List of talks and abstracts.

Talks are 50 minutes long with 10 minutes for questions at the end. Talks will take place in the room Pedro Nunes.

• Eloísa Grifo Symbolic powers ▼
  Hilbert’s Nullstellensatz provides a dictionary between solution sets of systems of polynomial equations and ideals in a polynomial ring. Symbolic powers, an ubiquitous topic within commutative algebra and algebraic geometry which arises naturally from a completely algebraic perspective, provide the key to give a higher order version of Hilbert’s dictionary. In this talk, we will introduce symbolic powers, tell a bit of their history, and discuss some of the many open questions on the subject.
• António Girão A walk through modern combinatorics ▼
  Combinatorics has seen great developments in the past decades which have changed it considerably. In this talk, I will give a brief but broad overview of several different aspects of modern research in combinatorics and its connections to other areas of Mathematics - ranging from Extremal and Probabilistic graph theory to Ramsey Theory as well as the study of random discrete structures.
• Gonçalo dos Reis Numerics for Self-stabilizing diffusions under super-measure growth, and early life electric batteries range prognostics ▼
  In this two-part talk, we cover recent development in the simulation of Self-stabilizing diffusions under super-linear growth assumptions on the equation's coefficients. These equations appear ubiquitously in interacting-particle system modelling. We discuss the phenomena of particle corruption in the simulations and illustrate our findings with a range of examples. In the second part of the talk, we discuss recent developments in Battery life modelling and lifetime prediction using data-driven tools, including insights on battery degradation.
• João Pereira The implicit method of moments ▼
  The focus of this talk is the multivariate method of moments, which consists of using the empirical moments to fit probability distributions to data. Most of the applications focus on using only the first two moments, the mean and covariance. That is with some reason, as higher order moments are huge multi-dimensional arrays that can be hard to store and use to do computations. I will introduce the idea of implicit method of moments, which consists of using samples directly to calculate quantities that involve the moment tensors, that way avoiding forming these very big multi-dimensional arrays. We show how we can use the implicit method of moments to estimate parameters of Gaussian Mixture Models and obtain an algorithm that is competitive with the state-of-the-art. On other point, I will talk about CP decomposition, which is a generalization of the eigen-decomposition for tensors, that has a few blessings and curses of its own. Here, I will mention an algorithm we recently proposed to calculate this decomposition, and how it can be used for obtaining the decomposition of moment tensors implicitly.
• Vanda Inácio de Carvalho Density regression via Dirichlet process mixture of normal structured additive regression models ▼
  There is an increased interest in studying how the distribution of a univariate, real- valued, continuous response changes with a set of covariates. Within a Bayesian nonparametric framework, Dependent Dirichlet process mixture of normal distributions provide a highly flexible approach for estimating the conditional density function. In this talk, I will review this class of models, showing the advantages and disadvantages of different model’s formulations. I will also detail a novel flexible, versatile, and computationally tractable model for density regression based on a dependent Dirichlet process mixture of normals model where an additive structure is assumed for the mean of each component and the effects of continuous covariates are modelled trough smooth functions. The major modelling components are penalised B-splines and their bivariate tensor product extension. The resulting model can easily deal with discrete covariates, nonlinear effects of continuous covariates, interaction surfaces, spatial effects, and varying coefficient terms. A practically important feature of our method is that, since the full conditional distributions for all model parameters are available in closed form, it allows for ready posterior simulation through Gibbs sampling. The results of the simulation study show that our approach successfully recovers the true conditional densities in a variety of challenging scenarios. Applications to a toxicology, disease diagnosis, and agricultural study are provided.
• André Guerra Morrey's problem and quasiconformal mappings ▼
  Quasiconvexity is the fundamental notion in the vectorial Calculus of Variations and is essentially equivalent to the existence of minimizers. A fundamental problem, considered by Morrey in the 50s and 60s, is whether quasiconvexity is equivalent to ellipticity (in the sense of Legendre-Hadamard). In 1992, Šverák showed that in 3 or higher dimensions they are not equivalent, but the two-dimensional case remains open. In this case one can hope for a "complex analysis miracle", and we will discuss deep connections of Morrey's problem to old questions in Complex Analysis.
• Rita Costa On the stability of rotating black holes ▼
  The black hole stability conjecture has been a long standing problem in General Relativity since the 1950s. This talk begins with a gentle introduction to the problem, and outlines some of my own contributions to the subject. Part of the talk is joint work with Yakov Shlapentokh-Rothman (Toronto) and Marc Casals (Leipzig/UCD/CBPF).
• Nuno Romão Topology of vortices with toric targets ▼
  The vortex equations are PDEs describing BPS field configurations for gauged sigma models in fibre bundles. In my talk, I will focus on the situation where the base is a Riemann surface and the fibre/target is a toric Kähler manifold, both assumed compact. Then the moduli spaces of solutions (up to unitary gauge) are generalised configuration spaces associated to a graph that can be extracted from the toric data. The fundamental groups of such spaces can be understood as certain groups of coloured braids, which I shall describe in very concrete terms. (Joint work with Marcel Bökstedt.)
• João Lourenço Geometric representation theory and p-adic geometry ▼
  One of Gauss' most celebrated results was his quadratic reciprocity law, treating the solvability of quadratic equations modulo primes. As a higher-degree generalization, Artin established a bijective correspondence of Galois and Hecke characters, via a map from the idèle class group of a number field to its abelianized Galois group. Then, Langlands proposed that there should be a general correspondence between higher-dimensional automorphic and Galois representations. This conjecture has kept number theorists busy for the last 4 decades and plenty of it remains unknown.
  
