Scientific Program

Eliana Duarte

University of Porto

Toric Varieties and Maximum Likelihood Estimation

In this minicourse I will introduce the audience to the class of toric varieties, these are algebraic varieties which are described as irreducible varieties arising as the vanishing of sets of binomial equations. These varieties play an important role in algebraic geometry and also in statistics. We will address the problem of maximum likelihood estimation for these varieties from the algebraic point of view. I will present several examples, theorems and open problems. All of the content will be introduced from basic principles.

Marina Ferreira

CNRS & University of Toulouse

Emergence of macroscopic phenomena in large particle systems

Macroscopic phenomena often arise in systems composed of many interacting particles. Examples include cloud formation in the atmosphere or the development of organs in a chick embryo. Intriguingly, such large-scale patterns are not encoded in the individual components—like molecules or cells—but instead emerge from their collective behavior when the system reaches a certain size. How do these patterns emerge? And how stable are they? Motivated by concrete applications, I will discuss two different scenarios: (1) systems of particles that coalesce upon collision,  relevant in atmospheric science, and (2) assemblies of rigid spheres connected by springs, modeling the mechanical behavior of cell tissues. The approach follows from ideas coming from mathematical physics, particularly statistical mechanics, and draws on a range of mathematical tools including optimization, scientific computing, functional analysis, measure theory and integro-differential equations.

Inês Rodrigues 

NOVA University of Lisbon

Crystals and quasi-crystals: local axioms and connections to quasi-symmetric functions

Crystal graphs are powerful combinatorial tools for working with the plactic monoid and symmetric functions. Quasi-crystal graphs are an analogous concept for the hypoplactic monoid and quasi-symmetric functions. We will introduce a characterization of quasi-crystal graphs by presenting a set of local axioms, and conclude that connected quasi-crystal graphs have a unique highest weight element, whose weight is a composition, and are isomorphic to quasi-crystal graphs of semistandard quasi-ribbon tableaux. We will also explore the interaction of fundamental quasi-symmetric functions and Schur functions, and the arrangement of quasi-crystal components within crystal components. 

Based on joint works with Alan Cain, António Malheiro and Fátima Rodrigues.

Gonçalo Oliveira

Instituto Superior Técnico of Lisbon

Counting Zeros of the Electric Field (joint with C. Fillmore and H. Edelsbrunner)

In 1873, James C. Maxwell conjectured that the electric field generated by n point charges in generic position has at most (n-1)^2 isolated zeroes. The first (non-optimal) upper bound was only obtained in 2007 by Gabrielov, Novikov, and Shapiro, who also posed two additional interesting conjectures. In this talk, I will report on joint work with Chris Fillmore and Herbert Edelsbrunner in which we derive the best upper bound known to date on the number of zeroes of the electric field, and construct a counterexample to one of the previously mentioned Conjectures. We also explore examples and construct configurations of charges achieving the highest ratios of the number of electric field zeroes by point charges found to this day.

Leonardo De Carlo

Lusófona University

Connections between Scaling Limits in Lattice Gases and the Hodge decomposition

Lattice Gases are toy models of Nonequilibrium Stastitical Mechanics, where a microscopic particle/energy configuration evolves according to  Markov Process constrained by certain physical rules (e.g. conservation laws). They are of interest because they offer a simple (often solvable) setting where to study Nonequilibrium Stationary States, since their macroscopic dynamics (called Hydrodynamics) can be derived in terms of the Law of Large Numbers for certain macroscopic observables.  Consequently,  their Central Limit Theorem and Large Deviations can also be studied.

In past work, we developed a functional Hodge Decomposition for such models, which showed new phenomelogy in the Hydrodynamic behavior in dimension greater than one. 

In this talk, we will present some standard Lattice Gases, remembering some basic facts about Markov Chains and the Law of Large Numbers, later we will explain the general scheme for proving their Hydrodynamics, and finally we will present the discrete Hodge Decomposition and its role in the Scaling Limits of Lattice Gases.

