Program

Abstract: Steinerberger (2019) discovered a surprising connection between the root distribution of derivatives of random polynomials and free additive convolution powers. This connection  was later rigorously established by Hoskins and Kabluchko (2021).

In this talk, we'll see how tools from finite free probability not only lead to remarkably simple and elegant proofs of this result, but also open the door to new results about polynomials and random polynomials. Along the way, we’ll draw on ideas from random matrix theory and free probability to guide our understanding.

This talk is based on  joint works with A. Campbell, K. Fujie, J. Garza-Vargas, D. Perales, and Y. Ueda and J. Vazquez Becerra.

Abstract: We consider a random Gaussian model of Laplace eigenfunctions on the hemisphere satisfying the Dirichlet boundary conditions, and random Laplace eigenfunctions on the unit square subject to Dirichlet boundary conditions. We test M. Berry’s ansatz on nodal deficiency in presence of boundary, in particular the square billiard is studied, where the high spectral degeneracies allow for the introduction of a Gaussian ensemble of random Laplace eigenfunctions.  Based on join works with Giacomo Cherubini, Domenico Marinucci, Igor Wigman. 

Abstract: 

Consider a centered real stationary Gaussian process $f(t)$. What is the probability of a significant deviation of the number of zeroes in a long interval $[0,T]$ from its expectation? Such events are known as overcrowding and undercrowding of zeroes (depending on whether the deviation is above or below expectation).

We show that when the spectral measure of the process is compactly supported, then there is a sharp phase transition in the overcrowding probability: There is a constant $c$ such that, for any $\epsilon>0$, the probability of having more than $(c+\epsilon)T$ zeroes has Gaussian decay in $T$, while having more than $(c-\epsilon) T$ has exponential decay in $T$.

If, in addition, the spectral measure has a spectral gap near the origin, then a similar phase transition occurs for the undercrowding probability.

The methods involve a combination of tools  from the theory of Gaussian processes and complex analysis. Joint work with Ohad Feldheim & Lakshmi Priya.

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Abstract: Finite free convolutions are operations on polynomials of fixed degree that serve as finite-dimensional counterparts of additive and multiplicative free convolution. They provide an algebraic way to assemble complicated polynomials from elementary building blocks while retaining control over their structure, zeros, and asymptotic behavior.

The first part of the talk will survey applications of this approach to special functions. Many multiparameter families of hypergeometric polynomials, as well as certain multiple orthogonal polynomials, admit representations as finite free convolutions of simpler factors. These representations yield results on the location and real-rootedness of their zeros, interlacing and monotonicity with respect to the parameters, and asymptotic zero distributions described through free convolutions of limiting measures. I will also briefly discuss how this construction extends, after an appropriate deformation, to the q-hypergeometric setting.

The second part will report on work in progress concerning the origins of such operations. Under suitable hypotheses, a linear operator on polynomials with a sufficiently rich family of eigenfunctions gives rise to a convolution, and the resulting convolution algebra can be canonically identified with a quotient algebra determined by the operator’s spectrum. This viewpoint places familiar finite free convolutions within a common framework, suggests a systematic way to construct new ones, and reduces the preservation of real-rootedness to a generalized coefficientwise-multiplier problem.

This talk is based on my ongoing collaboration with Rafael Morales, Daniel Perales, and Thomas Wolfs.

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Abstract: Consider polynomials which take integer values on the integers (IVP). Elkies and Speyer, answering a question of Dimitrov on MathOverflow, showed that there is an exponential growth threshold for IVPs on the natural numbers. Roughly speaking, there are infinitely many IVPs with a growth rate above the threshold and only finitely many IVPs below that threshold (of arbitrary degree).

I will provide some background on this problem and describe a more general problem, where there is a growth condition on the integers, and its connection with logarithmic capacity.

Joint work with Avner Kiro.

Abstract: This talk starts with my earlier work with Jürgen Angst and Guillaume Poly, where we proved that universality persists for random polynomials with weakly dependent coefficients. In this talk, I will present several open problems and conjectures for interpolating between weak dependence and strong dependence.

