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.

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.

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.

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.

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.

The HADES seminar on Tuesday, November 2nd, will be given by Sung-Jin Oh at 5 pm in 740 Evans.

Speaker: Sung-Jin Oh

Abstract: An alternative title for this talk could be “What I wish I knew about the divergence equation in graduate school.” Equations that resemble the prescribed divergence equation arise from many places in physics, such as the incompressibility condition in fluid mechanics, the Gauss law in electromagnetism and the (linearized) constraint equations in general relativity. I will describe a construction of solution operators for these equations with certain support properties based on a few simple ideas, such as manipulation of delta distributions, smooth averaging and standard harmonic analysis. Then I will discuss how such a construction leads to simplification (and improvement) of some theorems for the Yang-Mills and Einstein equations.

The HADES seminar on Tuesday, October 26th, will be given by Benjamin Pineau at 5 pm in 740 Evans.

Speaker: Benjamin Pineau

Abstract: In this talk, we study the well-posedness of the generalized derivative nonlinear Schrödinger equation (gDNLS)
$$iu_t+u_{xx}=i|u|^{2\sigma}u_x,$$
for small powers $\sigma$. We analyze this equation at both low and high regularity, and are able to establish global well-posedness in $H^s$ when $s\in [1,4\sigma)$ and $\sigma \in (\frac{\sqrt{3}}{2},1)$. Our result when $s=1$ is particularly relevant because it corresponds to the regularity of the energy for this problem. Moreover, a theorem of Liu, Simpson and Sulem (~2013) establishes the orbital stability of the gDNLS solitons, provided that there is a suitable $H^1$ well-posedness theory.

To our knowledge, this is the first low regularity well-posedness result for a quasilinear dispersive model where the nonlinearity is both rough and is of lower than cubic order. These two features pose considerable difficulty when trying to apply standard tools for closing low-regularity estimates. While the tools we developed are used to study gDNLS, we believe that they should be applicable in the study of local well-posedness for other dispersive equations of a similar character. It should also be noted that the high regularity well-posedness presents a novel issue, as the roughness of the nonlinearity limits the potential regularity of solutions. Our high regularity well-posedness threshold $s<4\sigma$ is twice as high as one might naively expect, given that the function $z\mapsto |z|^{2\sigma}$ is only $C^{1,2\sigma-1}$ Hölder continuous. Moreover, although we cannot prove $H^1$ well-posedness when $\sigma\leq \frac{\sqrt{3}}{2}$, we are able to establish $H^s$ well-posedness in the high regularity regime $s\in (2-\sigma,4\sigma)$ for the full range of $\sigma\in (\frac{1}{2},1)$. This considerably improves the known local results, which had only been established in either $H^2$ or in weighted Sobolev spaces. This is joint work with Mitchell Taylor.

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.