The HADES seminar on Tuesday, March 29 will be at 3:30 pm in Room 740.

Speaker: Yuchen Mao

Abstract: Unlike many other equations, initial data for the Einstein equation have to solve the constraint equations, which makes it an interesting problem to construct asymptotically flat localized initial data. Carlotto and Scheon proved the existence of gluing construction of such initial data supported in a cone through a functional analytic approach. We give a simpler proof by explicitly constructing a solution with conic support that achieves the optimal decay conjectured by Carlotto, and lower regularity. Another conjecture made by Carlotto is whether we can construct initial data localized in a smaller region without violating the positive mass theorem. As an application of our solution operator, we prove this is possible for the case of a degenerate sector. This is a joint work with Zhongkai Tao.

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

Speaker: Jeffrey Galkowski

Abstract: We discuss the typical behavior of two important quantities on compact Riemannian manifolds: the number of primitive closed geodesics of a certain length and the error in the Weyl law. For Baire generic metrics, the qualitative behavior of both of these quantities has been well understood since the 1970’s and 1980’s. Nevertheless, their quantitative behavior for typical manifolds has remained mysterious. In fact, only recently, Contreras proved an exponential lower bound for the number of closed geodesics on a Baire generic manifold. Until now, this was the only quantitative estimate on either the number of geodesics for typical metrics, and no such estimate existed for the remainder in the Weyl law. In this talk, we give stretched exponential upper bounds on the number of primitive closed geodesics for typical metrics. Furthermore, using recent results on the remainder in the Weyl law, we will use our dynamical estimates to show that logarithmic improvements in the remainder in the Weyl law hold for typical manifolds. The notion of typicality used in this talk will be a new analog of full Lebesgue measure in infinite dimensions called predominance.
Given recent results of myself and Canzani on the Weyl law, all of these estimates are reduced to a study of the closed geodesics on a typical manifold. We will recall these results on the Weyl law and discuss the ideas used to understand closed geodesics on typical manifolds.
Based on joint work with Y. Canzani.

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

Speaker: Arthur Touati

Abstract: In this talk, I will present recent work on high-frequency solutions
to the Einstein vacuum equations. From a physical point of view, these solutions
model high-frequency gravitational waves and describe how waves travel on a fixed
background metric. There are also interested when studying the Burnett conjecture,
which addresses the lack of compactness of the family of vacuum spacetimes. These
high-frequency spacetimes are singular and require to work under the regime of
well-posedness for the Einstein vacuum equations. I will review the literature on
the subject and then show how one can construct them in generalised wave gauge
by defining high-frequency ansatz.

The HADES seminar on Tuesday, March 1 will be at 3:30 pm in Room 740.

Speaker: Zirui Zhou

Abstract: In this talk, we will present a trilinear smoothing inequality of the form
$$\left|\int_{\mathbb R^2} \prod_{j=0}^2 (f_j\circ\varphi_j)(x)\,\eta(x)\,dx\right|
\leq C \prod_{j=0}^2 \|f_j\|_{W^{p,\sigma}}$$and two of its applications. Lebesgue space bounds are established for certain maximal bilinear functions. The proof combines the degenerate-case trilinear smoothing inequality with Calderón-Zygmund theory.

The second application gives a quantitative nonlinear Roth theorem, which recovers Roth-type theorems proved by Bourgain and Christ-Durcik-Roos. This talk is based on joint work with Michael Christ.

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.

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).

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.

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.

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.

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.