Morawetz inequalities, which provide global control for data in the defocusing case. In the final part of the talk, I will describe how these techniques can be adapted below the energy space using almost conservation laws (the I-method), and present a new global existence result for the one-dimensional defocusing septic NLS for a class of discontinuous and unbounded initial data.

The HADES seminar on Tuesday, September 30th, will be at 3:30pm in Room 740.

Speaker: Rui Liang

Abstract:In this talk, we will consider the Schrödinger equation with cubic nonlinearity on the circle, with initial data distributed according to the Gibbs measure.  We will discuss the challenges and strategies involved in establishing the Poincaré recurrence property with respect to the Gibbs measure in the full dispersive range. This work, using the theory of the random averaging operator developed by Deng-Nahmod-Yue ’19, addresses an open question proposed by Sun-Tzvetkov ’21. We will also explain why the Gibbs dynamics for the full dispersive range is sharp in some sense. Finally, we will see how the theory of random tensors works for extending this work to multi-dimensional settings.

The HADES seminar on Tuesday, September 23rd, will be at 3:30pm in Room 740.

Speaker: Sung-Jin Oh

Abstract: In this talk, I will present recent joint work with Philip Isett (Caltech), Yuchen Mao (UC Berkeley), and Zhongkai Tao (IHÉS) that introduces a new versatile approach to constructing integral solution operators (i.e., right-inverses up to finite rank operators) for a broad class of underdetermined operators, including the divergence operator, linearized scalar curvature operator, and the linearized Einstein constraint operator. They are optimally regularizing and, more interestingly, have prescribed support properties (e.g., produce compactly supported solutions for compactly supported forcing terms). My goal is to (1) describe our approach, (2) demonstrate how it generalizes the well-known construction of Bogovskii, which has proved very useful in fluid dynamics, and (3) explain how it connects underdetermined PDEs with the rich literature on the dual problem on overdetermined differential operators.