Xabier García Martínez

University of Santiago de Compostela

Categorical characterisations of algebraic structures

In this talk we will give some ideas on how to capture the essence of some algebraic structures using purely categorical notions. For instance, we will see how to differenciate groups from monoids using just universal properties, or how to find Lie algebras amongst all non-associative algebras by the existence of an adjoint functor.

Claudio Alexandre Piedade

University of Porto

Merging Coset Geometries: from group operations to fusing geometries

Incidence geometries are in the basis of Tits buildings and related structures. Coset geometries are incidence structures derived from group cosets, where points, lines, and higher-dimensional elements correspond to cosets of certain subgroups. These capture symmetry and combinatorial properties of groups, particularly in relation to buildings and flag complexes.

In this talk, we will describe distinct ways of building a coset geometry by standard operations on groups, such as free product (with amalgamation), HNN extension and split extension. Properties of coset geometries like flag-transitivity and residually connectedness are preserved under this operations.

These constructions allow us to apply this to geometrical structures like polytopes, giving operations of building new structures.

Hector Chang-Lara

CIMAT & University of Coimbra

Very-Degenerate PDEs Arising in Congested Transport

We address the problem of assigning optimal routes in a graph that transports two given densities across its nodes. The traffic flow along each edge at a given time induces a metric on the graph, with respect to which the routes must be geodesics. The resulting configurations are known as Wardrop equilibria. Additionally, a central planner may require the assignment to be efficient: that is, to minimize the Kantorovich functional associated with this metric. Under symmetry assumptions on the cost functions, the problem reduces to a class of variational problems with divergence constraints, originally studied by Beckmann. The corresponding partial differential equations are highly degenerate and belong to a fascinating class of problems that remains rich with open questions and opportunities for further exploration.

Part of this work is a collaboration with Sergio Zapeta Tzul, former master’s student at CIMAT and current PhD student at the University of Minnesota.

Tajani Asmae

University of Aveiro & ENS of Casablanca

Gradient controllability of fractional SIR  reaction diffusion models

In this work, we investigate the gradient controllability of a fractional SIR (Susceptible–Infected–Recovered) epidemic model governed by reaction-diffusion equations with Caputo time-fractional derivatives of order between 0 and 1. The inclusion of fractional derivatives accounts for memory effects in disease transmission, while the diffusion terms capture spatial spread within a heterogeneous environment. We study the ability to steer the gradient of the state variables—particularly the infected population—toward desired spatial distributions using distributed or boundary control strategies. 

Algebra

Room Pedro Nunes | Floor 1 (Piso 1) | Department of Mathematics

Guilherme Azevedo

NOVA University of Lisbon

Partition Regularity of Pythagorean Triples

A central question in Ramsey theory is concerned with determining the patterns that must appear in a single cell for every partition of the set of natural numbers N = {1, 2, ...} into finitely many cells. Equations satisfying the property that, given any finite partition of N, there is a solution with all the variables belonging to the same cell are called partition regular.

We will focus on the problem of determining the partition regularity of the Pythagorean equation x^2 + y^2 = z^2, by trying to understand it through the lens of multiplicative function theory.

Zafar Iqbal

NOVA University of Lisbon

Homological and Combinatorial Properties of BCH Codes

Coding theory, originated in the late 1940s, is driven by the need to ensure reliable communication over noisy channels. Linear error-correcting codes, in particular, became central to the field due to their algebraic structure and practical utility. These codes can be studied via combinatorial, algebraic, and geometric methods: to each code one associates a matroid from its generator or parity-check matrix, and the independent sets of the matroid define a simplicial complex whose Stanley–Reisner ring captures the code’s combinatorial structure. The Betti numbers of this ring encode key invariants of the code. Computing them is difficult in general, but they reflect important parameters and offer new perspectives on the code’s structure. In this talk, we will focus on the homological invariants of certain families of BCH codes. BCH codes are regarded as one of the most useful codes in the theory of error-correcting codes. Despite their prominence, basic properties such as dimension and minimum distance remain unresolved in many cases. Time permitting, we will discuss open problems such as Charpin’s conjecture regarding the minimum distance of primitive BCH codes.

