The HADES seminar on Tuesday, November 19th, will be at 2:00pm in Room 740.

Speaker: Arian Nadjimzadah (UCLA)

Abstract: In this talk, we describe the deep connection between oscillatory integrals and curved Kakeya problems that was observed by Bourgain. Then we sketch some of the key discoveries in the study of the classical Kakeya problem in $\mathbb R^3$, and see how they can inform an approach to solving curved Kakeya problems. The results we will discuss are Wolff’s hairbrush bound, the $SL_2$ Kakeya set bound of Katz-Wu-Zahl, and the multilinear Kakeya inequality of Bennett-Carbery-Tao.

The HADES seminar on Tuesday, November 12th, will be at 2:00pm in Room 740.

Speaker: Jose Lopez

Abstract: The Yang-Mills equations are important in physics because the equations of motion of the fundamental forces of particle physics are described by quantized versions of these equations. As an example, we will study the work of Mazzeo-Swoboda-Weiss-Witt on the asymptotics of Hitchin’s equations, a dimensional reduction of Yang-Mills.

The HADES seminar on Tuesday, November 5th, will be at 2:00pm in Room 740.

Speaker: Ning Tang

Abstract: The hyperbolic vanishing mean curvature (HVMC) equation in Minkowski space is a quasilinear wave equation that serves as the hyperbolic counterpart of the minimal surface equation in Euclidean space. This talk will concern the modulated nonlinear asymptotic stability of the $1+4$ dimensional hyperbolic catenoid, viewed as a stationary solution to the HVMC equation. This stability result is under a “codimension-one” assumption on initial perturbation, modulo suitable translation and boost (i.e. modulation), without any symmetry assumptions. In comparison to the $n \geq 5$ case addressed by Lührmann-Oh-Shahshahani, proving catenoid stability in $n = 4$ dimensions shares additional difficulties with its $3$ dimensional analog, namely the slower spatial decay of the catenoid and slower temporal decay of waves. To overcome these difficulties for the $n = 3$ case, the strong Huygens principle, as well as a miracle cancellation in the source term, plays an important role in the work of Oh-Shahshahani to obtain strong late time tails. Without these special structural advantages in $n = 4$ dimensions, our novelty is to introduce an appropriate commutator vector field to derive a new hierarchy with higher $r^p$-weights so that an improved pointwise decay can be established.

In this talk, I will first give an outline of the proof and then focus on a model case to illustrate our approach to achieving an improved decay in $4$ dimensions.

The HADES seminar on Tuesday, October 29th, will be at 2:00pm in Room 740.

Speaker: James Rowan

Abstract: Computers cannot do exact arithmetic with arbitrary real numbers. But as analysts, that shouldn’t bother us too much since we usually deal in inequalities; often, we only need upper and lower bounds. I will give an introduction to interval arithmetic and survey some ways to make rigorous use of numerics in PDE proofs. First, I will discuss an interval arithmetic version of Newton’s method for root-finding and illustrate its use to compute a threshold value of a physical parameter in a solitary water waves problem (from joint work with Lizhe Wan). Then I will discuss the use of interval arithmetic to “upgrade” approximate solutions to nearby exact solutions and give a survey of some recent developments in this area that have inspired some recent projects I have been working on.

The HADES seminar on Tuesday, October 22th, will be at 2:00pm in Room 740.

Speaker: Zhenhao Li

Abstract: The internal waves equation describes perturbations of a stable-stratified fluid. In an effectively 2D aquarium $\Omega \subset \mathbb{R}^2$, the equation is given by

$ (\partial_t^2 \Delta + \partial_{x_2}^2)u(x, t) = f(x) \cos(\lambda t), \quad t \ge 0, \quad x \in \Omega$

with Dirichlet boundary conditions. The behavior of the equation is intimately related to the underlying classical dynamics, and Dyatlov-Wang-Zworski proved that for $\Omega$ with smooth boundary, strong singularities form along the periodic trajectories of the underlying dynamics. Such phenomenon was first experimentally observed in 1997 by Maas-Lam in an aquarium with corners. We will discuss some recent work proving that corners contribute additional mild singularities that propagate according to the dynamics, matching the experimental observations.

The HADES seminar on Tuesday, October 8th, will be at 2:00pm in Room 740.

Speaker: Robert Schippa

Abstract: We consider -smoothing estimates for the wave equation with harmonic potential. For the proof, we linearize an FIO parametrix, which yields Klein-Gordon propagation with variable mass parameter. We obtain decoupling and square function estimates depending on the mass parameter, which yields local smoothing estimates with sharp loss of derivatives. The obtained range is sharp in 1D, and partial results are obtained in higher dimensions.

The HADES seminar on Wednesday, 4 September, will be at 3:30pm in Evans 736. (Note the unusual space and time)

Speaker: Jingbo Wan (Columbia)

Abstract: We establish boundedness and polynomial decay results for the Teukolsky system in the exterior spacetime of slowly rotating and strongly charged sub-extremal Kerr-Newman black holes, with a focus on axially symmetric solutions. The key step is deriving a physical-space Morawetz estimate for the associated generalized Regge-Wheeler system, without relying on spherical harmonic decomposition. The estimate is potentially useful for linear stability of Kerr-Newman under axisymmetric perturbation and nonlinear stability of Reissner-Nordstrom without any symmetric assumptions. This is based on a joint work with Elena Giorgi.