Nuno Arala, Leibniz Universität Hannover

A nonabelian circle method

The circle method, introduced around 100 years ago in papers of Ramanujan, Hardy and Littlewood, is a powerful tool in the quantitative study of Diophantine Equations. After giving an overview of the classical form of the method, we will introduce a new nonabelian variant that allows applications to equations with variables in non-commutative algebras, such as the quaternions. This is joint work with Jayce R. Getz, Jiaqi Hou, Chun-Hsien Hsu, Huajie Li and Victor Y. Wang.

João Baptista, Lisboa

Riemannian submersions and Kaluza-Klein models

In this talk I will describe the relation between Riemannian submersions and Kaluza-Klein models on higher-dimensional spacetimes M_4 x K. The setting will be more general than usual, as we consider submersions whose fibres need not be totally geodesic. In these conditions, the higher-dimensional metric encodes both massless and massive 4D gauge fields, as well as a non-trivial Higgs sector. I will discuss some of the new features that appear in these models. For example, how geodesic motion in higher-dimensions correctly indicates that the classical mass and charge of a test particle, as perceived in 4D, can change in regions of spacetime where massive gauge fields are present.

Francisco Blasques, Vrije Universiteit Amsterdam

Filtering Methods for Time-Varying Conditional Moments

This talk explores the development of filtering techniques used in time-series analysis for modeling time-varying conditional moments. We will focus on recent advances in score-driven models, a flexible class of observation-driven methods that adapt dynamically to new information. Special attention will be given to their role in robust statistics, modeling locally explosive dynamics, and advancing methods for dynamic causal inference. The talk will also cover applications in engineering, biology, economics, and the medical sciences.

Nikolaos Chalmoukis, Università di Milano Bicocca

Some recent developments in the theory of interpolating sequences

Interpolating sequence have been an instrumental part of the rapid development of function theoretical methods in the second half of the 20th century. In particular Carleson's interpolation theorem for the Banach algebra of bounded analytic functions in the unit disc was, on one side, a stepping stone towards the Corona theorem and  an inspiration for the development of techniques such as the d-bar method of P. Jones on the other. In this talk we will discuss some recent developments on the subject, in particular regarding interpolating sequences for a general class of space called complete Nevanlinna Pick spaces. We will see how the development of abstract functional analytic techniques and some recent breakthrough in the theory of C* algebras allow us to revisit the classical theorems of Carleson and later work of Shapiro and Shields in a new light and at the same time answer some questions of Agler and McCarthy.

Edgar Costa, Massachusetts Institute of Technology (ONLINE)

Effective computation of Hodge cycles

In this talk, we will overview several techniques to compute the Hodge cycles on surfaces. We will start by studying the self-product of a curve, with the goal of computing the endomorphism of its Jacobian. We will follow this with the problem of computing the Picard lattice of a K3 surface.

Susana Gomes, University of Warwick

Modelling and control of opinion dynamics on evolving networks

The field of opinion dynamics has recently seen a large interest from the mathematics community, both from modelling and control perspectives. Most works focus on the Hegselmann-Krause model, a bounded confidence model that assumes everyone can communicate with everyone else as long as their opinions are close enough. Typical results focus on analysing whether the system achieves consensus, and control strategies aim to steer a population towards consensus (or make it so more quickly) by using controls that act directly on agents. These models can be made more realistic if one introduces a social network that constrains communication between agents and if the controls are restricted.

In this talk, I will introduce recent work on co-evolving networks (i.e. a dynamical system where the underlying network evolves at the same time as, and influenced by, the individuals' opinions), and justify why the resulting models can be obtained as a limit of one-to-one interactions in the right conditions. I will then propose a new type of control based on adaptive networks: I will present various control strategies and analyse under which conditions opinions can or cannot be steered towards a given target, then corroborate and extend our analytical results with computational experiments and a study of optimal controls.

If time permits, I will also discuss recent work on introducing ageing effects, which allows us to explore the mean-field limit and long-time behaviour of these models, beyond just reaching consensus. 

Sophie Marques, Stellenbosch University

Contributions to the research on algebraic (geometric) structures

This talk traces my research journey through the study of certain algebraic (geometric) structures, often also observed through a categorical lens. My work spans group schemes, gluing techniques, jet schemes (algebras), algebraic varieties, invariant theory, field theory, Galois theory, classification problems, and ramification. More recently, my interests have expanded to include near-linear algebra, near-fields, the Ulam spiral, and questions concerning moduli spaces.

João Nuno Mestre, Universidade de Coimbra

Multiplicative Geometric objects via tensors

Geometric objects compatible with symmetries produce both rich phenomena and useful concrete tools to solve problems in geometry. The geometric objects we have in mind are tensors - objects like a smooth function, a vector field, a differential form (in particular volume forms), or a complex structure, and they are essential in Differential Geometry. The symmetries we will have in mind are those described by Lie groups and their generalizations.

In this talk we will see what are multiplicative tensors (tensors on Lie groups compatible with the multiplication), how to find them and how to describe them using an appropriate cohomology. We will see some examples of their phenomena, and indicate some applications, for example for solving Hamiltonian systems.

