Prossimi seminari del Dipartimento di Matematica

A quaternionic version of the classical Hilbert Nullstellensatz has recently been established in its strong form within the framework of slice regular polynomials. This result naturally led to the introduction of the notion of a quaternionic slice algebraic set, endowing (\mathbb{H}^n) with a Zariski-type topology. In this seminar, after a brief overview of the subject, we will discuss a characterization of the irreducibility of slice algebraic sets in terms of algebraic properties of the associated ideals of slice regular polynomials. The results presented are based on joint works with Giulia Sarfatti and Fabio Vlacci.

A 90 anni dalla scomparsa di Salvatore Pincherle, presentiamo una panoramica della sua attività scientifica. Questa ricerca si basa in parte sulla tesi di laurea magistrale in Matematica elaborata da Linda Canti, sotto la mia supervisione, nonché su altri documenti d'archivio relativi a Pincherle che ho recentemente raccolto. Ninety years after Salvatore Pincherle's death, we present an overview of his scientific work. This research is based in part on the Master's thesis in Mathematics written by Linda Canti, under my supervision, as well as on additional archival documents concerning Pincherle that I have recently collected.

In 1994, Simon conjectured that every pure shellable simplicial complex is shelling-completable, i.e. the independence complexes of uniform matroids are extendably shellable. In this seminar, after reviewing the fundamental concepts and providing an overview of the state of the art, I present a technique producing an infinite number of shellable complexes that are not shelling-completable—one for each dimension greater than or equal to 3—thereby disproving Simon's conjecture. This is joint work with Paolo Sentinelli.

We discuss the possibility to train a thermodynamic system, whose micro-variables are fully determined through the Hamiltonian by the usual statistical mechanical rules, to perform tasks such as memorization and generalization. Training is achieved by the application of suitable external fields, playing the role of {\it data}. At variance with conventional machine learning, no other logical or algorithmic rules are introduced. The system is amenable, in principle, to exact analytical calculations. We specialize this general approach to a prototypical Ising system with annealed dichotomous couplings and study its learning ability. Results indicate excellent memorization and good generalization capacity already for small systems, and a tendency to improve with system size.

These lectures cover the optimal control of McKean-Vlasov equations, a topic that has attracted growing interest in connection with mean-field game theory, large population stochastic control, and more recently machine learning. We start with mean-field Markov decision processes in discrete time, where the key ideas — propagation of chaos, lifting to the Wasserstein space, dynamic programming — can be introduced transparently. We then develop the continuous-time theory of controlled McKean-Vlasov SDEs, building the necessary tools: differentiation with respect to probability measures, Itô's formula along flows of marginal laws, the Master Bellman equation, and the stochastic maximum principle with its associated FBSDE system. Linear-quadratic problems serve as a running illustration throughout. The last part of the course is devoted to numerical methods and applications. We present neural network-based algorithms and reinforcement learning approaches for solving mean-field control problems on the Wasserstein space, and discuss recent extensions to non-exchangeable mean-field control for heterogeneous interacting systems. We conclude with an application to generative modeling, where learning stochastic dynamics from distributional observations is formulated as a McKean-Vlasov control problem, with optimality conditions given by a tractable FBSDE system.