Author Archives: yonah

A stochastic fluid-structure interaction problem describing Stokes flow interacting with a membrane

The HADES seminar on Tuesday, February 15 will be at 3:30 pm in Room 740.

Speaker: Jeffrey Kuan

Abstract: In this talk, we present a well-posedness result for a stochastic fluid-structure interaction model. We study a fully coupled stochastic fluid-structure interaction problem, with linear coupling between Stokes flow and an elastic structure modeled by the wave equation, and stochastic noise in time acting on the structure. Such stochasticity is of interest in applications of fluid-structure interaction, in which there is random noise present which may affect the dynamics and statistics of the full system. We construct a solution by using a new splitting method for stochastic fluid-structure interaction, and probabilistic methods. To the best of our knowledge, this is the first result on well-posedness for fully coupled stochastic fluid-structure interaction. This is joint work with Sunčica Čanić (UC Berkeley).

Almost Sure Weyl Law for Toeplitz Operators

The HADES seminar on Tuesday, February 8 will be at 3:30 pm in Room 740.

Speaker: Izak Oltman

Abstract: When computing eigenvalues of finite-rank non-self-adjoint operators, significant numerical inaccuracies often occur when the rank of the operator is sufficiently large. I show the spectrum of Toeplitz operators, with a random perturbation added, satisfy a Weyl law with probability close to one. I will begin with numerical animations, demonstrating this result for quantizations of the torus (a result proven by Martin Vogel in 2020). Then give a brief introduction to Toeplitz operator (quantizations of functions on Kahler Manifolds). And finally outline the main parts of the proof, which involve constructing an `exotic calculus’ of symbols on a Kahler manifold.

Complex absorbing potential method for calculating scattering resonances

The HADES seminar on Tuesday, February 1 will be at 3:30 pm in Room 736.

Speaker: Haoren Xiong

Abstract: Complex absorbing potential (CAP) method, which is a computational technique for scattering resonances first used in physical chemistry. The method shows that resonances of the Hamiltonian $P$ are limits of eigenvalues of CAP-modified Hamiltonian $P – it|x|^2$ as $t \to 0+$. I will show that this method applies to exponentially decaying potential scattering, and many other things will be presented, including the Davies harmonic oscillator and the method of complex scaling.

Attractive Coulomb-like Schrödinger operators at low energy: resolvent bounds

The HADES seminar on Tuesday, January 25, will be given by Ethan Sussman at 3:30 pm on Zoom.

Speaker: Ethan Sussman

Abstract: Using techniques recently developed by Vasy to study the limiting absorption principle on asymptotically Euclidean manifolds, we study the effect of an attractive Coulomb-like potential on the resolvent output at low energy. In contrast with the situation for Schrödinger operators with short-range potentials — as analyzed in detail by Guillarmou, Hassell, and Vasy — the spectral family of an attractive Coulomb-like Schrödinger operator fails to degenerate to the same degree as the Laplacian at zero energy. We will see how to use this observation to analyze the output of the limiting resolvent uniformly down to E=0.

No pure capillary solitary waves exist in 2D

The HADES seminar on Tuesday, November 16th, will be given by Mitchell Taylor at 5 pm in 740 Evans.

Speaker: Mitchell Taylor

Abstract: We prove that the 2D finite depth capillary water wave equations admit no solitary wave solutions. This closes the existence/non-existence problem for solitary water waves in 2D, under the classical assumptions of incompressibility and irrotationality, and with the physical parameters being gravity, surface tension and the fluid depth. Joint work with Mihaela Ifrim, Ben Pineau, and Daniel Tataru.

Observability for Schrodinger equation on the torus

The HADES seminar on Tuesday, October 19th, will be given by Zhongkai Tao at 5 pm in 740 Evans.

Speaker: Zhongkai Tao

Abstract: The Schrodinger equation describes the motion of a particle on a manifold. It is quite nice that the distribution of the particle is closely related to classical dynamics. I will introduce the observability estimate, the control result and describe how they are related to classical dynamics. At the end, I will talk about my attempt to make the estimates quantitative. No prerequisite in microlocal analysis is needed. This work comes from my undergraduate research mentored by Semyon Dyatlov.

Streak artifacts from non-convex metal objects in X-ray tomography

The HADES seminar on Tuesday, September 28th, will be given by Joey Zou at 5 pm in 740 Evans.

Speaker: Joey Zou (University of California, Santa Cruz)

Abstract: In X-ray CT scans with metallic objects, streak artifacts in the computed image may arise due to beam hardening effects, where the attenuation coefficient of metallic objects vary strongly with energy. A mathematical description of these artifacts using the notion of wavefront sets was given by Choi, Park, and Seo in 2014, followed by the work of Palacios, Uhlmann, and Wang, who gave quantitative descriptions of the artifacts that recovered qualitative observations from CT scans when the metallic objects are strictly convex. In this talk, I will discuss joint work with Yiran Wang which builds on the previous work by using microlocal analysis to study artifacts generated by non-convex metallic objects, as well as artifacts associated to a broader class of attenuation variations than was considered before. The problem relies on the analytic behavior of a nonlinear function composed with the image of the X-ray transform applied to certain functions, for which we use the work of Melrose, Ritter, Sa Barreto et al. on semilinear wave equations via the usage of iterated regularity spaces in which both the X-ray transform image and its nonlinear composition live.