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 (rschippa@berkeley.edu).

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 (rschippa@berkeley.edu).

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, 3/31, will be given by Anna Mazzucato (Penn State 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 (rschippa@berkeley.edu).

Title: On the Euler equations with in-flow and out-flow boundary conditions.

Abstract: I will discuss recent results concerning the well-posedness and regularity for the incompressible Euler equations when in-flow and out-flow boundary conditions are imposed on parts of the boundary, motivated by applications to boundary layers. This is joint work with Gung-Min Gie (U. Louisville, USA) and James Kelliher (UC Riverside, USA). I will also discuss energy dissipation and enstrophy production in the zero-viscosity limit at outflow, joint work with Jincheng Yang (U Chicago and IAS), Vincent Martinez (CUNY, Hunter College), and Alexis Vasseur (UT Austin).

The APDE seminar on Monday, 3/17, will be given by Rana Badreddine (UCLA) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Robert Schippa (rschippa@berkeley.edu).

Title: The Calogero-Moser derivative NLS equation

Abstract: We consider a type of nonlocal nonlinear derivative Schrödinger equation on the torus, called the Calogero-Sutherland DNLS equation.We derive an explicit formula to the solution of this nonlinear PDE. Moreover, using the integrability tools, we establish the global well-posedness of this equation in all the Hardy-Sobolev spaces $H_{+}^s(\mathbb{T}),s\ge 0$ down to the critical regularity space, and under a mass assumption on the initial data for the focusing equation, and for arbitrary initial data for the defocusing equation. Finally, a sketch of the proof for extending the flow to the critical regularity $L_{+}^2(\mathbb{T})$ will be presented.

The APDE seminar on Monday, 3/10, will be given by Pablo Shmerkin (University of Brisith Columbia)in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Anuj Kumar (anujkumar@berkeley.edu).

Title: Around the sum-product problem for fractals

Abstract: A fundamental result of Bourgain asserts, in its simplest form, that if $A$ is a Borel subset of the real line with Hausdorff dimension $s\in (0,1)$, then $\dim (A+A.A)\ge s+c(s)$ for some small constant $c(s)>0$. Note that this implies, in particular, that there exist no Borel subrings of the reals of intermediate dimension. I will present some recent variants and improvements on this result, including joint work (across several papers) with N. de Saxcé, T. Orponen and H. Wang.

The APDE seminar on Monday, 3/3, will be given by Gustav Holzegel (University of Münster) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Sung-Jin Oh (sjoh@math.berkeley.edu).

Title: Quasi-linear wave equations on black hole backgrounds

Abstract: After a brief introduction to the geometric features of black holes, I will discuss recent joint work with Dafermos, Rodnianski and Taylor introducing a new scheme to prove small data global existence results for quasilinear wave equations on sub-extremal Kerr black hole backgrounds. In the slowly rotating case (see arXiv:2212.14093), a key ingredient is a new purely physical, highly degenerate, spacetime estimate at the top order which avoids understanding the detailed structure of trapped geodesics. In the full sub-extremal case (see arXiv:2410.03639) this estimate needs to be tailored to a finite number of wave packets defined by an appropriate frequency decomposition in the azimuthal and stationary frequencies. The relation to the stability problem for black holes will also be discussed.

The APDE seminar on Monday, 2/10, will be given by Hannah Cairo (UC Berkeley) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Anuj Kumar (anujkumar@berkeley.edu).

Title: A Counterexample to the Mizohata-Takeuchi Conjecture

Abstract: We derive a family of $L^p$ estimates on the X-Ray transform of positive measures in ${\mathbb{R}}^d$, which we use to construct a $\log R$-loss counterexample to the Mizohata-Takeuchi conjecture for every $C^2$ hypersurface in ${\mathbb{R}}^d$ that does not lie in a hyperplane. In particular, endpoint multilinear restriction estimates cannot be derived from MT-type estimates.