Ágnes Backhausz, ELTE Eötvös Loránd University & HUN-REN Alfréd Rényi Institute of Mathematics
From graph limits to the structure of random matrices
Graph limit theory can be viewed as a combination of combinatorics, analysis and probability theory, which can be applied to understand structural properties of graphs. In the last 15 years, it turned out to be a powerful tool for spectral theory, extremal combinatorics, random graphs; furthermore, it also has connections to group theory. In the talk, a general overview of graph limit theory and its applications will be presented, including different approaches in the sparse, dense and intermediate edge density regime. In the second half of the talk, we will see the basic ideas of how this theory can be extended to matrices and operators, and how this theory can be used to understand the structure and spectral properties of random matrices, including the asymptotic behavior of the eigenvectors.
Alessandra Celletti, University of Rome Tor Vergata
The Interplay between mathematical perturbation methods and celestial
mechanics
Several mathematical theories provide fundamental insights into the
dynamical behaviour of celestial bodies. Conversely, a broad range of
problems in celestial mechanics and astrodynamics have historically
motivated the development and refinement of mathematical theories,
highlighting the deep and reciprocal relationship between mathematics and
the dynamics of the Solar system.
In this talk, I will discuss several problems concerning the motion of
celestial bodies, including the classical two- and three-body problems. I
will explain how small perturbations can affect the stability of their
motion and how mathematical theories can be used to predict their
long-term behaviour. One of the techniques I will discuss is
Kolmogorov–Arnold–Moser (KAM) theory.
I will also consider models that include dissipation as well as the
influence of random effects. Throughout the talk, mathematical results
and model problems from celestial mechanics will be presented side by
side, helping to clarify the main ideas and their applications.
Cristiana De Filippis , University of Parma
Nonuniform Ellipticity and Nonlinear Potentials
The representation formula for the Poisson equation gives an explicit expression of
solutions in terms of the ingredients, yielding optimal zeroth- and first-order pointwise bounds, thus allowing for a sharp regularity transfer from data to solutions. This way, nonlinear PDEs can be treated, up to the C^{1}-level, as if they were linear. After reviewing classical and more recent breakthroughs, I will discuss a novel approach to the (ir)regularity of solutions to certain PDEs arising in geometric and physical models.
Elisenda Feliu , University of Copenhagen
Algebra and reaction networks.
Biochemical reaction networks describe how interacting chemical species evolve over time. Their dynamics are commonly modelled by systems of ordinary differential equations whose right-hand sides are polynomial or rational functions. Fundamental questions include whether a network can admit multiple equilibria, whether these equilibria are stable, and how qualitative changes in the dynamics (bifurcations) can occur as parameters vary. Many of these questions can be reformulated as problems about the solutions of systems of polynomial equations and inequalities, making them amenable to methods from algebra and real algebraic geometry. This talk will introduce the mathematical framework of reaction networks and illustrate how algebraic techniques can be used to study several aspects of their dynamics.
Laura Monk, University of Bristol
What does a typical hyperbolic surface look like?
Hyperbolic surfaces are surfaces of constant curvature -1, studied for their rich algebraic, geometric and dynamical properties. They are not easy to visualise due to their negative curvature. The aim of this talk is to provide a description of the geometry of hyperbolic surfaces in the large scale limit, by estimating quantities such as their diameter, spectral gap and Cheeger constant. A modern probabilistic approach has led to multiple breakthroughs in recent years, emulating an extremely successful line of study of regular graphs and relying on the influential work of Fields medallist Maryam Mirzakhani.
Carola Bibiane Schönlieb, Cambridge UniversityCarola Bibiane Schönlieb
Mathematical Imaging in the Era of AI
Mathematical imaging has for decades been a powerful catalyst for new developments across the mathematical sciences. At its core lie fundamental questions about how information can be represented, reconstructed, and interpreted from incomplete, noisy, or indirect data. Addressing these questions has led to deep advances in areas such as functional and harmonic analysis, geometry, variational methods, inverse problems, partial differential equations, probability and statistics, and, more recently, learning theory. The resulting interplay between theory, computation, and applications has made imaging a uniquely fertile meeting point for pure and applied mathematics alike. Applications span an extraordinary range of domains, including biomedical and materials imaging, astronomy, environmental science, cultural heritage, and emerging technologies such as autonomous systems and data-driven diagnostics.
The rapid rise of artificial intelligence is now reshaping the landscape of imaging science once again. Data-driven approaches, in particular deep learning, have achieved remarkable empirical success in tasks such as reconstruction, segmentation, and synthesis. At the same time, their opaque nature and heavy reliance on data raise fundamental mathematical questions concerning stability, generalisation, interpretability, uncertainty quantification, and the incorporation of prior knowledge and physical constraints.
In this talk, I will present mathematical imaging as a continuing source of new mathematical ideas and challenges in the age of AI. I will highlight how modern approaches seek to blend data-driven models with structure, geometry, and physical principles, giving rise to novel analytical frameworks and computational paradigms. This perspective positions imaging not merely as an application area, but as a driver for the next generation of mathematical theory at the interface of analysis, computation, and learning.
Maria Vlasenko , Kyiv School of Economics
p-adic periods from naive viewpoint
The goal of the lecture is to demonstrate the connection between period integrals and counting solutions to algebraic equations over finite fields. We will start with an introduction to periods, an old concept in algebraic geometry. Their connection to counting solutions over finite fields was discovered by Bernard Dwork in the 1960s and gave rise to fundamental developments in arithmetic geometry. We will explore this fascinating topic in a way that is accessible to non-specialists, using only widely familiar concepts and simple examples.
Jessica Wade, Imperial College London
Fixing the system, not the women