On the spectrum of the clamped round plate. Daniel Rosenblatt
Abstract: Siegel has shown in 1929 that all multiplicities in the spectrum of an elastic round membrane are due to its rotational symmetry, i.e. every eigenfunction is of a separated form. We give some background on his solution and then we discuss the case of a clamped round plate. Is every eigenfunction of the clamped round plate of a separated form?
We prove that any eigenfunction is a linear combination of at most two separated ones, and we relate this problem to Schanuel's conjecture from Transcendental Number Theory. The talk is based on joint work with Dan Mangoubi.

Small ball estimates: what, how, and why. Dan Mikulincer
Abstract: Small ball estimates are a quantitative measure of how a function can concentrate near zero, often reflecting singularities in the function. In this talk, we focus on such estimates for functions in high dimensions. We will present known results, outline common proof techniques, and discuss their limitations. These limitations will naturally lead to more recent advances as well as some open questions. Finally, we discuss potential applications in different fields such as Harmonic Analysis and Optimization.

Graph theory and Scrambling of Quantum Information. Uzy Smilansky
Abstract: The introduction of quantum computing brought with it the need to adopt classical and well - founded concepts from Information theory to the Quantum Mechanical world. In particular, concepts like Information Scrambling, Chaos and its measure in terms of Lyapunov exponents had to be reformulated to coexist with tenets of Quantum theory such as Heisenberg's uncertainty principle, while complying with Bohr's correspondence principle. I will try to show how a new approach based on concepts and results from Graph Theory offers a solution to the above mentioned difficulties. This talk is based on a joint work with Sven Gnutzmann.

Cohomology for Linearized Boundary-Value Problems in Riemannian Geometry. Roee Leder
Abstract: In this two-part lecture, I shall present a framework I developed for casting the solvability and uniqueness conditions of linearized geometric boundary-value problems in cohomological terms. The theory is designed to be applicable without assumptions on the underlying Riemannian structure and provides tools to study the emergent cohomology explicitly. To achieve this generality, Hodge theory is extended to sequences of Douglas–Nirenberg systems that interact via Green’s formulae, overdetermined ellipticity, and a condition I call the order-reduction property, replacing the classical requirement that the sequence form a cochain complex. This property typically arises from linearized constraints and gauge equivariance, as demonstrated by several examples, including the linearized Einstein equations with sources, where the cohomology encodes geometric and topological data. In my talks, I shall cover the background and details required for the construction and will emphasize the key points of analytical interest. I will also present the necessary geometric concepts. The lectures are based on my manuscript arXiv:2504.18494.

Exploring Orthogonal Polynomial Ensembles via Recurrence Relations. Daniel Ofner
Abstract: Determinantal Point Processes (DPPs) are stochastic point processes whose probability distributions can be expressed in terms of the determinant of a correlation kernel. In many cases, the correlation kernel of a given DPP can be represented as a squared Vandermonde determinant. By applying elementary column operations, this correlation kernel can be identified with the Christoffel–Darboux kernel, a well-known object in the theory of orthogonal polynomials. Such processes are commonly referred to as Orthogonal Polynomial Ensembles (OPEs). OPEs naturally arise in random matrix theory and statistical mechanics. Two extensively studied special cases of these processes are the Circular Unitary Ensemble (CUE) and the Gaussian Unitary Ensemble (GUE). In this talk, we discuss the connection between DPPs and orthogonal polynomials, highlighting how known results about orthogonal polynomials can be leveraged to study the DPP corresponding to the OPE. In particular, we explore the well-known recurrence relations of the orthogonal polynomials and demonstrate that, through the bijection between the recurrence coefficients and the underlying measure, one can derive results regarding the asymptotic distribution of a given OPE. Based on a joint work with J.Breuer.