Abstract: Let p_n be a degree n random polynomial with independent, identically distributed, rotationally invariant roots. I will present a result concerning the zeros obtained after differentiating p_n approximately tn times, where 0<t<1. Under a finite logarithmic moment assumption, the empirical distribution of the zeros converges to a deterministic rotationally invariant measure, which can be computed explicitly from the initial radial distribution. This establishes a conjecture of Hoskins and Kabluchko. 

Abstract: Recent works of Galligo, Najnudel, and Vu (2025) and Najnudel and Vu (2026) study repeated differentiation for polynomials of the form P(z)=p(z^m), where p is a deterministic polynomial of degree n with real, non-negative roots, in the regime where m and n are large. If m>> log(n) and the root distribution of P converges to a compactly supported, radial probability measure mu_0, these works show that for 0< t<1, the root distribution of the nmt-th derivative of P converges to a compactly supported probability measure mu_t given by an explicit formula for its radial quantile function. 

In this talk I will:

1) present a simplified proof of this result,

2) extend it to repeated applications of the differential operator z^a(d/dz)^b, and

3) compute the limiting root distribution in the case when m is fixed and n tends to infinity.

Based on a joint work with Brian Hall (arXiv 2607.16954).

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Abstract: I will discuss recent developments concerning moments of characteristic polynomials in non-Hermitian random matrix ensembles, focusing on truncations of Haar-distributed unitary matrices. The corresponding characteristic polynomial moments exhibit a rich structure, including connections with Painlevé transcendents. Depending on the regime considered, Painlevé IV, V or VI can occur. These moments are also naturally related to a class of orthogonal polynomials in the complex plane.

For these polynomials, planar orthogonality can be reformulated as a Riemann–Hilbert problem on suitable contours. I will describe some aspects of the resulting asymptotic analysis, based on the Deift–Zhou steepest descent method, and explain how it yields strong asymptotics for both the orthogonal polynomials and the characteristic polynomial moments. This is based on joint work with Alfredo Deaño and, more recently, with Kenneth McLaughlin and Leslie Molag.

Abstract: Given a compact set K in the plane, the Chebyshev polynomials for K are the monic polynomials of given degree with minimal supremum norm over K. Understanding the connection between the geometry of K and the analytic asymptotic properties of the Chebyshev polynomials is a classical question in approximation theory. 

The fundamental case when K is an analytic (closed) Jordan curve was understood by Faber more than a century ago, and the ""multi-cut"" case of several Jordan curves was described by Widom. This talk is about the case when K is a single Jordan arc, which has proven surprisingly elusive. 

We will discuss recent work on a conjecture on the asymptotics of the norms (“Widom factors”) of the Chebyshev polynomials for analytic Jordan arcs, going back to Widom's work. Our approach relies on a version of Faber polynomials and a connection to discrete orthogonal polynomials due to Remez and Vidensky.

This is based on joint work with Benedikt Buchecker, Benjamin Eichinger and Olof Rubin.

Abstract: For a monic polynomial p of degree n, the t-lemniscate is defined as the sub-level set Λₚ(t) := {z ∈ ℂ : |p(z)| < t}. For t = 1, this is referred to as the unit lemniscate or simply lemniscate. The typical topological behavior for random lemniscates exhibits a fascinating dependence on the root distribution μ. In this work, we investigate two contrasting regimes:

• Disk Model: For roots i.i.d. in 𝔻, the expected number of components grows sub-linearly as C√n (C ≈ 0.453).

• Circle Model: For roots i.i.d. on S¹, the expected number of components is drastically higher, growing linearly at a rate of n/2.

We further characterize the topological phase transition as the level t evolves from 0 to ∞, identifying the critical thresholds where the lemniscate's connectivity undergoes a fundamental shift.