Ângelo Manuel

University of Aveiro

Cubic Reciprocity Law and Its Application

This presentation explores the Cubic Reciprocity Law, a fundamental result in number theory that generalizes the classical Quadratic Reciprocity Law. Focusing on rational primes, we investigate the conditions under which a given integer is a cubic residue modulo a rational prime p≡1(mod3). Key topics include the role of Eisenstein integers, the characterization of cubic residues, and Jacobi's conjectures, which provide explicit criteria for cubic residues. We also discuss applications of these results, such as the construction of cubic Paley graphs and cryptographic implications. The talk will highlight the interplay between algebraic structures and arithmetic properties, offering insights into higher reciprocity laws.

Geometry

Room 2.4 | Floor 2 (Piso 2)  | Department of Mathematics

Miguel Garcia

NOVA University of Lisbon

An introduction to microlocal equivalence of quasi-ordinary surfaces

For germs of plane curves, Zariski proved that an irreducible plane curve germ is analytically equivalent to an irreducible plane curve germ with finite Puiseux parametrization. Germs of plane curve germs represent the simplest instance of quasi-ordinary hypersurfaces in dimension two. Zariski’s algebraic techniques cannot be applied to the general case. We show that, for some types of quasi-ordinary surfaces, these techniques can indeed be applied in the context of contact geometry. This is an ongoing work for my thesis.

Alejandro Calleja

Instituto de Ciencias Matemáticas

Hodge structures on configuration spaces of orbits

Given an algebraic variety X with an action of an algebraic group G, we define the n-th configuration space of orbits as the set of n-tuples of points of X such that the orbits through G of these points are pairwise disjoint. In this talk we introduce these spaces and their applications to Knot Theory. We will also show how we can study the Hodge structure of these spaces by relating it to the one of X.

Salah Chaib

University of Minho

Symmetries of low-dimensional metric Lie groups

We investigate the symmetries of low-dimensional Lie groups equipped with a left-invariant pseudo-Riemannian metric (metric Lie groups). We begin by exploring the isometry groups of well-known 2-dimensional homogeneous spaces to provide geometric context and motivation. We then present a unified Lie-theoretical approach to identify all connected, simply connected 3-dimensional metric Lie groups whose full isometry group has dimension at least four. For each such pair $(G,g)$, we also determine the isotropy type. This is joint work with Ana Cristina Ferreira and Abdelghani Zeghib.

Mathematical Physics

Room 2.5 | Floor 2 (Piso 2)  | Department of Mathematics

Tomé  Graxinha

University of Lisbon

Regularity of Lyapunov Exponents for random cocycles

The notion of Lyapunov exponent is a central concept in ergodic theory and in the theory of smooth dynamical systems, which measures the sensitivity of the dynamical system with respect to the initial conditions. We study the regularity of the Lyapunov exponents in the context of linear cocycles, which is a class of dynamical systems over fiber bundles that preserves the linear bundle structure and induces a measure preserving dynamical system on the base. The dependency of the Lyapunov exponents as a function of the law that determines the system can be extremely irregular. We present some results with respect to some specific class of dynamics regarding the regularity of the Lyapunov exponents, namely that in the non-invertible two-dimensional case the Lyapunov exponent is either analytic or discontinuous (P. Duarte, M. Durães, T.G. and S.Klein). Furthermore we prove that, under some sharp moment conditions, the Lyapunov exponent for non-compact and non-invertible random cocycles is locally H¨older continuous (P. Duarte and T.G.). In an example evolving Schr\""odinger cocycles we show that the moment conditions above are crucial to have the H\""older continuity (P. Duarte and T.G). We also present some statistical properties on these settings such as large deviation type estimates (P. Duarte, M. Durães, T.G. and S.Klein).

