The APDE seminar on Monday, 10/13, will be given by Gigliola Staffilani (MIT) 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: Non radial blow up for defocusing supercritical NLS equation.
Abstract: In this paper we construct smooth, non-radial solutions of the defocusing nonlinear Schrodinger equation that develop an imploding finite time singularity, both in the periodic setting and the full space. The result is obtained by transforming the NLS into a compressible Euler type equation via the Madelung transformation and use imploding solutions for them. This is joint work with Conzalo Cao, Javi Gomez-Serrano and Jia Shi.
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.
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.
The APDE seminar on Monday, 4/21, will be given by Ovidiu-Neculai Avadanei (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 Robert Schippa ().
Title: Counterexamples to Strichartz estimates and gallery waves for the irrotational compressible Euler equation in a vacuum setting.
Abstract: We consider the free boundary problem for the irrotational compressible Euler equation in a vacuum setting. By using the irrotationality condition in the Eulerian formulation of Ifrim and Tataru, we derive a formulation of the problem in terms of the velocity potential function, which turns out to be an acoustic wave equation that is widely used in solar seismology. This paper is a first step towards understanding what Strichartz estimates are achievable for the aforementioned equation. Our object of study is the corresponding linearized problem in a model case, in which our domain is represented by the upper half-space. For this, we investigate the geodesics corresponding to the resulting acoustic metric, which have multiple periodic reflections next to the boundary. Inspired by their dynamics, we define a class of whispering gallery type modes associated to our problem, and prove Strichartz estimates for them. By using a construction akin to a wave packet, we also prove that one necessarily has a loss of derivatives in the Strichartz estimates for the acoustic wave equation satisfied by the potential function. In particular, this suggests that the low regularity well-posedness result obtained by Ifrim and Tataru might be optimal, at least in a certain frequency regime. To the best of our knowledge, these are the first results of this kind for the
irrotational vacuum compressible Euler equations.
The APDE seminar on Monday, 4/14, will be given by Ben Dodson (Johns Hopkins 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: The conformal nonlinear wave equation with radially symmetric initial data.
Abstract: In this talk, we prove a scattering result for the conformal nonlinear wave equation with radially symmetric initial data. This result is sharp.
The APDE seminar on Monday, 4/07, will be given by Ruixiang Zhang (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 Robert Schippa ().
Title: Introduction to weighted restriction estimates.
Abstract: Weighted (Fourier) restriction estimates is ubiquitous in
subjects such as analysis, number theory and geometric measure theory.
We will use a few examples to introduce these estimates and
applications and talk about progress on a few problems. We will also
compare the problems to the classical Fourier restriction estimate and
discuss their key connections and differences.