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.
The APDE seminar on Monday, 9/30, will be given by Koji Ohkitani (RIMS, Kyoto University) 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: Numerical determination of the self-similar profile
for the 3D Navier-Stokes equations and its applications
Abstract: We present the forward self-similar profile for the 3D Navier-Stokes
equations, representing the late stage of decaying Navier-Stokes flows.
The existence of such a profile has been known, but its precise functional
form has not been determined numerically, let alone mathematically.
Here we determine the profile for the first time using numerical methods.
This has been achieved by a combination of two things; a numerical method
of solving the Navier-Stokes equations in the whole space and the explicit
form of the linearised solution. Taking the initial data from the
linearised solution, we solve the fully-nonlinear Navier-Stokes equations
to observe its convergence to a steady solution in the dynamically scaled
space. We have confirmed that the nonlinear correction is small,
consistent with the previous perturbative analysis. Applications of the
self-similar profile are briefly discussed.
The APDE seminar on Monday, 9/23, will be given by Anuj Kumar (UC Berkeley) 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: Nonuniqueness of solutions to the Euler equations with integrable vorticity
Abstract: Yudovich established the well-posedness of the two-dimensional incompressible Euler equations for solutions with bounded vorticity. DiPerna and Majda proved the existence of weak solutions with vorticity in L^p ( p > 1). A celebrated open question is whether the uniqueness result can be generalized to solutions with L^p vorticity. In this talk, we resolve this question in negative for some p > 1. To prove nonuniqueness, we devise a new convex integration scheme that employs non-periodic, spatially-anisotropic perturbations, an idea that was inspired by our recent work on the transport equation. To construct the perturbation, we introduce a new family of building blocks based on the Lamb-Chaplygin dipole. This is a joint work with Elia Bruè and Maria Colombo.
The APDE seminar on Monday, 9/16, will be given by Warren Li (Princeton University) 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: BKL bounces outside homogeneity
Abstract: In the latter half of the 20th century, physicists Belinski, Khalatnikov and Lifshitz (BKL) proposed a general ansatz for solutions to the Einstein equations possessing a (spacelike) singularity. They suggest that, near the singularity, the evolution of the spacetime geometry at different spatial points decouples and is well-approximated by a system of autonomous nonlinear ODEs, and further that general orbits of these ODEs resemble a (chaotic) cascade of heteroclinic orbits called “BKL bounces”. In this talk, we present recent work verifying the validity of BKL’s heuristics in a large class of symmetric, but spatially inhomogeneous, spacetimes which exhibit (up to one) BKL bounce on causal curves reaching the singularity. In particular, we prove AVTD behavior (i.e. decoupling) even in the presence of inhomogeneous BKL bounces. The proof uses nonlinear ODE analysis coupled to hyperbolic energy estimates, and one hopes our methods may be applied more generally.