Resonances on hyperbolic surfaces… and Berkovich space

The HADES seminar on Tuesday, February 20th, will be at 3:30pm in Room 939.

Speaker: Zhongkai Tao

Abstract: Hyperbolic surfaces are surfaces with constant negative curvature -1. They appear in many places: number theory, PDE, geometry, and topology… and they have many special properties. Despite a lot of studies and efforts put into this subject, the spectral theory of hyperbolic surfaces remains mysterious, especially in the infinite volume case. I will introduce some basic notions of the spectral theory on hyperbolic surfaces, and advertise some open problems. Then I will talk about recent developments on degeneration of hyperbolic surfaces, which uses some new tools from non-Archimedean geometry, and how this would potentially help us understand hyperbolic surfaces.

Asymptotic behavior of global solutions to a scalar quasilinear wave equation satisfying the weak null condition

The HADES seminar on Tuesday, February 13th, will be at 3:30pm in Room 939.

Speaker: Dongxiao Yu

Abstract: I will discuss the long time dynamics of a scalar quasilinear wave equation in three space dimensions. This equation satisfies the weak null condition and has global existence for sufficiently small $C_c^\infty$ initial data. In the talk, I will first present an asymptotic completeness result which describes the asymptotic behavior of global solutions to the scalar quasilinear wave equation near the light cone ($|x|\approx t$). Then, I will discuss a work in progress on the asymptotic behavior inside the light cone  ($|x|\ll t$).

Local well-posedness and smoothing of MMT kinetic wave equation

The HADES seminar on Tuesday, January 30th, will be at 3:30pm in Room 939 (not in 740 this semester!).

Speaker: Joonhyun La

Abstract: In this talk, we will prove local well-posedness of kinetic wave equation arising from MMT equation, which is introduced by Majda, Mclaughlin, and Tabak and is one of the standard toy models to study wave turbulence. Surprisingly, our result reveals a regularization effect of the collision operator, which resembles the situation of non-cutoff Boltzmann. This talk is based on a joint work with Pierre Germain (Imperial College London) and Katherine Zhiyuan Zhang (Northeastern).

Wave dynamics and semiclassical analysis: from graphs to manifolds

The HADES seminar on Tuesday, December 5th, will be at 3:30pm in Room 748 (not in 740 this week!).

Speaker: Akshat Kumar

Abstract: Graph Laplacians and Markov processes are intimately connected and ubiquitous in the study of graph structures. They have led to significant advances in a class of geometric inverse problems known as “manifold learning”, wherein one wishes to learn the geometry of a Riemannian submanifold from finite Euclidean point samples. The data gives rise to the geometry-encoding neighbourhood graphs. Present-day techniques are dominated primarily by the low spectral resolution of the graph Laplacians, while finer aspects of the underlying geometry, such as the geodesic flow, are observed only in the high spectral regime.

We establish a data-driven uncertainty principle that dictates the scaling of the wavelength $h$, with respect to the density of samples, at which graph Laplacians for neighbourhood graphs are approximately $h$-pseudodifferential operators. This sets the stage for a semiclassical approach to the high-frequency analysis of wave dynamics on weighted graphs. We thus establish a discrete version of Egorov’s theorem and achieve convergence rates for the recovery of geodesics on the underlying manifolds through quantum dynamics on the approximating graphs. I will show examples on samples of model manifolds and briefly discuss some applications to real-world datasets.

Mode stability for Kerr(-de Sitter) black holes

The HADES seminar on Tuesday, November 28th, will be at 3:30pm in Room 740.

Speaker: Rita Teixeira da Costa

Abstract: The Teukolsky master equations are a family of PDEs describing the linear behavior of perturbations of the Kerr black hole family, of which the wave equation is a particular case. As a first essential step towards stability, Whiting showed in 1989 that the Teukolsky equation on subextremal Kerr admits no exponentially growing modes. In this talk, we review Whiting’s classical proof and a recent adaptation thereof to the extremal Kerr case. We also present a new approach to mode stability, based on uncovering hidden spectral symmetries in the Teukolsky equations. Part of this talk is based on joint work with Marc Casals (CBPF/UCD).

This talks complements yesterday’s Analysis & PDE seminar, but will be self-contained.

Methods for sharp well-posedness for completely integrable PDE

The HADES seminar on Tuesday, November 14th, will be at 3:30pm in Room 740.

Speaker: Thierry Laurens

Abstract:We will describe some of the methods used to prove sharp well-posedness for the Benjamin–Ono equation in the class of H^s spaces, namely, the method of commuting flows. Since its introduction by Killip and Visan in 2019, this groundbreaking approach to completely integrable systems has been adapted to a wide variety of models in order to prove sharp well-posedness results that were previously inaccessible. In this talk, we will describe some of the overarching principles of the method of commuting flows, with a focus on how these ideas were implemented in the case of the Benjamin–Ono equation. This is based on joint work with Rowan Killip and Monica Visan.

Strichartz estimates for Schroedinger evolutions

The HADES seminar on Tuesday, November 7th, will be at 3:30pm in Room 740.

Speaker: Daniel Tataru

Abstract: I will provide a broad introduction to the topic of dispersive and Strichartz estimates for Schroedinger evolutions on curved backgrounds, with the final goal of describing the new Strichartz estimates proved jointly with Mihaela Ifrim in the context of 1D quasilinear Schroedinger flows.

Optimal enhanced dissipation for geodesic flows

The HADES seminar on Tuesday, October 31st will be at 3:30pm in Room 740.

Speaker: Maciej Zworski

Abstract: We consider geodesic flows on negatively curved compact manifolds or more generally contact Anosov flows (all these concepts will be pedagogically explained). The object is to show that if $ X $ is the generator of the flow and $ \Delta $, a (negative) Laplacian, then solutions to the convection diffusion equation, $ \partial_t u = X u +  \nu \Delta $, $ \nu \geq 0 $,  satisfy \[    \| u ( t )   –  \underline u \|_{L^2 ( M) } \leq C \nu^{-K} e^{ – \beta t } \| u( 0 )  \|_{L^2 ( M) }, \] where $ \underline u $ is the (conserved) average of $ u (0) $ with respect to the contact volume form and $K $ is a fixed constant. This provides many examples of very precise {\em optimal enhanced dissipation} in the sense of recent works of Bedrossian–Blumenthal–Punshon-Smith and Elgindi–Liss–Mattingly. The proof is based on results by Dyatlov and the speaker on stochastic stability of Pollicott–Ruelle resonances,  another concept which will be introduced and explained. The talk is based on joint work with Zhongkai Tao.

Sharp Furstenberg Sets Estimate in the Plane

The HADES seminar on Tuesday, October 24th will be at 3:30pm in Room 740.

Speaker: Kevin Ren

Abstract: Fix a real number 0 < s <= 1. A set E in the plane is a s-Furstenberg set if there exists a line in every direction that intersects E in a set with Hausdorff dimension s. For example, a planar Kakeya set is a special case of a 1-Furstenberg set, and indeed we know that 1-Furstenberg sets have Hausdorff dimension 2. However, obtaining a sharp lower bound for the Hausdorff dimension of s-Furstenberg sets for any 0 < s < 1 has been a challenging open problem for half a century. In this talk, I will illustrate the rich connections between the Furstenberg sets conjecture and other important topics in geometric measure theory and harmonic analysis, and show how exploring these connections can fully resolve the Furstenberg conjecture. Joint works with Yuqiu Fu and Hong Wang.