Gabriele Degano

University of Lisbon

Ground state of quantum KdV via ODE/IM correspondence

We present an anharmonic oscillator known as the ground state potential for the quantum KdV model. Our focus is on the spectral determinant arising from the associated central connection problem, whose zeros we analyze in several asymptotic regimes—specifically, in the limits of large energy, angular momentum, and anharmonicity degree. We derive explicit expressions for the leading-order behavior in each case. These results, which recently appeared in the literature, contribute to the ongoing exploration of the deep relationship between integrable models and ordinary differential equations, that is the celebrated ODE/IM correspondence.

Geometry

Room Pedro Nunes | Floor 1 (Piso 1)  | Department of Mathematics

Mohammed Sabak

Hassan II University of Casablanca

A Numerical Invariant for Checkerboard Colorable Virtual Links

We introduce a numerical invariant for checkerboard colorable virtual links derived entirely from combinatorial data. By examining this invariant, we provide a geometric interpretation for specific classes of these links, including classical, almost classical, and alternating virtual links. Our findings suggest a broader applicability, leading us to conjecture that this geometric interpretation holds for all checkerboard colorable virtual links.

Algebra

Room Pedro Nunes | Floor 1 (Piso 1)  | Department of Mathematics

Norihiro Yamada

University of Coimbra

A graphical calculus for linear categories

Symmetric monoidal closed categories are ubiquitous in mathematics, and so are the linear-nonlinear adjunction. In particular, the extension of the former with the latter, known as linear categories, forms the categorical counterpart of intuitionistic linear logic. In the literature of monoidal categories and linear logic, geometric or graphical methods have been extensively used for more than 35 years, but out of reach of linear categories. As a result, it remains nontrivial to decide the equality between morphisms in a linear category. In this talk, I present my recent solution to this long-standing problem and illustrate its applications in category theory. 

Rustam Turdibaev 

New Uzbekistan University

Invariants of anti-commuting pairs of matrices

We investigate the invariants of anti-commuting matrix pairs under simultaneous conjugation. Using geometric invariant theory, we determine a minimal generating set and develop an algorithmic approach for larger cases. Computations confirm that our defining ideals are radical and align with known results, refining the structure of the invariant anti-commuting variety. We also find primary invariants for matrix sizes not exceeding 5.

Faustino Maciala 

University of Minho

Characterizing the Drazin Index in Directed Double Star Graphs​

A fundamental challenge in spectral graph theory is identifying when matrices associated with graphs are invertible and formulating explicit expressions for their inverses. This challenge extends naturally to the study of pseudo-invertibility.

In this work, we utilize the concept of Drazin invertibility—characterized by a specific nonsingular matrix—to ascertain the index of matrices linked to particular graph structures, notably double star digraphs. Our approach generalizes previous findings, offering a broader understanding of the conditions under which such matrices are Drazin invertible.​

Analysis

Room 2.4 | Floor 2 (Piso 2)  | Department of Mathematics

Makson Santos

University of Lisbon

Regularity theory for a class of degenerate normalized p-laplacian equations

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.

Edhin Mamani Castillo

Federal University of Minas Gerais

Thermodynamic formalism for geodesic flows

The proposal belongs to the area that studies the qualitative geometric and statistical properties of systems that evolve over time, i.e, dynamical systems, ergodic theory and thermodynamic formalism. In particular, we study the abstract mathematical model of the movement of N particles that move freely but are restricted to a fixed surface. This model is called the geodesic flow associated with a surface (or in general with a Riemannian manifold). There exist certain probability measures that maximize the complexity of the system's trajectories, called measures of maximal entropy. The goal is to present a recent result on the uniqueness of this measure for general contexts (N particles) and manifolds admitting zero and positive curvature (manifolds without conjugate points). This is a joint work with Rafael Ruggiero. 

João Fontinha

University of Lisbon

The measure transition problem for the Laplace transform with connections to number theory

Using Phragmén-Lindelöf principles, we prove via a Cauchy contour argument how certain classes of bilateral Laplace representations possessing an infinite number of poles on their abscissas of convergence extend meromorphically across a vertical strip of definition. We apply this technique to 1/\zeta(s) and connect the domain of validity of its Laplace representation to conjectured bounds for the Mertens function.