  On the other hand, Serre, Grothendieck, Deligne developed powerful methods in algebraic geometry and applications began to emerge in number theory and representation theory. For instance, it became possible to exploit geometric properties of flag varieties or classifying spaces of bundles over curves to answer questions arising from representation theory. Examples of this are the theory of Kazhdan-Lusztig polynomials or the geometric Satake equivalence of Mirkovic-Vilonen.
  
  Consequently, the Langlands program acquired a geometric flavor over function fields, culminating in Lafforgue's construction of Langlands' parameters.
  
  For a long time, it was not possible to carry out this geometrization in mixed characteristic, essentially because a more analytic theory was lacking. Even though Shimura varieties were available and led to tremendous progress, they were insufficient if one wanted a satisfying abstract theory. Recently, most of this became suddenly possible over p-adic fields thanks to advances in p-adic geometry. These include Scholze's theory of perfectoids or the Fargues and Fontaine's invention of a certain special curve in p-adic Hodge theory. The foundations of the geometrization program have now been carried out by Fargues and Scholze.
  
  In my talk, I will try to cover the main ideas among these topics aiming at a general audience and hopefully explain how my own work fits (admittedly very modestly) into the picture.
• Miguel Moreira Enumerative geometry, wall-crossing and Virasoro constraints ▼
  Enumerative geometry is a very classical subject and it has been greatly influential in the development of Algebraic Geometry since the 19th century. In modern terms, enumerative geometry is essentially the study of the (virtual) intersection theory of moduli spaces. In my talk I will explain some of the exciting developments in enumerative geometry in the last 30 years, focusing on the Virasoro constraints. These were born from physical ideas by Witten and were crucial in the development of Gromov-Witten theory. More recently (in joint works with A. Oblomkov, A. Okounkov, R. Pandharipande and with A. Bojko, W. Li) we have been studying analogous Virasoro constraints in the parallel world of "sheaf counting theories" and have developed tools to understand and prove them in many cases.
• Teresa Conde Rank functions for modules and categories ▼
  The dimension of a vector space is an important numerical invariant given by the cardinality of a basis over the ground field. When passing to modules, bases over the underlying ring do not necessarily exist and a different approach is needed to define the rank of a module. Motivated by work of Cohn and Schofield, we will discuss the notion of a Sylvester rank function on a ring and its interplay with ring morphisms into skew-fields. We will then take these ideas to the general context of categories, namely to triangulated categories. We shall see that certain structural information on the given category is captured by rank functions, and I will report on some progress made in this direction in joint work with Mikhail Gorsky, Frederik Marks and Alexandra Zvonareva.
• Margarida Melo Tropicalizing moduli spaces and applications ▼
  In algebraic geometry, understanding the geometry of moduli spaces is a beautiful and powerful way to understand the classes of objects that they parametrize. In the last few years, it has been understood that these moduli spaces can often be “tropicalized” via modular maps which allow to study many properties of the original spaces by looking at their tropical counterpart. In the talk, I will try to explain this interplay in the case of the moduli space of curves and the moduli space of abelian varieties. Special attention will be given to applications to the study of the topology of these moduli spaces, and in particular to their top weight rational cohomology.