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.

Solution operators for divergence-type equations with prescribed support properties

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.

Global well-posedness for the generalized derivative nonlinear Schrödinger equation

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.

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.

Well-Posedness For The Dispersive Hunter-Saxton Equation

The HADES seminar on Tuesday, October 12th, will be given by Ovidiu-Neculai Avadanei at 5 pm in 740 Evans.

Speaker: Ovidiu-Neculai Avadanei

Abstract: This talk represents a first step towards understanding the well-posedness for the dispersive Hunter-Saxton equation. This problem arises in the study of nematic liquid crystals, and its non-dispersive version is known to be completely integrable. Although the equation has formal similarities with the KdV equation, the lack of $L^2$ control gives it a quasilinear character, with only continuous dependence on initial data. Here, we prove the local and global well-posedness of the Cauchy problem using a normal form approach to construct modified energies, and frequency envelopes in order to prove the continuous dependence with respect to the initial data. This is joint work with Albert Ai.

A stationary set method for estimating oscillatory integrals

The HADES seminar on Tuesday, October 5th, will be given by Ruixiang Zhang at 5 pm in 740 Evans.

Speaker: Ruixiang Zhang (UC Berkeley)

Abstract: Given a polynomial $P$ of constant degree in $d$ variables and consider the oscillatory integral $$I_P = \int_{[0,1]^d} e(P(\xi)) \mathrm{d}\xi.$$ Assuming $d$ is also fixed, what is a good upper bound of $|I_P|$? In this talk, I will introduce a “stationary set” method that gives an upper bound with simple geometric meaning. The proof of this bound mainly relies on the theory of o-minimal structures. As an application of our bound, we obtain the sharp convergence exponent in the two dimensional Tarry’s problem for every degree via additional analysis on stationary sets. Consequently, we also prove the sharp $L^{\infty} \to L^p$ Fourier extension estimates for every two dimensional Parsell-Vinogradov surface whenever the endpoint of the exponent $p$ is even. This is joint work with Saugata Basu, Shaoming Guo and Pavel Zorin-Kranich.

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.