Giuliano Zugliani

State University of Campinas

Tube structures on closed manifolds

We study a locally integrable tube structure defined by a differential operator L associated with complex 1-forms defined on closed manifolds. Locally, we have a system of linear first order PDEs. We will present the results obtained so far on the global solvability of L, which are related to homological properties involving such forms, and the current open questions.

Applied Mathematics

Room 2.5 | Floor 2 (Piso 2)  | Department of Mathematics

João Marcelino

University of Coimbra

Bayesian Network structure learning for homogeneous decomposable models

A Bayesian Network (BN) is a probabilistic graphical model that uses a directed acyclic graph (DAG) to represent variables and their conditional dependencies. BNs can be described using parent sets or characteristic imsets. In the score-based approach, the goal is to find the BN that maximizes a quality score, which lead to an Integer Linear Programming (ILP) formulation under certain conditions. Decomposable and homogeneous decomposable models are Markov random fields, which are undirected models. These models can be learned similarly to BNs, using ILP formulations based on characteristic imsets and specific constraint sets named Chordal graph polytope and homogeneous graph polytope. Due to the complexity of these polytopes relaxing the polytopes is often necessary and effective.

Paulo Rocha

University of Lisbon

Portfolio Problem with Consumption Under the α-Hypergeometric Stochastic Volatility Model

In this session, we will look at the dynamic programming method to solve a variant of the portfolio problem in which consumption is allowed. We assume that the agent makes his investment and consumption decisions based on a power utility function.

Considering a simple portfolio, composed of a bond and a single stock on a market modeled by the α-Hypergeometric Stochastic volatility model, we will derive the correspondent Hamilton-Jacobi-Bellman equation and discuss the existence of a classical solution and the techniques used to find such a solution.

André Gide Bédiang

University of Évora

Mathematical analysis of a cholera epidemic model with control treatment

We unveil a SITR cholera model with two control mechanisms: a vaccination, and a reduced contact rate. Through the analysis of the corresponding characteristic equations, we obtain a disease-free equilibrium and an endemic equilibrium. We have got a threshold known as the control reproduction number Rv. If Rv < 1, sufficient conditions for the global asymptotic stability of the disease-free equilibrium are obtained, and the disease will be eliminated from the community. Also, for the case Rv > 1, the endemic equilibrium is globally asymptotically stable. We finally perform some sensitivity analysis of Rv, on the parameters in the aim of determining their relative importance to disease transmission and confirming that the two control mechanisms used in our model are always beneficial in controlling the disease.

Adriana Cardoso

University of Porto

Non-Commutative Principal Ideal Domains in Quaternion Algebras

The main goal of this presentation is to explain the concept of a Non-Commutative Principal Ideal Domain in Quaternion Algebras and also present a algorithm to check if an arbitrary quaternion order is a Non-Commutative PID. We will end the talk by exploring some examples.

Afonso Costa

University of Coimbra

Discontinuous Galerkin finite element method for the Kelvin-Voigt Viscoelastic Mathematical Model

This presentation introduces a space-time numerical scheme for simulating viscoelastic materials modelled by the Kelvin-Voigt equation. We begin by generalizing the well-known discontinuous Galerkin finite element method (DG-FEM) for elliptic problems to our viscoelastic model with non-homogeneous Dirichlet conditions, and establish its fundamental properties—coercivity and boundedness of the bilinear form. Building on this framework, we develop a coupled DG-FEM in space and finite difference method (FDM) in time for the viscoelastic model. Stability and theoretical error estimates are studied, demonstrating second-order convergence in space and first-order convergence in time. Numerical simulations have been conducted to validate the theoretical error estimates, confirming them and supporting the accuracy and effectiveness of the proposed scheme.

