Talks (titles and abstracts)

Pedro Abdalla Teixeira: tba

Antonio Avilés Lopez: Compact spaces associated to Banach lattices
We review some results and problems from a joint work with Martínez Cervantes, Rueda Zoca and Tradacete, that establishes a connection between the structure of Banach lattices and the topology of compact spaces.

Peter Balazs: tba

Radu Victor Balan: Sorting based embeddings of quotient metric spaces
Consider a finite dimensional real vector space and a finite group acting unitarily on it. We study the general problem of constructing Euclidean stable embeddings of the quotient space of orbits. Our embedding is based of sorted coorbits. We obtain sufficient conditions for injective and stable embeddings. In particular, we show that, whenever such embeddings are injective, they are automatically bi-Lipschitz. Additionally, we demonstrate that stable embeddings can be achieved with reduced dimensionality, and that any continuous or Lipschitz G-invariant map can be factorized through these embeddings. This talk is based on joint works with Efstratios Tsoukanis and Matthias Wellershoff. arXiv: 2308.11784 , 2310.16365, 2410.05446

Francesca Bartolucci: tba

María J. Carro: Connection between Boundary Value problems for the Laplacian and Muckenhoupt weights
We shall present several classical boundary value problems for the Laplacian on Graph Lipstchitz Domains in the complex plane which can be solved using Harmonic Analysis techniques via the use of Muckenhoupt weights. We will start by introducing these weights and their main properties, present the original technique due to C. Kenig in the 80’s dealing with the Dirichlet problem and move up to a recent result related with the solution of a transmission problem.
The talk will be based on a number of works in collaboration with Teresa Luque, from Universidad Complutense de Madrid, Virginia Naibo from Kansas State University, Carmen Ortiz-Caraballo from Universidad de Extremadura and María Soria from Rutgers University.

Dan Edidin: Stability of the generalized phase retrieval problem
The classical phase retrieval problem involves estimating a signal from its Fourier magnitudes (power spectrum) by leveraging prior information about the desired signal. This paper extends the problem to compact groups, addressing the recovery of a set of matrices from their Gram matrices (the second moment). In this broader context, the missing phases in Fourier space are replaced by missing unitary or orthogonal matrices arising from the action of a compact group on a finite-dimensional vector space. This generalization is driven by applications in multi-reference alignment and single-particle cryo-electron microscopy, a pivotal technology in structural biology. The main mathematical result we will present identifies priors for which the (square root) of the second moment is a bi-Lipschitz map.

Frank-Dieter Filbir: tba

Dorsa Ghoreishi: tba

João Gonçalves Ramos: tba

Philipp Grohs, Universität Wien: tba

Philippe Jaming: Gabor phase retrieval via semidefinite programming
In this talk we consider the problem of reconstructing a signal $f\in L^2(\mathbb{R})$ from from sampled phaseless measurements of its Gabor transform

$Gf(x,\omega)=\int_{\mathbb{R}} f(t)e^{-\pi (t-x)^2}e^{2i\pi t\omega}\,\mbox{d}t.$

It is now rather well understood that $Gf$ should be "well-connected" for the problem of reconstructing $f$ from $\{|Gf(x,\omega)|, (x,\omega)\in R^2\}$ to be stable.

Here we propose an algorithm that allows to solve the reconstruction problem of $f$ on some part of $A \subset R$ from sampled measurements $\{|Gf(x,\omega)|, (x,\omega)\in\Omega\}$ where $\Omega$ is a sufficiently dense part of $R^2$ on which $Gf$ is well-connected.

The algorithm amounts to solving two convex problems and is, as such, amenable to numerical analysis. We show,that the scheme accurately reconstructs from noisy data within the connected regime. The approach is based on ideas from complex analysis, Gabor frame theory as well as matrix completion.

This is joint work with Martin Rathmair

Felix Krahmer: tba

Denny Ho-Hon Leung: Stability of isometries between the positive cones of ordered Banach spaces
Let $U$ and $V$ be respective subsets of Banach spaces $E$ and $F$. A mapping $T:U\to V$ is an $\varepsilon$-isometry if

$\bigl|\|Tx_1 - Tx_2\| - \|x_1-x_2\|\bigr| \leq \varepsilon \quad \forall x_1,x_2\in U.$

The Hyers-Ulam stability problem for given $U$ and $V$ asks if there is a constant $K$ so that every $\varepsilon$-isometry can be approximated by a true isometry $L$ up to $K\varepsilon$, i.e., $\|Tx - Lx\| \leq K\varepsilon$ for all $x\in U$.
In this talk, I will describe a general scheme for establishing Hyers-Ulam stability of $\varepsilon$-isometries mapping between the positive cones of certain ordered Banach spaces, including $C(X)$,$C^*$-algebras, $L^p$ and the Schatten $p$-classes, $1<p<\infty$. If time permits, I will also discuss the case of $L^1$ and a partial solution for the case of trace class operators.

