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.

The APDE seminar on *Wednesday* 10/08, will be given by Nataša Pavlović (UT Austin) in-person in Evans 732, and will also be broadcasted online via Zoom from 4:00pm to 5:00pm PDT. **Please note the unusual day and location.** To participate, please email Adam Black (adamblack@berkeley.edu).

Title: What happens when bosons are mixed with fermions

Abstract: Investigating mixtures of bosons and fermions is an extremely active area of research in experimental physics for constructing and understanding novel quantum bound states such as those in superconductors, superfluids, and supersolids. These ultra-cold Bose-Fermi mixtures are intrinisically different from gases with only bosons or fermions. Namely, they show a fundamental instability due to energetic considerations coming from the Pauli exclusion principle. Inspired by this activity in the physics community, recently we started exploring the mathematical theory of Bose-Fermi mixtures.

• One of the main challenges is understanding the physical scales of the system that allow for suitable analysis.  We will describe how we overcame this challenge in the joint work with Esteban Cárdenas and Joseph Miller by identifying a novel scaling regime in which the fermion distribution behaves semi-clasically, but the boson field remains quantum-mechanical. In this regime, the bosons are much lighter and more numerous than the fermions.
• Time permitting, we will also describe new results obtained with Esteban Cárdenas, Joseph Miller and David Mitrouskas inspired by recent experiments by DeSalvo et al.  on mixtures of light fermionic atoms and heavy bosonic atoms. A key observation – and this has been theoretically long predicted – is the emergence of an attractive fermion-mediated interaction between the bosons. We give a rigorous derivation of fermion-mediated interactions and prove the associated stability-instability transition.

The APDE seminar on Monday, 9/29, will be given by Andrea Nahmod (UMass Amherst) 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: Probabilistic scaling, propagation of randomness and invariant Gibbs measures

Abstract: In this talk, we will start by describing how classical tools from probability
offer a robust framework to understand the dynamics of waves via appropriate ensembles
on phase space rather than particular microscopic dynamical trajectories. We will continue
by explaining the fundamental shift in paradigm that arises from the “correct” scaling in this
context and how it opened the door to unveil the random structures of nonlinear waves that
live on high frequencies and fine scales as they propagate. We will then discuss how these
ideas broke the logjam in the study of the Gibbs measures associated to nonlinear
Schrödinger equations in the context of equilibrium statistical mechanics and of the
hyperbolic Phi^4_3 model in the context of constructive quantum field theory.
We will end with some open challenges about the long-time propagation of randomness
and out-of-equilibrium dynamics.

The APDE seminar on Monday, 9/22, will be given by Adam Black (UC Berkeley) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PDT. To participate, please email Ryan Unger (runger@berkeley.edu).

Title: Quantum diffusion and random matrix theory

Abstract: The Schrödinger equation with a random potential serves as a simple model for the propagation of waves in a disordered medium. It is conjectured that when the strength of the potential is weak, the solution should evolve diffusively as time goes to infinity. In this talk, I will explain a new proof of this phenomenon for long but finite times. The proof combines ideas from random matrix theory with resolvent estimates for the Laplacian. This is joint work with Reuben Drogin and Felipe Hernández.