Dynamics of a Maximally Open Quantized Cat Map

The HADES seminar on Tuesday, February 22 will be at 3:30 pm in Room 740.

Speaker: Yonah Borns-Weil

Abstract: Quantum dynamics is concerned with quantum analogues of classical dynamical systems. A common situation is scattering theory, in which the Hamiltonian dynamics of scattered particles are replaced by wavefunctions obeying the Schrödinger equation. When time is discretized, the analogues are open quantum maps, which are Fourier integral operators arising from phase space diffeomorphisms that are then “opened” by sending some regions to “infinity.” In this talk, we analyze a simple open quantum map, based on the classical Arnol’d cat map. We shall show using the method of Grushin problems that the spectrum has a very simple form in the semiclassical regime as h approaches 0. Emphasis will be given to motivation and interpretations of the result.

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.

Scattering and Pointwise Decay of Some Linear and Nonlinear Wave Equations

The HADES seminar on Tuesday, December 7th, will be given by Shi-Zhuo Looi at 5 pm on Zoom.

Speaker: Shi-Zhuo Looi

Abstract: We discuss the proof of sharp pointwise decay for linear wave equations, and then scattering and sharp pointwise decay for power-type nonlinear wave equations. These results hold on a general class of asymptotically flat spacetimes, which are allowed to be either nonstationary or stationary. The main ideas for the linear problem include local energy decay and commuting vector fields, while the nonlinear problem uses r-weighted local energy decay and Strichartz estimates. For either problem, the initial data are allowed to be large and non-compactly supported.

Introduction to Wave Front Sets Using Diffeomorphisms of the Circle

The HADES seminar on Tuesday, November 30th, will be given by Maciej Zworski at 5 pm in 740 Evans.

Speaker: Maciej Zworski

Abstract: With motivation coming from mathematical study of internal waves (which for the sake of time will make an appearance as movies only) I will discuss wave front set properties of distributions invariant under circle diffeomorphisms. The concept of the wave front set will be explained in the simplest 1D setting and various other things will be presented, including the proof of the easiest case of Sternberg’s linearization theorem. Part of a project with S Dyatlov and J Wang.

Decoupling for some convex sequences in $\mathbb R$

The HADES seminar on Tuesday, November 23rd, will be given by Yuqiu Fu at 5 pm on Zoom.

Speaker: Yuqiu Fu (MIT)

Abstract: If the Fourier transform of a function $f:\mathbb R \rightarrow \mathbb C$ is supported in a neighborhood of an arithmetic progression, then $|f|$ is morally constant on translates of a neighborhood of a dual arithmetic progression.
We will discuss how this “locally constant property” allows us to prove sharp decoupling inequalities for functions on $\mathbb R$ with Fourier support near certain convex/concave sequence, where we cover segments of the sequence by neighborhoods of arithmetic progressions with increasing/decreasing common difference. Examples of such sequences include $\{\frac{n^2}{N^2}\}_{n=N+1}^{N+N^{1/2}}$ and $\{\log n\}_{n=N+1}^{N+N^{1/2}}.$
The sequence $\{\log n\}_{n=N+1}^{2N}$ is closely connected to Montgomery’s conjecture on Dirichlet polynomials but we see some difficulties in studying the decoupling for $\{\log n\}_{n=N+1}^{2N}.$ This is joint work with Larry Guth and Dominique Maldague.

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.

Modified Scattering in One Dimensional Dispersive Flows

The HADES seminar on Tuesday, November 9th, will be given by Daniel Tataru at 5 pm in 740 Evans.

Speaker: Daniel Tataru

Abstract: For a nonlinear flow, scattering is the property that global in time solutions behave like solutions to the corresponding linear flow. In this talk, we will examine this property for generic one dimensional dispersive flows.