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 ().

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 ().

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.

The APDE seminar on Monday, 2/03, will be given by Sameer Iyer (UC Davis) 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 ().

Title: Beyond the Goldstein Singularity & Recent Advances in the Boundary Layer Theory

Abstract: The Prandtl system describes the flow in the boundary layer that forms near the boundary when taking the inviscid limit in the Navier-Stokes system. It was first derived in 1904 by Prandtl. Many important questions related to the Prandtl system and the inviscid limit are still open. We will review some recent advances in this direction. Reversal occurs after the separation point, and is characterized by regions in which the velocity changes sign. The classical point of view of treating the stationary Prandtl system as an evolution equation in the tangential variable x completely breaks down. Instead, we view the problem as a quasilinear, mixed-type, free-boundary problem. This is a joint work with Nader Masmoudi.

The APDE seminar on Monday, 1/27, will be given by Iqra Altaf (Chicago) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Mengxuan Yang ().

Title: A one-dimensional planar Besicovitch set.

Abstract: A Γ-Besicovitch set is a set that contains a rotated copy of Γ in every direction.
Our main result is the construction of a non-trivial 1-rectifiable set Γ in the plane, for which there exists a 1-dimensional Γ-Besicovitch set.

The APDE seminar on Monday, 11/25, will be given by Ciprian Demeter (Indiana U) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang ().

Title: Fourier decay of fractal measures and a Szemeredi-Trotter theorem for tubes

Abstract: I will prove a natural analogue of the celebrated Szemeredi-Trotter theorem for lines in the case of tubes satisfying non-concentration assumptions. As an application, I will analyze the Fourier transform of Frostman measures supported on the parabola. This is joint work with Hong Wang.

The APDE seminar on Monday, 11/18, will be given by Arian Nadjimzadah (UCLA) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang ().

Title: Improved bounds for intermediate curved Kakeya sets in R^3.

Abstract: A central problem in harmonic analysis is to understand the L^p bounds of oscillatory integral operators. Bourgain showed that if many wavepackets can be arranged to cluster tightly in space, then the operator has poor L^p bounds. The classical Kakeya Conjecture says that this clustering cannot happen for the extension operator for the paraboloid. At the other extreme, there are operators for which the full length of the wavepackets can cluster near a surface.

In this talk we discuss the intermediate case, where the wavepackets through a small ball can cluster near a surface. Most operators exhibit this behavior. For a large class of such operators, we show improved bounds for the corresponding curved Kakeya problems.

The main tools are Wolff’s hairbrush argument, the multilinear Kakeya inequality of Bennett-Carbery-Tao, and a variant of Wolff’s circular maximal function theorem due to Pramanik-Yang-Zahl. The geometric conditions we will call “coniness’’ and “twistiness’’ play a central role.

The APDE seminar on Monday, 11/4, will be given by Zhenhao Li (MIT) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang ().

Title: Quantum–classical correspondence past Ehrenfest time

Abstract: The evolution of a quantum system coupled to an external environment can be described by the Lindblad master equation. We consider such systems described by an at most quadratically growing classical Hamiltonian and self-adjoint jump operators (satisfying certain growth and nondegeneracy conditions). The classical counterpart to the Lindblad equation is given by a corresponding Fokker–Planck equation. We show that the quantum evolution remains close to the quantization of the classical evolution in trace norm for much longer than Ehrenfest time. The time of correspondence and the trace norm comparison improves upon recent works by Galkowski–Zworski and Hern{\’a}ndez–Ranard–Riedel in the constant to strong coupling strength regime.

The APDE seminar on Monday, 10/28, will be given by Semyon Dyatlov (MIT) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang ().

Title: Control of eigenfunctions in higher dimensions

Abstract: Semiclassical measures are a standard object studied in quantum chaos, capturing macroscopic behavior of sequences of eigenfunctions in the high energy limit. In previous work with Jin and Nonnenmacher we showed that for Laplacian eigenfunctions on negatively curved surfaces, semiclassical measures have full support. This was restricted to dimension 2 because the key new ingredient, the fractal uncertainty principle (proved by Bourgain and the speaker), was only known for subsets of the real line.

I will present several recent results on the support of semiclassical measures in higher dimensions, both on manifolds and in the toy model of quantum cat maps, contained in joint work with Jézéquel, joint work with Athreya and Miller, and work in progress by Kim. Some of these use the higher dimensional fractal uncertainty principle recently proved by Cohen. Others rely on separating the stable/unstable directions into fast and slow directions, and only applying the fractal uncertainty principle in the fast directions.

The APDE seminar on Monday, 10/14, will be given by Gabriele Benomio (GSSI) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang ().

Title: A new gauge for gravitational perturbations of Kerr spacetimes

Abstract: I will present a new geometric framework to address the stability of the Kerr solution to gravitational perturbations in the full sub-extremal range. Central to the framework is a new formulation of nonlinear gravitational perturbations of Kerr in a geometric gauge tailored to the outgoing principal null geodesics of Kerr. The main features of the framework will be illustrated in the context of the linearised theory, which serves as a fundamental building block in nonlinear applications.

The APDE seminar on Monday, 10/7, will be given by Andrej Zlatos (UCSD) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang ().

Title: Stable regime singularity for the Muskat problem

Abstract: The Muskat problem on the half-plane models motion of an interface between two fluids of distinct densities in a porous medium that sits atop an impermeable layer, such as oil and water in an aquifer above bedrock.  We develop a local well-posedness theory for this model in the stable regime (lighter fluid above the heavier one), which includes considerably more general fluid interface geometries than even existing whole plane results and allows the interface to touch the bottom.  The latter applies to the important scenario of the heavier fluid invading a region occupied by the lighter fluid along the impermeable layer.  We also show that finite time singularities do arise in this setting, including from arbitrarily small smooth initial data, by obtaining maximum principles for the height, slope, and potential energy of the fluid interface.