Fall 2025

We investigate uniqueness sets on the real line for \(L^2\) functions whose Fourier transforms exhibit super-exponential decay. Such functions can be analytically extended to entire functions of a specified order \(\rho>1\). We establish two results concerning the density of their zeros. This theoretical framework generalizes existing results for the 2nd order case (\(\rho=2\)) to any order \(\rho>1\). Finally, we apply these findings to the Short-Time Fourier Transform (STFT) phase retrieval problem, demonstrating that if the window function belongs to this class, uniqueness of a signal can be guaranteed from its STFT intensity measurements on a sufficiently dense \(\rho\)-root lattice.

We investigate the thermodynamic formalism for Viana maps—skew products obtained by coupling an expanding circle map with a slightly perturbed quadratic family on the fibers. For every Hölder potential \(\varphi\) whose oscillation is below an explicit threshold, we show that an equilibrium state not only exists but is unique and satisfies an upper level-2 large-deviation principle. All of these conclusions persist under sufficiently small perturbations of the reference map.

The 1914 and 1915 Nobel Prizes in Physics—awarded for the discovery of X-ray diffraction and Bragg’s law—gave humanity, for the first time, a way to observe the lattice structure of matter by reading the dual lattice from the peaks in a diffraction experiment.

For the next seventy years it was believed that only periodic atomic arrangements (true lattices) could exhibit long-range order, meaning a pure-point diffraction measure. This view was overturned in 1984 with Dan Shechtman’s discovery of quasicrystals—non-periodic solids displaying sharp diffraction peaks—a breakthrough recognized by the 2011 Nobel Prize in Chemistry. Mathematicians such as Bombieri, Taylor, and Meyer developed the theory explaining these aperiodic structures. Unlike periodic lattices, whose diffraction is supported exactly on a lattice, a quasicrystal’s diffraction lives on a dense set of peaks that becomes effectively discrete only after an intensity cut-off.

Fourier Quasicrystals (FQs) sharpen this concept: they are sets whose diffraction (Fourier transform of the counting measure) is pure point with genuinely discrete support. Trivial examples include lattices and finite unions or translates of lattices.

For two decades it was widely believed that no non-trivial FQs exist, a belief encapsulated in Lagarias’s conjecture and proved by Lev and Olevskii in 2016. This changed in 2020 when Kurasov and Sarnak constructed the first truly non-periodic one-dimensional FQ and introduced a general method based on Lee–Yang polynomials.

In this talk I will present our result showing that all one-dimensional FQs arise from the Kurasov–Sarnak construction. I will then describe our recent work extending the theory to all dimensions via a new class of high-codimension algebraic varieties that we call Lee–Yang varieties.

This talk is based on joint work with Alex Cohen, Cynthia Vinzant, Mario Kummer, and Pavel Kurasov, and is intended for a broad mathematical audience—no prior expertise required.

Given i.i.d observations on a space \(S\), we study the spectral properties of the associated empirical graph Laplacian. Our main results shows that the eigenspaces of the empirical graph Laplacian are close to the eigenspaces of a Laplacian on \(S\). In our analysis, we connect empirical graph Laplacians to a resistance form via electric network theory, and to kernel PCA and consider the heat kernel of \(S\) as a reproducing kernel feature map, or the Green’s function in terms of a resistance metric. This approach leads to novel points of view.

Let \(S^n\) be the \(n\)-sphere in \(\mathbb R^{n+1}\). The first part of the talk concerns the modified wave equation on \(S^n\) \[ \Delta u=\left(\frac{\partial^2}{\partial t^2} +\left(\frac{n-1}{2}\right)^2\right) u, \] for a function \(u(x,t)\in C^\infty(S^n\times\mathbb R)\), where \(\Delta\) is the Laplace-Beltrami operator. This equation was first studied by Lax and Phillips in 1978. We will discuss their solution in terms of spherical harmonics, and how that leads to Huygens’ principle. We show how this series solution, as well as a Radon transform approach, leads to a closed form solution in terms of mean values. Finally we discuss the snapshot problem for the above wave equation.

The second part of the talk concerns the generalized Euler-Poisson-Darboux equation on \(S^n\), given by \[ \Delta u=\left(\frac{\partial^2}{\partial t^2}+(n-1+2\alpha)\,(\cot t)\,\frac{\partial}{\partial t}+\alpha(n-1+\alpha)\right)u, \] where \(\alpha\) is an arbitrary complex parameter. If \(\alpha=0\) or \(\alpha=1\), this equation characterizes the range of the mean value operator over balls and spheres, respectively, of radius \(t\). We will discuss the solution of this equation in terms of a fractional convolution operator on \(S^n\).

It is well known that a discrete sum can be viewed as an integration with respect to a suitable measure. In this talk, we explore how this simple yet powerful idea can unify perspectives in signal processing and machine learning.

In the first part, we generalize the classical least-squares problem to a probabilistic function approximation problem over a reproducing kernel Hilbert space (RKHS). This formulation extends the kernel method widely used in machine learning and statistics. We establish the existence and uniqueness of the optimal solution under mild assumptions and derive a probabilistic representer theorem, showing that the optimizer admits an integral representation. In particular, when the measure is finitely supported or the Hilbert space is finite-dimensional, the problem reduces to a measure quantization problem, revealing a natural connection between probabilistic approximation and sampling theory.

In the second part, we turn to finite frames and their generalization to probabilistic frames, obtained by interpreting finite frames as discrete probability measures. This probabilistic viewpoint enables the use of tools from optimal transport and the Wasserstein distance to study frame properties and related minimization problems. We will introduce the fundamental concepts and discuss frame perturbations and energy minimization problems. If time permits, we will conclude with some applications and open questions arising from this integration framework.

In the modern landscape of machine learning and artificial intelligence, deep neural networks (DNNs) have demonstrated remarkable success across various domains. However, despite their widespread empirical effectiveness, our theoretical understanding of DNNs remains in its early stages. In this talk, I will present results on the estimation of non-parametric functions using deep neural networks, emphasizing how these models adapt to underlying low-dimensional compositional structures in data. Specifically, we show that when the true data-generating function possesses such a structure, DNNs can exploit it to achieve faster convergence rates. Additionally, I will discuss the minimax optimality of DNNs, with respect to both the sample size and the covariate dimensionality. Finally, I will highlight our recent findings on the analysis of Physics-Informed Neural Networks (PINNs), providing theoretical justification for their effectiveness in learning physical processes. These results contribute to a deeper understanding of why and when DNNs excel in complex function estimation tasks.