Abstract: We study the complex partition function of the continuous random energy model (CREM) on supercritical Galton--Watson trees in the regime of strong correlations, characterized by strict concavity of the covariance profile. We rigorously identify all non-glassy phases of the model and confirm the high-temperature part of the phase diagram conjectured by Kabluchko and Klimovsky (2014). In contrast to the complex random energy model, the strongly correlated CREM exhibits three distinct high-temperature regimes: an expectation-dominated phase, a fluctuation-dominated phase, and a mixed fluctuation--expectation phase that is absent in the REM and in homogeneous branching Brownian motion, and that arises here from the strict concavity of the covariance profile.

The latter regime arises from an interior saddle point in the second-moment analysis and reflects a genuinely multiscale interference phenomenon induced by the curvature of the covariance profile. We prove central limit theorems with explicit, phase-dependent normalizations and random variances given by martingale limits. The analysis relies on two main tools of independent interest: a Lindeberg--Feller-type central limit theorem for random recursive structures, and a coupling of CREMs with different covariance profiles that replaces Gaussian comparison arguments unavailable in the complex setting. The talk is based on a joint work with Lisa Hartung and Maximilian Fels. 

Abstract: A distinguishing feature of real non-Hermitian random matrices is presence of purely real eigenvalues with high probability. In this talk, we discuss the counting statistics for these real eigenvalues for the real Ginibre ensembles. In particular, we discuss its intermediate large deviation regime, which captures the asymptotic behavior bridging typical fluctuations and the large deviation regime characterized by the macroscopic deformation of the spectral droplet. This talk is based on joint work with Sung-Soo Byun, Gregory Schehr, and Jonas Jalowy.

Abstract: This talk explores the distribution of zeros in two distinct settings: deterministic entire functions and stochastic disordered systems. In both cases, zero locations encode structural properties -- analytic in the former, and thermodynamic in the latter.

Part I: The Geometry of Coefficients. We begin with the Laguerre--Pólya class of entire functions that serve as uniform limits of real-rooted polynomials. We present necessary conditions and sufficient conditions for real-rootedness in terms of the Taylor coefficients aₖ. Specifically, we characterise membership in this class via the second quotient sequence qₙ(f) := aₙ₋₁²/(aₙ₋₂aₙ). We show that the asymptotic behaviour of qₙ(f) determines whether f is real-rooted. We conclude by discussing operators that preserve real-rootedness. This is joint work with Anna Vishnyakova.

Part II: The Physics of Zero-Freeness. Motivated by Lee-Yang theory, we study the spherical Sherrington-Kirkpatrick (SSK) model at complex temperatures. We provide a rigorous derivation of the phase diagram conjectured by Obuchi & Takahashi (2012). Using eigenvalue rigidity and steepest descent analysis, we identify three distinct regimes in the complex β-plane: a high-temperature phase (|β| < 1/2), a glassy phase (|Re β| > 1/2), and a fluctuation phase (|β| > 1/2, |Re β| < 1/2) where Lee-Yang (Fisher) zeros of the partition function accumulate with constant density 2/π. We characterise the macroscopic zero density and prove that, in the fluctuation phase, the microscopic zero process converges in distribution to the zero set of a planar Gaussian analytic function. This is joint work with Anton Klimovsky.

Together, these results illustrate how zero distributions -- whether determined by coefficient constraints or statistical fluctuations -- encode structural information in both analysis and statistical physics.

Abstract: I will discuss the asymptotic distribution of zeros of orthogonal polynomials whose recurrence coefficients are unbounded and periodically modulated. In this setting, some zeros may escape to infinity, so the classical normalized zero counting measures are no longer the right object.

Under natural assumptions, we identify the correct scaling and prove convergence to an explicit infinite Radon measure. A key feature is that the limiting measure is determined by the essential spectrum of the underlying Jacobi matrix, giving a spectral description of zero distribution beyond the bounded-coefficient case.

The talk will be based on joint work with Bartosz Trojan: “Asymptotic zeros’ distribution of orthogonal polynomials with unbounded recurrence coefficients,” Journal of Functional Analysis 289 (2025), Article 111162.

