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.

Bounds for spectral projectors on Riemannian manifolds

The HADES seminar on Tuesday, April 26 will be at 3:30 pm in Room 740.

Speaker: Pierre Germain

Abstract: On a Riemannian manifold, consider the spectral projector on a thin spectral band $[\lambda , \lambda + \delta]$ for the Laplace-Beltrami operator. What is its operator norm from $L^2$ to $L^q$? Or, to put it in semiclassical terms, how large can the $L^p$ norm of a quasimode normalized in $L^2$ be? This is a fascinating problem, which is closely related to a number of fundamental analytic questions. I will try and describe what is known, and some recent progress that have been made. There will be some overlap with my talk at the Analysis seminar, but not much.

“Free” constructions in analysis

The HADES seminar on Tuesday, April 19 will be at 3:30 pm in Room 740.

Speaker: Mitchell Taylor

Abstract: We will discuss some aspects of the nonlinear geometry of function spaces, and how “free” constructions can shed light on such problems. The goal will be to show that certain nonlinear properties of function spaces are actually equivalent to linear ones.

Forty Years of Calderón’s Problem

The HADES seminar on Tuesday, April 12 will be at 3:30 pm in Room 740.

Speaker: Gunther Uhlmann

Abstract: Calderón’s problem (also called electrical impedance tomography) asks
the question of whether one can determine the electrical conductivity
of a medium by making voltage and current measurements at the
boundary. I will give a survey of some of the progress made on this
problem, including the more recent progress on solving similar
problems for nonlinear equations and nonlocal operators.

Galilean Theory of Dispersion and Scattering: Conservation laws, Blind Cones and the Increase of Uncertainty

The HADES seminar on Tuesday, April 5 will be at 3:30 pm in Room 740.

Speaker: Nima Moini

Abstract: In this talk, I will sketch a new approach to the study of kinetic equations solely under the assumption of conservation laws. The new idea is based on an uncertainty principle, the introduction of blind cones with respect to an observer and the Galilean invariance of different inertial frames of reference. In fact, as the uncertainty inevitably increases with time, particles will move away in an asymptotically radial manner from any fixed observer thereby establishing a new notion of dispersion. The generality of this approach reveals a mathematical relationship between the Landau and Boltzmann equations in the context of “the grazing collisions”, which until now was solely phenomenological. Moreover, I will discuss a new scattering theory for the kinetic equations and demonstrate its utility in the case of the Boltzmann equation for hard spheres. The new framework improves upon the existing results by proving the asymptotic completeness of the solutions of the Boltzmann equation near an equilibrium in the  $L^\infty$ setting. In particular, for any solution to the transport equation, there are arbitrarily close in $L^\infty$  norm, scattered solutions of the Boltzmann equation, this implies that solutions of the Boltzmann equation defined over the whole space will not converge to the state of thermodynamic equilibrium.

Construction of Initial Data for the Einstein Equation

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.

Weyl laws and closed geodesics on typical manifolds

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.

Construction of high-frequency spacetimes

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.

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.