Seminari del Dipartimento di Matematica

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

ore 11:30

presso Aula Vitali

algebra e geometria

I will present an explicit construction of closed oriented aspherical smooth 4-manifolds realizing every positive Euler characteristic. This family of manifolds saturates the Hopf-Thurston conjecture on the Euler characteristic of aspherical manifolds, as well as the Gromov-Lück inequality on their geography.

The Mandelbrot set and its Satellite copies

ore 15:00

presso Seminario II

algebra e geometria

analisi matematica

didattica della matematica

interdisciplinare

sistemi dinamici

We consider dynamical systems on the Riemann sphere whose evolution rule is given by a complex polynomial. Here infinity is a (super-)attracting fixed point, and the filled Julia set is the set of points with bounded orbit. The quadratic family, P_c(z)=z^2+c, gives rise to the Mandelbrot set M: the set of parameters c such that the filled Julia set of P_c is connected. Computer experiments quickly reveal the existence of small homeomorphic copies of M inside itself; the existence of such copies was proved by Douady and Hubbard. Each little copy is either primitive (with a cusp on the boundary of its main cardioid region) or a satellite (without a cusp). Lyubich proved that the primitive copies of M satisfy a stronger regularity condition: they are quasiconformally homeomorphic to M. The satellite copies are not quasiconformally homeomorphic to M (as we cannot straighten a cusp quasiconformally), but are they mutually quasiconformally homeomorphic? In joint work with C. Petersen we prove that the answer is negative in general, but positive in some cases (when the rotation numbers have the same denominators).

Insiemi algebrici slice regolari e irriducibilità

ore 11:30

presso Aula Vitali

algebra e geometria

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.

L’EREDITÀ DI VITO VOLTERRA

ore 20:30

presso Cinema Perla - via san Donato 38

storia della matematica

The multiple cover formula conjecture for Gromov-Witten invariants of abelian surfaces

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

ore 11:30

presso Aula Vitali

algebra e geometria

Abelian surfaces are complex tori whose enumerative invariants satisfy remarkable regularity properties. The computation of their (reduced) Gromov-Witten invariants for the so called primitive classes is fairly well understood and many complete computations are available. A few years ago, G. Oberdieck conjectured a multiple cover formula expressing in a very simple way the invariants for the non-primitive classes in terms of the primitive one. In this talk, I'll sketch the proof of multiple cover formula conjecture obtained in a joint work with T. Blomme . This effectively solves completely the GW theory of abelian surfaces. The argument relies on a reduced degeneration formula and on the computation of correlated Gromov-Witten invariants for trivial bundles on elliptic curves.

Simon's Conjecture Disproved

ore 11:30

presso Aula Vitali

Seminario di Algebra e Geometria

algebra e geometria

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.

Thermodynamic learning

ore 11:00

presso - Aula Da Stabilire -

fisica matematica

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.

Brill-Noether loci of pencils with prescribed ramification

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

ore 11:30

presso Aula Vitali

algebra e geometria

Classical Brill-Noether theory studies projective algebraic curves through maps to projective space, using the language of linear series. A natural refinement of this framework considers linear series with prescribed ramification. In this talk, I will address the problem of the dimension of Brill–Noether varieties of pencils (i.e., linear series of rank one) with prescribed ramifications. I will compare this refined theory to the classical Brill–Noether problem, and present recent joint work with Andreas Leopold Knutsen and Ciro Ciliberto, which yields new results and improves upon the previous state of the art.

Mean Field Games: From Foundations to Singular Controls and Optimal Stopping

nel ciclo di seminari: SEMINARI DI PROBABILITÀ

ore 14:30

presso Centro Congressi "Alla Sosta dell'Imperatore" Strada dei Margeni, 18 38025 Dimaro Folgarida (TN)

Webpage della 3nd Dolomites Winter School on Mean-field Control, Mean-field Games and Applications

probabilità

Sito web Winter School

This short course provides an introduction to mean field games, with particular emphasis on their probabilistic formulation and on models involving singular control and optimal stopping. We will start from the basic idea of mean field interaction: a representative agent optimizes against the distribution of a large population, while equilibrium requires consistency between the agent’s optimal behavior and the induced population flow. After discussing this fixed point structure and some analytical tools, we will focus on classes of models motivated by economics and finance. Special attention will be devoted to mean field games in which agents make irreversible, monotone, or stopping decisions, both in stationary and non-stationary settings. In these models, a central theme is the interplay between free-boundary methods, reflected or absorbed diffusions, and fixed point arguments in the characterization of equilibria. We will also discuss how structural properties of the interaction, such as monotonicity or submodularity, can be exploited in suitable models to obtain comparison results and to gain insights into uniqueness or multiplicity of equilibria. The course will conclude by highlighting connections between mean field games and mean field control problems, as well as recent perspectives related to learning-based approaches.

Mean-Field Control: Theory, Methods, from Foundations to Learning

ore 15:30

presso Centro Congressi "Alla Sosta dell'Imperatore" Strada dei Margeni, 18 38025 Dimaro Folgarida (TN)

Webpage della 3nd Dolomites Winter School on Mean-field Control, Mean-field Games and Applications

probabilità

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.

Prossimi seminari del Dipartimento di Matematica