Schedule & Abstracts

Titles & Abstracts

Giorgio Cipolloni:

A tale of large random matrices and logarithmically correlated fields

We will review recent results in random matrix theory, with a focus on spectral properties of large non-Hermitian matrices with independent, identically distributed entries. We will then discuss an intriguing connection of such matrices with the theory of logarithmically correlated fields and with the fluctuations of their extremes.

Cristiana De Filippis:

Nonuniform ellipticity and nonlinear potentials

The representation formula for the Poisson equation gives an explicit expression of solutions in terms of the data, yielding zeroth- and first-order pointwise bounds via convolution with suitable Riesz potentials. Their mapping properties allow for a sharp regularity transfer from data to solutions, so that nonlinear PDEs can be treated, up to the C^{1}-level as they were linear. I will then discuss a novel potential-theoretic approach to the (ir)regularity of solutions to certain nonuniformly elliptic PDEs arising in geometric and physical models.

Rodica Dinu:

From maximum likelihood to chromatic polynomials

This presentation explores the interplay between Algebraic Geometry, Combinatorics, Algebraic Statistics, and Topology, using the concept of linear space of matrices as a unifying link. We begin by illustrating how the coefficients of a graph's chromatic polynomial are recovered through the geometry of Cremona transformations.

The talk then addresses Gaussian concentration models, defining the ML-degree as a measure of complexity and relating it to the degree of rational maps. We then highlight key conjectures, in particular a conjecture of Drton, Sturmfels, and Sullivant, that we have recently proved. Furthermore, we connect the Euler characteristic of projective hypersurfaces to the multidegree of gradient maps, providing topological insights into determinantal varieties.

Jacek Jendrej:

Recent progress on the problem of soliton resolution

Dispersive partial differential equations are evolution equations (that is, involving the time variable) whose solutions preserve the energy, but can still decay in large time due to the fact that various frequencies propagate with distinct velocities. In some cases, there exist special solutions called solitons, which do not change their shape as time passes. The Soliton Resolution Conjecture predicts that, apart from exceptional cases, solitons are the only obstruction to the decay of solutions. More precisely, every solution eventually decomposes into a superposition of solitons and a decaying term called radiation.

We will discuss the conjecture in the context of the critical wave maps equation, which is the analog of the wave equation for maps from R2 to S2. The solitons correspond to harmonic maps, which were classified by Eells and Wood in 1976. We consider equivariant solutions, which are solutions having a specific symmetry preserved by the flow. In a joint work with Andrew Lawrie, we prove that soliton resolution holds for these solutions. Our proof hinges on an analysis of collisions of solitons and an appropriate localized Lyapunov functional, which together allow to prove a no return lemma for multisoliton configurations.

Building on some of these ideas, we solve, in a joint work with Andrew Lawrie and Wilhelm Schlag, an analogous problem for the harmonic map heat flow of Eells and Sampson (1964) without assuming any symmetry of the initial data.

Relinde Jurrius:

q-Analogues in combinatorics

Roughly speaking, a q-analogue in combinatorics is what happens if we generalize from sets to finite dimensional vector spaces over finite fields. For example, a combinatorial design consists of a finite sets of points, and a family of subsets of this set that all have the same size, such that every pair of points is in exactly one set of this family. For the q-analogue, we start with a finite dimensional vector space, and define a family of subspaces that all have the same dimension, such that every two-dimensional space is in exactly one set of this family.

At first sight, this might look like a rather straightforward exercise. And sometimes that is true. But also sometimes q-analogues are very nontrivially, or do not even exist. Furthermore, it can happen that two statements about sets are equivalent, while their q-analogues are not.

In this talk we will see many examples and non-examples of q-analogues, and we will dive into linear algebra over finite fields to get some intuition on why q-analogues can be difficult, but also fun.

Carla Rizzo:

Polynomial Identity Theory: Ideas, Methods, and Open Problems

Algebra is built on identities—rules like commutativity and associativity that remain true no matter what values we plug in. More generally, an identity is an expression that always holds within a given algebraic structure. In this talk, we focus on a rich and subtle type called polynomial identities: noncommutative polynomials that vanish no matter how their variables are replaced by elements of a given algebra.

The study of such identities—known as polynomial identity theory—offers a powerful way to understand and classify algebraic structures. Instead of examining an algebra directly, we explore the “laws” it satisfies, revealing deep structural information. Since its development in the mid-20th century, this approach has led to deep results connecting ring theory with combinatorics and representation theory. At the same time, some fundamental problems remain open. A notable example is the longstanding challenge of describing all polynomial identities of matrix algebras of size greater than two, which continues to motivate much of the field.

In this talk, I will present an overview of polynomial identity theory, focusing on its central ideas and methods, and highlighting how combinatorial techniques play a central role in both classical results and current research.

Discussion session topics:

• Publications

• Teaching

• Well-being

• Funding and opportunities

• AI

• Interdisciplinary research

• Career pathways

• Popularisation of maths

(These are indicative broad areas right now. We will list specific topics in the coming weeks.)