Author Archives: zirui

Examples of Hölder-Stable Phase Retrieval

The HADES seminar on Tuesday, May 10 will be at 3:30 pm in Room 1015 (Notice the room change).

Speaker: Benjamin Pineau

Abstract: Let $(X, \mathcal A, \mu)$ be a measure space. Let $V$ be a closed subspace of the (real or complex) Hilbert space $L^2 = L^2 (\mu)$. We say that $V$ does Holder-stable phase retrieval if there exists a constant $C < \infty$ and $\gamma \in (0, 1]$ such that \begin{equation}\label{eq} \min_{|z|=1} \|f − zg\|_{L^2} \leq C\||f| − |g|\|_{L^2}^\gamma (\|f\|_{L^2} + \|g\|_{L^2} )^{1−γ}\,\forall f, g \in V,(*)\end{equation}

Recently, Calderbank, Daubechies, Freeman, and Freeman have studied real subspaces of real-valued $L^2$ for which (*) holds with $\gamma = 1$ and constructed the first examples of such infinite-dimensional subspaces. In this situation, if $|f|$ is known then $f$ is uniquely determined almost everywhere up to an unavoidably arbitrary global phase factor of $\pm 1$. Moreover, if $|f|$ is known within a small tolerance in norm then up to such a global phase factor, f is determined within a correspondingly small tolerance. This issue arises for instance in crystallography, where one seeks to recover an unknown function $F \in L^2 (\mathbb R)$ from the absolute value of its Fourier transform $\hat F$.

In this talk, I will discuss a set of simple sufficient conditions for constructing infinite-dimensional (real and complex) subspaces $V \subset L^2 (\mu)$ which satisfy (*) and show how to construct some natural examples in which (*) holds. These examples include certain variants of Rademacher series and lacunary Fourier series. This is a joint work with Michael Christ and Mitchell Taylor.

Implicitly Oscillatory Multilinear Integrals

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

Speaker: Michael Christ

Abstract: An archetypal (bilinear) oscillatory integral inequality states that $$ \Big| \iint_{\mathbb{R}^d\times\mathbb{R}^d} f(x)\,g(y)\,e^{i\lambda\phi(x,y)}\,\eta(x,y)\,dx\,dy\Big|\le C|\lambda|^{-\gamma} \|{f}\|_{L^2}\|{g}\|_{L^2}$$ where $\lambda\in\mathbb{R}$ is a large parameter, $\phi$ is a smooth real-valued phase function which is nondegenerate in a suitable sense, $f,g$ are arbitrary $L^2$ functions, $\eta$ is a  smooth compactly supported cutoff function, and $\gamma>0$ and $C<\infty$ depend on $\phi$ but not on $f,g,\lambda$. Its main features are the decaying factor $|\lambda|^{-\gamma}$, the absence of any smoothness hypothesis on the measurable factors $f,g$, and the interplay between the structure of $\phi$ and the product structure of $f(x)\,g(y)$. If $\phi$ is nonconstant then $e^{i\lambda\phi}$ oscillates rapidly, creating cancellation that potentially results in smallness of the integral.

Implicitly oscillatory integrals involve no overtly oscillatory factor $e^{i\lambda\phi}$; instead, the measurable factors $f_j$ are themselves assumed to be oscillatory, but in a less structured way. A multilinear integral of this type takes the form \[ \int_{\mathbb{R}^2} \prod_{j=1}^N (f_j\circ\varphi_j)(x)\,\eta(x)\,dx\] where $\varphi_j:\mathbb{R}^2\to\mathbb{R}^1$ are smooth submersions, and the functions $f_j$ are merely measurable. The desired upper bound is expressed in terms of strictly negative order Sobolev norms of these functions. Thus the functions $f_j$ are rapidly oscillatory in the sense that they consist mainly of high frequency components.

I will give an introduction to recent (and ongoing) work on this topic, and on associatedsublevel set inequalities.

Trilinear Smoothing Inequalities and a Class of Bilinear Maximal Functions

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.

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.

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)
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.

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.