The HADES seminar on Tuesday, December 16th, will be at 3:30pm in Room 762.

Speaker: Shi-Zhuo Looi

Abstract: The classical wave equation is a basic model for the propagation of waves. In even space dimensions, solutions are known to develop long-lived polynomially decaying tails inside the region where the wave has passed, in contrast with the sharp finite propagation of disturbances in odd dimensions.

In this talk, I will discuss how such even-dimensional tails behave in the presence of forcing and nonlinear effects, as well as on non-stationary spacetime backgrounds.

The HADES seminar on Tuesday, December 9nd, will be at 3:30pm in Room 740.

Speaker: Steven Flynn

Abstract: Sub-Riemannian geometries arise naturally in quantum mechanics and control theory, yet fundamental questions about quantum dynamics remain open, suggesting that new microlocal tools are needed to extend classical results to these singular geometries.

I will present a pseudodifferential calculus for filtered manifolds with operator-valued symbols built using representation theory of nilpotent groups. The key innovation is an explicit quantization procedure for noncommutative symbols adapted to the filtration, extending the Van Erp-Yuncken calculus while maintaining essential properties: closure under composition, parametrices, and Sobolev continuity.

This framework enables systematic microlocal analysis on equiregular sub-Riemannian manifolds. This is joint work with Véronique Fischer and Clotilde Fermanian-Kammerer.

The HADES seminar on Tuesday, November 25th, will be at 3:30pm in Room 740.

Speaker: Joe Miller

Abstract: Interacting systems of particles and waves are foundational in many natural phenomena. This talk will outline mathematical approaches for deriving effective, statistical descriptions of such many-body dynamics by connecting them to solutions of nonlinear partial differential equations. Key examples include (i) the Boltzmann equation, which emerges as a limit of interacting hard spheres, (ii) the nonlinear Schrödinger equation, which describes quantum particle dynamics initialized near a Bose-Einstein condensate, (iii) the Vlasov equation, which is an effective model for both non-collisional particles evolving under Newtonian dynamics or as a semiclassical limit of fermionic interactions, and (iv) the kinetic wave equations, which model the statistical behavior of interacting waves. I will discuss my joint work on each of these equations, highlighting how to frame these PDEs as limits of the underlying particle or wave dynamics. Time permitting, I will discuss ongoing work on deriving a Boltzmann mean field game from a jump diffusion process.

The HADES seminar on Wednesday, November 12th, will be at 4:00pm in Room 732.

Speaker: Chanwoo Kim

Abstract: We construct global-in-time classical solutions to the nonlinear Vlasov-Maxwell system in a three-dimensional half-space beyond the vacuum scattering regime. We also prove dynamical asymptotic stability under general perturbations in the full three-dimensional nonlinear Vlasov-Maxwell system.

The HADES seminar on Tuesday, October 28st, will be at 3:30pm in Room 740.

Speaker: Beomjong Kwak

Abstract: In this talk, we present an optimal $L^4$-Strichartz estimate for the Schrödinger equation on the two-dimensional rational torus $\mathbb{T}^2$. We first recall the previously known results and counterexamples on the Strichartz estimates on the torus. Then we present our new Strichartz estimate, which has an optimal amount of loss, and the small-data global well-posedness of (mass-critical) the cubic NLS in $H^s,s>0$ as its consequence. An intuition for the relation between them is then provided. Our Strichartz estimate is based on a combinatorial proof. We introduce our key proposition, the Szemerédi-Trotter theorem, and explain the idea of the proof. This is a joint work with Sebastian Herr.