Schroedinger Type Operators with a Random Decaying Potential - a Review. Jonathan Breuer
Abstract: Random Schroedinger operators are models for quantum Hamiltonians of particles in disordered media. As such, their spectral theory has been a topic of intense interest in mathematical physics for several decades. The case of one dimensional operators with a random decaying potential has been introduced in the early 1980's as one where a transition from localized to extended states (still an open problem for extensive randomness in high dimensions) can be rigorously proven. Since then, this model and various related ones have been found to exhibit rich spectral properties and surprising connections to random matrix theory. In these two talks I plan to give an overview of the spectral theory of Schroedinger type operators with a random decaying potential. We shall present the various models, describe connections to random matrix theory and orthogonal polynomials, and review the main results and tools concerning both the infinite and asymptotic spectral theory.

Non-Euclidean Elasticity. Raz Kupferman
Abstract: I will present an introduction to the field of so-called incompatible elasticity, which apart of its applications in physics, can be viewed as a subfield of geometric analysis. The lecture will be as non-technical as possible, presenting main ideas and some open problems.

Laguerre Manin-Mumford Theorem. Avner Kiro
Abstract: This talk presents results analogous to the classical Manin-Mumford conjecture concerning roots of Laguerre polynomials. We consider "Laguerre special points" (roots of P_k(x)) and "Laguerre special subvarieties" in (C*)^n (defined by equations x_i=x_j or x_k=p, where p is a special point). Our main results state that an algebraic subvariety V ⊂ (C*)^n containing a Zariski-dense set of Laguerre special points must be a Laguerre special subvariety. The proof employs the Pila-Zannier strategy, combining o-minimal point counting (using Gevrey asymptotics and steepest descent for definability) with an Ax-Schanuel theorem tailored for the Laguerre differential equation. Joint work with G. Binyamini and J. Pila.

Nodal volume of harmonic functions. Lakshmi Priya M.E.

Abstract: Nadirashvili conjectured that the nodal volume (i.e., the appropriate Hausdorff measure of the zero set) of a non-constant harmonic function on \mathbb{R}^n is infinite, for n\geq 2. While the conclusion is straightforward in two dimensions, it proved to be a very difficult question in higher dimensions. Nearly a decade ago, in a remarkable feat, Logunov resolved this conjecture in all higher dimensions. Recently, together with Logunov and Sartori, we obtained almost sharp local estimates for the nodal volume of harmonic functions. This talk aims to provide an overview of a closely related topic: nodal volume of Laplace eigenfunctions on smooth compact manifolds, delve into the main ideas of Logunov that helped resolve Nadirashvili's conjecture, and conclude with a discussion of the recent result on local estimates.

Past Talks: Fall 2024:

Eigenfunctions of periodic Jacobi operators on trees. Eyal Seelig

Abstract: The theory of periodic Jacobi matrices on Z is rich and well-established, with diagonalization achieved through a set of special eigenfunctions known as Floquet solutions or Bloch waves. These eigenfunctions are governed by a phase parameter, which, as it varies from 0 to pi, reveals the band structure of the spectrum. In this talk we explore the generalization of this theory to periodic Jacobi operators on trees, namely lifts of Jacobi operators on finite graphs to their universal covers. We construct Floquet solutions that are governed by a single phase in certain tree models, despite the non-commutative symmetries of the tree. This contrasts with a general formula we prove for the density of states measure, where a different phase arises and plays a role in a new proof for gap labeling. Based on joint works with J. Banks, J. Breuer, J. Garza-Vargas, and B. Simon.

Shapes of trees. Oleg Ivrii

Abstract: A finite tree in the plane is *conformally balanced* if every edge has the same harmonic measure as seen from infinity, and harmonic measures on the two sides of every edge are identical. It is well known that a finite tree has a conformally balanced shape, which is unique up to scale. In this talk, we study shapes of infinite trees, focusing on the case of an infinite trivalent tree. To conformally balance the infinite trivalent tree, we truncate it at level n, form the true tree T_n and take n to infinity. We show that the Hausdorff limit of the T_n contains the boundary of the developed deltoid, the domain obtained by repeatedly reflecting the deltoid in its sides. This is joint work with P. Lin, S. Rohde and E. Sygal.

