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.

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.

The APDE seminar on Monday, 9/15, will be given by our own Ely Sandine (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 Sung-Jin Oh ().

Title: On Self-Similar Blow-up for the Euler-Poisson System

Abstract: The Euler-Poisson system describes the evolution of a self-gravitating compressible fluid. I will present recent work proving the existence of certain self-similar implosion profiles for this system. These solutions belong to a family numerically conjectured by Hunter. I will discuss related PDEs, previous results for this system and aspects of the proof.

The APDE seminar on Monday, 5/5, will be given by Björn Bringmann (Princeton University) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PDT. To participate, please email Robert Schippa ().

Title: Global well-posedness of the stochastic Abelian-Higgs equations in two dimensions.

Abstract: There has been much recent progress on the local solution theory for geometric singular SPDEs. However, the global theory is still largely open. In this talk, we discuss the global well-posedness of the stochastic Abelian-Higgs model in two dimensions, which is a geometric singular SPDE arising from gauge theory. The proof is based on a new covariant approach, which consists of two parts: First, we introduce covariant stochastic objects, which are controlled using covariant heat kernel estimates. Second, we control nonlinear remainders using a covariant monotonicity formula, which is inspired by earlier work of Hamilton.

This is joint work with S. Cao.

The APDE seminar on Monday, 4/28, will be given by Kiril Datchev (Purdue University) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PDT. To participate, please email Robert Schippa ().

Title: Low frequency scattering and decay of waves.

Abstract: Low frequency waves are sensitive to the large-scale geometry of the environment through which they travel and of scatterers with which they interact. Their analysis has implications for wave evolution, and for the scattering matrix and phase. We study these using resolvent asymptotics, and present a robust method for deriving such asymptotics, based in part on an identity of Vodev and on boundary pairing. We focus on two-dimensional Euclidean scattering because of the rich phenomena observed in this setting, but other dimensions work just as well, and the method also applies to more general geometric situations.

This project is joint work with Tanya Christiansen, and parts are also joint work with Colton Griffin, Pedro Morales, and Mengxuan Yang.