Abstract: I will introduce a new notion of characteristic polynomials for tensors via Grassmann integrals and distributions of roots of random tensors. As anticipated, the Fuss-Catalan distribution appears. Some important points of our work are that there are only N roots for tensors of dimension N, and that our formulation covers general tensors (complex, real, Grassmann, etc) including totally antisymmetric tensors, but excluding totally symmetric tensors. It is based on arXiv:2510.04068[math-ph].

Abstract: Two symmetric integer matrices are cospectral if they share characteristic polynomial. Cospectrality is believed to be atypical in high-dimensional settings, but proving this remain a bit of a mystery. 

I will discuss a notion where the potential orthogonal matrix realizing cospectrality is restricted to have rational entries. Sufficient conditions to rule out this type of cospectrality are available in terms of the prime factorization of the discriminant, the latter being an integer expressed as the product of squared distances between eigenvalues. Some progress on understanding how often such conditions are applicable was recently made by Nikita Lvov and myself in arXiv:2603.26932. I will discuss the probabilistic methods that we developed and the conjectures they lead one to.

Abstract: Definition.

A sequence of nonnegative numbers (aₖ)ₖ₌₀^∞ is called a totally positive sequence, if all minors of the infinite matrix

\begin{array}{ccccc}

   a_0 & a_1 & a_2 & a_3 &\ldots \\

   0   & a_0 & a_1 & a_2 &\ldots \\

   0   &  0  & a_0 & a_1 &\ldots \\

   0   &  0  &  0  & a_0 &\ldots \\

   \vdots&\vdots&\vdots&\vdots&\ddots

 \end{array}

are non-negative.

Concept of total positivity found numerous applications and was studied from many different sides. It has applications in distribution of zeros of polynomials and entire functions, Pólya frequency sequences, unimodality and log-concavity, stochastic processes and approximation theory, mechanical systems, planar networks, combinatorics and representation theory.

Let P, P(0) > 0, be a real polynomial of degree m. It is easy to check that

∑ₖ₌₀^∞ P(k)xᵏ = Q(x)/(1 − x)ᵐ⁺¹,

where Q is a real polynomial of degree at most m.

We will discuss the following problem: for which P the sequence (P(k))ₖ₌₀^∞ is totally positive? Due to the famous theorem by Aissen, Schoenberg, Whitney and Edrei it happens if and only if all the zeros of Q are real and non-positive.

We also consider the sequences of the form

((1 − c₁qᵏ)(1 − c₂qᵏ) · … · (1 − cₘqᵏ))ₖ₌₀^∞,

where 0 < q < 1. We investigate the following problem: for which values of parameters such sequences are totally positive.

Abstract: We consider the biorthogonal polynomials pₙ, qₙ subject to the orthogonality

∫ pₘ(x) qₙ(f(x)) W(x) dx = δₘ,ₙ hₘ.

They are generalizations of orthogonal polynomials, with f(x) = x being the special case. If f(x) = xᶿ, these biorthogonal polynomials are related to the Muttalib-Borodin ensemble. We show that the asymptotics of the biorthogonal polynomials with f(x) = x² can be used to study the local statistics of the eigenvalues of the Hermitian random matrix model with external source. In the case where the potential of the random matrix model is V(x) = x⁴/4 − tx²/2 and the external source has two distinct eigenvalues ±a of equal multiplicities, we find a new family of limiting correlation kernels that are constructed from solutions of the Riemann-Hilbert problem associated with the Boussinesq hierarchy. This talk is based on arXiv:2512.20343 and is joint work with Shui-Xia Xu.

Abstract: I will explain how the average characteristic polynomials of derivative type ensembles (de)compose naturally under certain polynomial convolutions. Derivative type ensembles were originally introduced because they provided the natural framework to describe the eigenvalues of sums and products of random matrices. In those settings, the relevant polynomial convolutions are the finite free additive and multiplicative convolution. Our framework goes beyond and leads to a variety of polynomial convolutions, including those compatible with discrete grids.