Léonard Monsaingeon, Instituto Superior Técnico, Universidade de Lisboa

Optimal transport, entropy, and dynamics

Optimal transport is a very versatile theory that allows to lift the geometry on a given underlying space to the overlying Wasserstein space of probability measures. Based on this rich interpretation, Jordan, Kinderlehrer and Otto showed 25 years ago that the (linear!) heat equation can be seen as the (highly nonlinear!) Wasserstein gradient flow of the Boltzmann entropy in the space of probabillty measures, thus providing a strong form of the second principle of thermodynamics. This sparked tremendous interest due to the interconnection between various mathematical branches, ranging from applied PDEs, numerical analysis, probability and interacting particle systems, metric geometry, functional analysis, and more. In this talk I will try to review the main ideas behind this formalism, and show how classical optimal transport can be extended to study a broad class of evolution equations from a variational standpoint (eg. Hele-Shaw dynamics, reaction-diffusion, multiphase flows, evolutionary genetics, etc.)

Giuseppe Negro, Instituto Superior Técnico, Universidade de Lisboa

The Penrose transform in Harmonic Analysis

The Penrose transform is a classical conformal mapping of Minkowski spacetime, introduced by Sir Roger Penrose in the 1960s in the context of gravitation. In this talk we will present it in detail and explain some unexpected connections with problems concerning oscillatory operators, such as the Fourier restriction operator to the cone. 

Pedro Pinto, Technische Universität Darmstadt

Proof-theoretic unwinding of Mathematics

Proof mining is a research program in mathematical logic that seeks to uncover hidden computational content within existing mathematical proofs. Many classical arguments, particularly in areas like functional analysis, demonstrate the existence of objects without offering any explicit constructions or quantitative information. By applying tools from proof theory, proof mining systematically analyzes such proofs to extract precise numerical data, such as rates of convergence or of metastability, which are not evident in the original argument. This method not only deepens our understanding of established results but also often yields new, stronger theorems with practical relevance.

This talk will present the aims, methods, and impact of proof mining, with a focus on applications in nonlinear functional analysis. I will focus on two specific topics that illustrate its power and versatility: first, how infinitary reasoning can be replaced by finitary arguments through the use of principles that, while set-theoretically false, are sound within a carefully controlled logical framework; and second, how logical analysis can facilitate the transfer of results from linear to nonlinear settings, revealing unexpected structural uniformities. Time permitting, I will also discuss some exciting recent developments in the field.

Sérgio Rodrigues, Radon Institute, Austrian Academy of Sciences

Stabilization of semilinear parabolic-like equations

A crucial task  in control applications is the design of a feedback operator allowing us to compute a control input which is able to stabilize a given dynamical system. Feedback inputs are given as a function of the state of the system, which is often not fully available in real world applications. Thus, another crucial task is the design of a dynamic Luenberger observer providing us with an estimate for the unknown state, by using the output of sensor measurements; now, the task is to find an operator that injects the output into the dynamics of the observer.

In this talk, we discuss recent developments on the design of such feedback-input  and output-injection operators. The focus is put on the design of simple and explicit operators. Both theoretical and numerical aspects are discussed, including a comparison to more classical operators obtained through optimal control tools and involving the solution of Riccati or Hamilton-Jacobi-Bellman equations.

Makson Santos, Faculdade de Ciências, Universidade de Lisboa

Regularity for degenerate normalized p-laplacian equations and the C^p'-conjecture

We study the regularity properties of viscosity solutions to a class of degenerate normalized p-laplacian equations.

In particular, we prove that the gradient of viscosity solutions are Hölder continuous, and we give the optimal exponent. Moreover, we also show that viscosity solutions to equations with very general degeneracy laws are differentiable. We apply our results to prove a particular case of the so-called C^p'-conjecture.

Delia Schiera, Instituto Superior Técnico, Universidade de Lisboa

On an anisotropic spectral optimization problem arising in population dynamics

In 1937, Fisher and Kolmogorov-Petrovskii-Piskunov independently introduced a reaction diffusion equation with logistic nonlinearity as a model to describe the dispersal of a population in a heterogeneous environment. Since then, there has been growing interest in this model

and its generalizations within the mathematical community.

I will discuss what happens when, instead of the classical diffusion induced by the Laplace operator, one considers an anisotropic diffusion, modeling a population moving in different directions with different probabilities.

We will mainly be interested in optimizing the spatial arrangement of resources for a species to survive in the habitat: from a mathematical point of view, this translates into a spectral optimization problem, precisely, our main goal will be to minimize the principal eigenvalue of the anisotropic diffusion operator with respect to the sign-changing weight that models resource allocation.

We show that the optimal weights are of bang-bang type, namely piece-wise constant functions that take only two values, and we completely solve the optimization problem in one dimension, in the case of homogeneous Dirichlet or Neumann boundary conditions. Additionally, I will present some partial results and ongoing work in the higher dimensional case.

Joint work with B. Pellacci and G. Pisante.

Pedro M. Silva, Faculdade de Ciências, Universidade de Lisboa

Higgs bundles, Non-abelian Hodge theory and geometric structures

Higgs bundles have now become central objects in understanding representations of the fundamental group of a closed surface into a Lie group. In this talk, we will present a geometric overview of their theory,  and how the Hitchin–Kobayashi correspondence can be used to study both individual representations and their moduli spaces.

Time permitting, we will also touch on a variation known as the conformal limit, which, together with the correspondence, allows one to build complex projective structures naturally associated with these representations. This is joint work with Peter B. Gothen.