Aline Leite Vilela D'Oliveira

University of Illinois

Moduli Space of Flat MECs

A classical result due to Atiyah and Bott establishes that the moduli space of flat principal bundle connections admits a symplectic structure under certain conditions. The notion of principal connection was generalized to groupoids - first by Laurent-Gengoux, Stienon and Xu, and later by Fernandes and Marcut - through what they called multiplicative Ehresmann connections (MECs).

In this work, we define two versions of the moduli space for flat MECs and take the first steps toward understanding how these generalized moduli spaces carry some geometric structure that generalizes the symplectic structure in the classical case.

António Leite

University of Porto

The Metacommutation Problem in Quaternion Algebras

We will describe what is known about the Metacommutation Problem, and describe some properties about the cycle structure of the Metacommutation map, which is a permutation.  We talk about the underlying relationships between the primes above a rational prime and the points in a non-degenerate conic.

Duarte Costa

University of Lisbon

Conic Sheaves through a bird's eye view

Sheaves allow us to assign and/or track information over a topological space. Conic Sheaves are a special class of sheaves in which the assigned data must be compatible with a pre-provided action of R^{+} on the space in question. I will discuss some different approaches to this concept through different contexts and definitions. As an end to the talk, I will describe how this concept can be reproduced in the framework of o-minimal structures, but with some stark differences from the previous approaches.

Juliana Carvalho De Souza

University Paris Dauphine-PSL

A Brief Overview Of Deep Learning Approaches For Climate Data

Extreme weather events have become more frequent, intense, and long-lasting due to global warming, with climate change intensifying phenomena such as droughts and heavy precipitation. These events can cause severe disruptions when they exceed critical thresholds in ecological, social, or physical systems. Remote sensing plays a key role in monitoring and understanding these climatic shifts, while traditional forecasting relies heavily on computationally intensive Numerical Weather Prediction (NWP) models. Machine learning presents an alternative, using data-driven methods to either complement or replace physics-based models, particularly through supervised learning on datasets like ERA5. Due to the complexity of climate patterns, unsupervised and weakly supervised methods are increasingly used to classify and analyze these phenomena. This work reviews deep learning approaches—supervised, semi-supervised, and self-supervised—for handling labeled and weakly labeled climate data, outlines foundational concepts, discusses existing solutions, and highlights future research directions.

Round Table: "Women in Mathematics"

With: Sofia Castro (Moderator) | Lina Coelho | Aline D'oliveira | Milene Santos

This round table brings together women from different fields of mathematics and an expert in gender studies to discuss the challenges and opportunities faced by women at the early stages of their careers. The conversation will focus on real experiences, common obstacles, and practical strategies to navigate academic and professional life in mathematics. The session aims to create a supportive space for students, PhD candidates, and early-career researchers to exchange ideas, ask questions, and find inspiration. All are welcome to join this open discussion on building a more inclusive and equitable mathematical community. 

Round Table: "Science Communication"

With: Gabriel Guimarães (Moderator) | Marta D. Santos | Adérito Araújo | Edgard Pimentel

Effective communication of mathematical research plays a crucial role in advancing the discipline, fostering interdisciplinary collaboration, and enhancing public understanding of mathematics. This round table will provide a platform for discussing the specific challenges and responsibilities associated with communicating mathematics to diverse audiences, including the general public. Participants will exchange ideas on best practices, strategies for outreach, and the role of mathematicians in shaping the public perception of the field. The session aims to encourage critical reflection and practical approaches to improving science communication within the mathematical community.

Panel: "Career Opportunities"

With: Carla Rizzo (Moderator) | Stéphane Clain | Marina Ferreira | Eduardo Dias | Leonardo de Carlo | Liliana Garrido da Silva | Beatriz Santos

In this panel, mathematicians working in academia, industry, and education will share their career paths and personal experiences. The panelists will discuss the challenges and opportunities of pursuing careers in different sectors, strategies for transitioning between academia and industry, work-life balance, and insights into building a successful academic application. The session will include an open Q&A, giving attendees the opportunity to engage directly with the speakers.