The HADES seminar on Tuesday, December 10th, will be at 2:00pm in Room 740.

Speaker: Shrey Aryan

Abstract: In this talk, we will discuss the proof of the Soliton Resolution Conjecture for the radial critical non-linear heat equation in dimension $d\ge 3$.

The HADES seminar on Tuesday, December 3rd, will be at 2:00pm in Room 740.

Speaker: Elena Kim

Abstract: We consider a quantum cat map $M$ associated to a symplectic matrix $A$ acting on the torus ${\mathbb{T}}^{2n}$, a popular model in quantum chaos. The semiclassical limit of the mass of eigenfunctions of $M$ is characterized by the semiclassical measure.

For the analogous model on hyperbolic manifolds, the quantum unique ergodicity conjecture posits that the Liouville measure is the only semiclassical measure; however, the corresponding statement for quantum cat maps is known to be false. It is thus an open question to otherwise describe semiclassical measures for quantum cat maps.

In this talk, I will explain how the higher-dimensional fractal uncertainty principle of Cohen can be used to characterize the supports of semiclassical measures $\mu$, including cases where $\mu$ has full support.

The HADES seminar on Tuesday, November 26th, will be at 2:00pm in Room 740.

Speaker: Ciprian Demeter (Indiana University Bloomington)

Abstract: Partition the unit square into $N^2$ tiny squares with side length $1/N$. Can a selection of roughly N such tiny squares be made, in such a way that each line intersects at most O(1) of them? Related questions will also be discussed.

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 $S{L}_2$ 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 $n\ge 5$ 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 $\mathrm{\Omega }\subset {\mathbb{R}}^2$, the equation is given by

$({\partial }_t^2\mathrm{\Delta }+{\partial }_{x_2}^2)u(x,t)=f(x)\cos (\lambda t),t\ge 0,x\in \mathrm{\Omega }$

with Dirichlet boundary conditions. The behavior of the equation is intimately related to the underlying classical dynamics, and Dyatlov-Wang-Zworski proved that for $\mathrm{\Omega }$ 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 15th, will be at 2:00pm in Room 740.

Speaker: Andrea Nuetzi

Abstract: We construct a chain homotopy for the de Rham complex of relative differential forms on compact manifolds with boundary. This chain homotopy has desirable support propagation properties, and satisfies estimates relative to weighted Sobolev norms, where the weights measure decay near the boundary. The estimates are optimal given the homogeneity properties of the de Rham differential under boundary dilation. When applied to the radial compactification of Euclidean space, this construction yields, in particular, a right inverse of the divergence operator that preserves support on large balls around the origin, and satisfies estimates that measure decay near infinity. I will explain how this right inverse can then be used to also obtain a right inverse of the divergence operator on symmetric traceless matrices in three dimensions.

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.