Seminários e cursos curtos

When studying systems of particles, the very first step before any qualitative analysis is to establish the well-posedness of the dynamics of the system. In the case of hard spheres, whose trajectories are piecewise affine, the singularities arising at collision events prevent the direct use of Cauchy-Lipschitz-type of arguments. This issue was addressed by Alexander (1975) in the elastic case, where the kinetic energy is conserved during the collisions. For dissipative systems, the question remains largely open, due to the possibility that infinitely many collisions take place in finite time, a phenomenon known as inelastic collapse. We will discuss the case of a particular class of inelastic hard sphere systems, in which a fixed amount of kinetic energy is lost in each sufficiently energetic collision. The results were obtained in collaboration with Juan J. L. Velázquez (Universität Bonn), and are published in arXiv:2403.02162v2 (to appear in Communications in Mathematical Physics).

Given any open, bounded set $\Omega$, we consider suitable combinations, via a reference function $\Phi$, of the first $p$-eigenvalue of the Dirichlet Laplacian of partitions of $\Omega$. We give two different formulations of the problem, one geometrical and one functional. We prove relations among the two formulations, existence and regularity of optimal partitions, convergence, and stability with respect to $p$ and to $\Phi$. Based on a joint work with G. Stefani (Padova).

I will begin with a concise review of the fundamental ideas underlying optimal transport theory, before illustrating how these concepts naturally emerge in cosmological contexts and provide a powerful framework for addressing the problem of cosmological reconstruction. I will then highlight how optimal transport extends beyond its analytical utility in handling big data, and demonstrate intriguing connections with certain theories of modified gravity.

Existing machine learning frameworks operate over the field of real numbers ($\mathbb{R}$) and learn representations in real (Euclidean or Hilbert) vector spaces (e.g., $\mathbb{R}^d$). Their underlying geometric properties align well with intuitive concepts such as linear separability, minimum enclosing balls, and subspace projection; and basic calculus provides a toolbox for learning through gradient-based optimization.

But is this the only possible choice? In this seminar, we study the suitability of a radically different field as an alternative to $\mathbb{R}$ — the ultrametric and non-archimedean space of $p$-adic numbers, $\mathbb{Q}_p$. The hierarchical structure of the $p$-adics and their interpretation as infinite strings make them an appealing tool for code theory and hierarchical representation learning. Our exploratory theoretical work establishes the building blocks for classification, regression, and representation learning with the $p$-adics, providing learning models and algorithms. We illustrate how simple Quillian semantic networks can be represented as a compact $p$-adic linear network, a construction which is not possible with the field of reals. We finish by discussing open problems and opportunities for future research enabled by this new framework.

Based on:
André F. T. Martins, Learning with the $p$-adics

Recent results on black hole interiors suggest a failure of strong cosmic censorship for charged black holes in the presence of a positive cosmological constant. In this talk we show that, in the context of the Einstein-Maxwell-real scalar field system, such violations are non-generic in a larger moduli space of non-smooth (spherically symmetric) initial data.

Categorical algebra is a fundamental branch of mathematics that lies at the intersection of category theory and algebra. On the one hand, it captures the fruitful properties and structures studied in algebra via category theory. On the other hand, it investigates the global categorical properties that algebraic objects enjoy when collected together. Both these endeavors are essential to extend and transport the fundamental concepts and theorems of algebra to different and broader settings. In this talk, we present an innovative theory that generalizes categorical algebra to the framework of 2-dimensional category theory. This has the notable advantage that the second dimension can be used both to weaken conditions that are too strict in nature and to refine algebraic invariants, obtaining a richer theory which encompasses a broader range of examples. Furthermore, 2-dimensional categorical algebra is essential to effectively compare different algebraic categories with each other.

This talk is based on a joint work with Elena Caviglia and Zurab Janelidze.

Abelian categories and triangulated categories provide fundamental frameworks to study homological and cohomological problems across algebraic geometry, topology and representation theory.

In this talk we will explain how we can study the 2-category AbCat of abelian categories and the 2-category Triang of triangulated categories through the lenses of 2-dimensional categorical algebra. Surprisingly, through these lenses, AbCat and Triang look extremely similar.

We will show that the important notions of Serre subcategories and Serre quotients of abelian categories precisely correspond respectively with the 2-dimensional kernels and cokernels in AbCat. In a similar way, thick triangulated subcategories and Verdier localizations of triangulated categories are exactly the 2-kernels and the 2-cokernels in Triang.

Furthermore, even more striking similarities between the two contexts arise when characterizing these 2-kernels and 2-cokernels in terms of categorical properties satisfied by their underlying functors.

These results will allow us to show that both the 2-categories AbCat and Triang are exact, in appropriate 2-dimensional senses. In particular, AbCat is 2-Puppe exact in a 2-dimensional sense, while Triang satisfies the weaker exactness property of a 2-homological category.

This talk is based on a joint work in progress with Zurab Janelidze, Luca Mesiti and Ulo Reimaa.