Category Archives: Fall 2023

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.

Wellposedness for Quasi-linear Problems and the Modified Energy Method

The HADES seminar on Tuesday, October 17th will be at 3:30pm in Room 740.

Speaker: Ryan Martinez

Abstract: We give an exposition of the Hadamard wellposedness and explain the modified energy method through the use of the Kirchhoff type Wave Equation as an example. We use the ideas from Daniel and Mihaela’s “Local Wellposedness for Quasilinear Problems: A Primer” as well as from their work with John K.  Hunter and Tak Kwong Wong, “Long Time Solutions for a Burgers-Hilbert Equation via a Modified Energy Method.”

Existence of more self-similar implosion profiles for the Euler-Poisson system

The HADES seminar on Tuesday, October 10th will be at 3:30pm in Room 740.

Speaker: Ely Sandine

Abstract: I will discuss implosion for the equations describing a gas which is compressible, isothermal and self-gravitating. Under the hypotheses of radial symmetry and self-similarity, the equations reduce to a system of ODEs which has been extensively studied by the astrophysics community using numerical methods. One such solution, discovered by Larson and Penston in 1969, was recently rigorously proved to exist by Guo, Hadžić and Jang. In this talk, I will discuss rigorous existence of a subset of the discrete family of solutions found numerically by Hunter in 1977.

Construction of nonunique solutions of the transport and continuity equation for Sobolev vector fields in DiPerna–Lions’ theory

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

Speaker: Anuj Kumar

Abstract: In this talk, we are concerned with DiPerna–Lions’ theory for the transport equation. In the first part of the talk, I will discuss a few results regarding the nonuniqueness of trajectories of the associated ODE. Alberti ’12 asked the following question: are there continuous Sobolev vector fields with bounded divergence such that the set of initial conditions for which the trajectories are not unique is of full measure? We construct an explicit example of divergence-free H\”older continuous Sobolev vector field for which trajectories are not unique on a set of full measure, which then answers the question of Alberti. The construction is based on building an appropriate Cantor set and a “blob flow” vector field to translate cubes in space. The vector field constructed also implies nonuniqueness in the class of measure solutions. The second part to talk is a more recent work jointly with E. Bruè and M. Colombo. We construct nonunique solutions of the continuity equation in the class L^\infty in time and L^r in space. We prove nonuniqueness in the range of exponents beyond what is available using the method of convex integration and sharply match with the range of uniqueness of solutions from Bruè, Colombo, De Lellis’ 21.

On a nonlinearly coupled stochastic fluid-structure interaction model.

The HADES seminar on Tuesday, September 26th will be at 3:30pm in Room 740.

Speaker: Krutika Tawri

Abstract: In this talk, we will present a constructive approach to investigate
the existence of martingale solutions to a benchmark fluid-structure interaction
problem that involves an incompressible, viscous fluid interacting with a linearly
elastic membrane subjected to a multiplicative stochastic force. The fluid flow
is described by the Navier-Stokes equations while the elastodynamics of the
thin structure is modeled by the Koiter shell equations. We will discuss the
challenges arising due to the random motion of the time-dependent fluid domain
and present our recent findings. This is joint work with Sunčica Čanić.