Distribution of powers of random unitary matrices through singularities of hyperplane arrangements. Itay Glazer

Abstract: Let X be a n by n unitary matrix, drawn at random according to the Haar measure on U_n, and let m be a natural number. What can be said about the distribution of X^m and its eigenvalues? The density of the distribution \tau_m of X^m can be written as a linear combination of irreducible characters of U_n, where the coefficients are the Fourier coefficients of \tau_m. In their seminal work, Diaconis and Shahshahani have shown that for any fixed m, the sequence (tr(X),tr(X^2),...,tr(X^m)) converges, as n goes to infinity, to m independent complex normal random variables (suitably normalized). This can be seen as a statement about the low-dimensional Fourier coefficients of \tau_m. In this talk, I will focus on high-dimensional spectral information about \tau_m. For example: (a) Can one give sharp estimates on the rate of decay of its Fourier coefficients? (b) For which values of p, is the density of \tau_m L^p-integrable? In the first part of the talk, I will answer (a) and (b). In addition, using works of Rains about the distribution of X^m, I will show how Items (a) and (b) are closely related (and that (b) is in fact equivalent) to a geometric problem about the singularities of certain varieties called (Weyl) hyperplane arrangements. In the second part of the talk, I will sketch the proof of this geometric problem, building on a known algorithm for resolution of singularities of hyperplane arrangements. Based on joint works with Julia Gordon and Yotam Hendel ([GGH]), and with Nir Avni and Michael Larsen ([AGL]).

The number variance of dilations of integer sequences. Nadav Yesha

Abstract: Let (x_n) be a sequence of positive integers. We will discuss the fluctuation of the number of elements modulo 1 of dilations (\alpha x_n) in short intervals, for generic values of alpha. The main motivation is to compare statistics such as the number variance with the random model, thereby observing pseudo-random behaviour for some interesting examples, such as x_n = n^2. Based on joint works with Zonglin Li and with Christoph Aistleitner (in progress).

Riesz bases of exponentials. Gady Kozma

Abstract: A Riesz basis is the second best thing after an orthonormal basis. We will survey the question: for a subset S of the real line, when does the space of functions on S have a Riesz basis of functions of the form exp(itx)? Some results presented are joint with Shahaf Nitzan and/or Alexander Olevskii.

Creating periodic points using holomorphic curves Shira Tanny

Abstract: An old question of Poincaré concerns creating periodic points via perturbations of Hamiltonian diffeomorphism. Hamiltonian diffeomorphisms are solutions of a certain ODE and arise from classical mechanics. In the first part of the talk, I will provide an overview of results related to this problem. In the second part, I will discuss how holomorphic curves—complex 1-dimensional submanifolds—can be used to study the existence and properties of periodic points of Hamiltonian diffeomorphisms. The talk is based on a joint work with Julian Chaidez.

An introduction to weighted polynomial approximation in the complex domain – part 2 Sasha Sodin

Abstract: This is the second part of the talk, but it should be mostly independent of the first part. We shall discuss the conditions under which a measure on a union of rays is uniquely determined by its Fourier transform. The result (based on joint work with Gady Kozma) is a generalisation of a theorem of Vul from 1959. The proof is based on a new Phragmén-Lindelöf principle, which may be of independent interest.

Irreducibility of Random Polynomials with Rademacher Coefficients Lior Bary-Soroker

Abstract: Consider a random polynomial whose coefficients are independent Rademacher random variables (taking the values ±1 with equal probabilities). A central conjecture in probabilistic Galois theory predicts that such polynomials are irreducible asymptotically almost surely as their degree approaches infinity. Here irreducibility is considered over the field of rational numbers. In the first part of the talk I will discuss the recent progress that has shown that this conjecture follows from the Generalized Riemann Hypothesis and that the limiting infimum of the irreducibility probability is positive. In the second part of the talk, we will explore ideas from the proof of the following result: the limiting supremum of the irreducibility probability is 1, unconditionally. Specifically, we demonstrate that along special sequences of degrees, the polynomial is irreducible asymptotically almost surely. This result is based on joint work with Hokken, Kozma, and Poonen.

An introduction to weighted polynomial approximation in the complex domain Sasha Sodin

Abstract: We shall review some, mostly classical, results pertaining to approximation of continuous functions on closed sets in the complex plane by polynomials. The talk will also serve as an introduction to a later talk, in which we shall, hopefully, discuss some results obtained in a joint work in progress with Gady Kozma.