Schedule & Abstracts

Titles & Abstracts (Plenary, alphabetical)

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.

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.

Dalimil Peša:

Rearrangement-invariant quasi-Banach function spaces and their amalgams

The amalgam approach to function spaces is based on the following idea: to examine the "local" behaviour of functions separately from their "global" behaviour. Intuitively speaking, "local" behaviour is the behaviour on sets that are in some sense small, i.e. the blow-ups of functions, while "global" behaviour is the behaviour "near infinity", i.e.the decay of functions. Those two types of behaviours typically do not interact with each other, whence their separate examination often yields valuable insights and understanding.

While this approach is classical in e.g. harmonic analysis, in the field of function spaces it appeared only sporadically until rather recently, when it started attracting more attention. This talk will present some of the recent results that benefited from the amalgam way of thinking, as well as explain some of the tools that are currently available in this field.

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.

Titles & Abstracts (Short talks, in order of appearance)

Tuesday

Olena Atlasiuk:

Title: On approximation properties of inhomogeneous multipoint boundary-value problems

We study a wide class of linear inhomogeneous boundary-value problems for $r$th-order systems of ODEs. The solutions belong to the Sobolev spaces $(W^{n+r}_p)^m$, $n\in\mathbb{N}\cup\{0\}$, $m, r \in \mathbb{N}$, $1\leq p\leq \infty$. The boundary conditions are of the most general form $By=c$, where $B$ is an arbitrary continuous operator from $(W^{n+r}_p)^m$ to $\mathbb{C}^{rm}$. Thus, they may contain derivatives of the unknown vector function of integer and/or fractional orders $\geq r$. We prove that the solutions of the original problems can be approximated in the space $(W^{n+r}_p)^m$, $p<\infty$, by solutions of ODE systems with polynomial coefficients, polynomial right-hand sides of the equation, and multipoint boundary conditions, which are independent of the right-hand sides of the original problem.

Dragos Manea:

Title: Mesh-free numerical method for Dirichlet eigenpairs of the Laplacian with potential

This presentation is concerned with the numerical approximation of Dirichlet eigenpairs of the Schrödinger operator with a radial potential on simply connected, smooth, bounded two-dimensional domains.

We propose a mesh-free method inspired by the Method of Particular Solutions and generalise the approach from the Laplacian to the case in which an additional $C^1$ radial potential is present. The main challenge is the lack of explicit basis functions analogous to Bessel functions. We overcome this difficulty by switching to polar coordinates and considering, for each candidate eigenvalue $\lambda$, the corresponding Bessel-type equations with a potential, in the radial variable. In this way, we obtain a family of basis functions that solve the eigenvalue equation (without boundary conditions) on a ball enclosing the domain of interest. These functions can be approximated using a one-dimensional finite element method.

The Dirichlet eigenvalues are then approximated by minimising, over linear combinations of the basis functions, the trace on the boundary of the domain of interest and identifying those values of $\lambda$ for which the computed minimum is sufficiently small. Compared with the standard Galerkin method, the proposed method is highly memory-efficient.

Makson Santos:

Title TBA

We would like to discuss regularity theory for the normalized p-Laplacian and its connection with the C^{p'} and Aronsson's conjectures.

Vladimír Švígler:

Title: Perfect stationary solutions of reaction–diffusion equations on lattices and regular graphs

Spatially periodic solutions in the context of lattice differential equations have already been studied. We presents the concept of perfect stationary solutions: solutions which attain finite number of values and the neighborhood structure of each of the vertices is given by its value. As such, perfect stationary solutions generalize periodic stationary solutions. In this talk we aim to present the connection to perfect colorings on graphs and show certain constructions which lead to infinite aperiodic stationary solutions with two values.    

Wednesday

Bogdan Djordjevic:

Title: Some applications of Banach algebras and modules

Abstract TBA

Cristina Molero-Río:

Title TBA

The widespread adoption of machine learning models in high-stakes domains highlights the growing need for interpretable and trustworthy predictive systems. Mathematical optimization has emerged as a powerful framework for designing such models, as it provides explicit control over model structure and constraints. In this talk, we will explore several optimization-based approaches to explainable supervised learning. These methods illustrate how optimization can be used to develop models that are not only accurate, but also transparent and aligned with domain expertise.

Róisín Neururer:

Title TBA

I will examine mathematicians’ views on the teaching and learning of mathematics at second-level in Ireland. In particular, the data indicates that research mathematicians perceive the subject in very different ways to how the subject is taught at second level. Moreover, they are concerned about the disconnect between their interpretation of mathematics with the perceptions and experiences of doing mathematics in the second-level classroom, as well as the impact his may have on the transition to third-level mathematics for learners.  I hope to outline suggestions, following from this data, for classroom practice and practitioner networking that may benefit mathematical learning and the wider mathematics community.

Belén Pulido:

Title TBA

When working with functional data a problem arises when the aim is to order functions. There are different concepts available in the literature to tackle this problem. The statistical depth provides a criterion to order them from center to outwards, while the epigraph and hypograph indices give an ordination from top to bottom or vice versa. 

In this talk, new definitions of these indices are proposed based on areas between curves. This new approach better isolates the outlying curves and can be considered in several data analysis problems, such as outlier detection or clustering. Finally, the good performance of these indices is presented through synthetic and real datasets focusing on different real problems

Thursday

Irina Bobrova

Title: On some non-commutative sequences

Abstract TBA

Ahmed Elshafei:

Title: On the application of mathematics in architecture (Architectural Geometry)

Architectural geometry is relatively new field of interdisciplinary research focusing on the application of differential geometry, topology, discrete-computational geometry, as well as mathematical physics in the process of architectural design and manufacturing. In this talk, we will illustrate some examples of such applications, in particular of differential geometry and of integrable systems. We will show the crucial role that, certain curves networks (parameterizations) on surfaces, together with transformations leaving them invariant, play in architectural fabrication. Moreover, we also show some connections between local surface theory and integrable equations arising in mathematical physics, and how these can have useful applications in architecture.  

Antoni Piotr Kodzis:

Title: Stratified Univalence Program; What Connects : Chromatic Homotopy Theory, Local Shtuka Diamonds and Infinite Matroid Trees

How one of the most common forms of ordering and progression, links nearly (as currently can be said) all of the fields of mathematics; via its deeper characteristics and properties? How does it connect so many different branches, which at first glance have near to none connections to one another? And at what direction does the program head.

Jesse Railo:

Title: Geometric inverse problems and related topics

Abstract TBA

Discussion sessions:

• Well-being (Lead: TBA; Format: TBA)

• Career pathways and opportunities (Lead: TBA; Format: Panel discussion and Q&As)

• Publications (Lead: TBA; Format: TBA)

• Women in mathematics (Lead: TBA; Format: TBA)

Other topics of significant interest:

The following topics will instead have their own channel on the WhatsApp Community for this EMYA Workshop. We encourage open discussion and sharing of ideas on these channels (and in-person of course!):

• What are the challenges faced by early-career mathematicians and how can EMYA address these?

• Interdisciplinary research

• Teaching/Maths education

• AI and mathematics

• Popularisation of maths