[Joint with Y. Dong and L. Li.]

Lukas Liehr: Non-uniform phaseless sampling
For a given function space, we consider the question of whether every function in the space can be uniquely determined, up to a phase factor, by samples of its modulus. We present recent results showing that unique determination fails when the sampling set possesses a rigid structure. On the other hand, uniqueness can be achieved by employing non-uniform sampling sets.

Dustin Mixon: tba

Timur Oikhberg: Phase retrieval in Banach lattices: some finite dimensional phenomena
We consider Stable Phase Retrieval (SPR) in the setting of Banach lattices; specifically, we say that a subspace E of a Banach lattice X does SPR if an element of E can be reconstructed "in a robust manner" from its modulus. We examine some finite dimensional phenomena, for instance:
(i) Given a finite dimensional Banach lattice X,what is the largest dimension of its SPR subspace? More generally, if F is a finite dimensional of a Banach lattice, what is the largest dimension of an SPR subspace contained in F?
(ii) Conversely, suppose X is a Banach lattice of dimension m. It is known that, if E is an SPR subspace of X of dimension n, then m>2n-2. How bad can the SPR constant be when the dimension of E is close to the largest possible?

Benjamin Pineau: Constructing Infinite-Dimensional SPR Subspaces
In this series of talks, I will discuss a simple construction which can be used to generate several natural examples of infinite dimensional subspaces of $L^2(\mu)$ which do stable phase retrieval (SPR). More precisely, we aim to construct closed subspaces $V \subset L^2(\mu)$ which satisfy the following stability estimate,

$\min_{|\lambda| = 1} \|f - \lambda g\|_{L^2} \leq C \||f| - |g|\|_{L^2}, \quad \forall f,g \in V$

for some constant $C > 0$. The algorithm I will outline was used to identify the first examples of complex infinite dimensional SPR subspaces of $L^2(\mu)$. This talk is based on joint works, each co-authored with a combination of Michael Christ, Daniel Freeman, Timur Oikhberg and Mitchell Taylor.

Palina Salanevich: PtyGenography: generative priors as regularizers for the phase retrieval problem
In phase retrieval and similar inverse problems, the stability of a solution under different noise levels is crucial for practical applications. To address instabilities, one often employs regularization techniques. However, Tikhonov and other conventional regularizers tend to smooth out high-frequency components, which can be problematic when trying to capture detailed features of a signal. Recently, generative models have emerged as a powerful alternative, allowing the incorporation of prior information on the signal into the problem and thereby enhancing reconstruction stability. The rationale here is that the conditioning of the composition of the generative model and the measurement map is more favorable than that of the measurement map alone, albeit at the cost of introducing a bias in the reconstruction. It has indeed been observed in numerical experiments that for high signal-to-noise ratio, the conventional reconstruction model performs better, while in the case of low signal-to-noise ratio, the generative reconstruction model outperforms it. In this talk, we will explore and compare the reconstruction properties of classical and generative inverse problem formulations and propose a new unified reconstruction approach that mitigates overfitting to the generative model for varying noise levels.
The talk is based on the joint work with Selin Aslan, Tristan van Leeuwen, and Allard Mosk.

Alberto Salguero: Free p-Banach lattices
The introduction of free objects in Banach lattices has been carried out by several groups of authors and proved to be  useful to study the properties of Banach lattices and its interaction between Banach spaces. However, when one eliminates local convexity from the picture, the situation changes drastically and many questions appear. In this talk we explore the not so well-known world of quasi-Banach and p-Banach lattices (for 0<p<1). In particular, we focus on the existence and properties of free p-Banach lattices associated to a p-Banach space, and some of its applications to the theory of p-Banach lattices, which has yet to be explored.
This is part of a joint work with P. Tradacete (ICMAT) and N. Trejo Arroyo (ICMAT-UCM).

Vladimir G. Troitsky: tba

Matthias Wellershoff: Uniqueness of phase retrieval from sampled STFT measurements: a short overview
This talk explores the problem of reconstructing a function from sampled magnitudes of its short-time Fourier transform (STFT). We will examine scenarios in which recovery is possible and ones in which it is not. The presentation aims to provide a broad overview of the developments in this area over the past five years, making the material accessible to newcomers. If time allows, we will also discuss some recent advancements.

Yu Xia: tba