Progetto formativo per il XL ciclo del Dottorato in Scienze Matematiche
| DOCENTE | CORSO | MESE DI INIZIO/DURATA | ABSTRACT |
| Prof. Luciano Campi, Unimi e Prof.ssa Giorgia Callegaro, Università di Padova |
Stochastic control and games with different information structures |
Gennaio 2025
15 ore (3 crediti) |
We will start by giving a short introduction of the most important approaches for solving stochastic control problems and games based on PDEs and BSDEs. In those problems an agent or a group of players are willing to optimize some objectives, which depend on the state of a (controlled) stochastic dynamic system. In this part, the latter will be assumed to be fully observable. Then, we will tackle the more realistic case where the agent/players have only a partial information on the state variable. Filtering theory will play a crucial role in this class of problems. After that, we will pass to the case when the agent/players can have different information structures on the state. Finally, we will apply the techniques previously developed in the lectures to some relevant applications in energy economics, Bayesian persuasion and finance. A good knowledge of the basis of stochastic calculus is required to fully understand the content of the course. |
| Prof. Alexander Volberg, Michigan State University | Boolean harmonic analysis with application to learning of classical and quantum objects |
Aprile 15 ore (3 crediti) Anno accademico 2024-25 |
Harmonic analysis on discrete structures similar to Hamming cube plays a crucial role in numerous questions of theoretical computer science and modern learning theory. I will explain the connection and demonstrate the power of the approach on several examples. There are many unsolved problems. Some of them will be explained. |
| Prof. Louis Dupaigne, Université Claude Bernard Lyon 1 | Partial differential equations under curvature bounds |
Maggio 15 ore (3 crediti) Anno accademico 2024-25 |
Curvature and PDEs have a long history together. PDEs can be used to give an answer to geometric questions, e.g. Yamabe’s problem. Conversely, geometry, specifically curvature, appears when studying qualitative properties of solutions of certain PDEs, such as the symmetry of extremal functions in optimal functional inequalities or the convergence to equilibrium of nonlinear diffusion flows. The course aims at providing a gentle introduction to optimal functional inequalities, Gamma calculus, Otto’s calculus, n-conformal geometry and elliptic regularity theory |
| Prof. Zaher Hani, University of Michigan | An introduction to mathematical wave kinetic theory |
Giugno 15 ore (3 crediti) Anno accademico 2024-25 |
This course will present the recent progress in the rigorous justification of the wave kinetic theory. This is the theory of non-equilibrium statistical physics for nonlinear waves. The main object to justify is the so-called “wave kinetic equation”, which stands as the wave analog of Boltzmann’s equation; this is Hilbert’s sixth problem for waves. We shall explain this progress in the context of the nonlinear Schrodinger equation, as a representative and universal model for nonlinear dispersive systems. The main methodology is the use of Feynman diagram (or Dyson series) expansions, and the key is to perform a deep analysis of the diagrams that allows to prove the convergence of the expansion. We shall discuss all the new ideas that were developed in this context to resolve this longstanding open problem. |
| Prof. Luciano Mari, Unimi | Il problema di Bernstein |
Aprile 15 ore (3 crediti) Anno accademico 2024-25 |
This course aims to introduce to the Berstein problem and to its generalizations. The “Bernstein problem” is the question whether minimal hypersurfaces in R^{n+1} which can globally be written as a graph must necessarily be Euclidean hyperplanes. Its complete solution (a positive answer if and only if n < 8) is a cornerstone in the development of Geometric Measure Theory and Geometric Analysis, and fits into a line of research that recently received new impulse, with the solution to a longstanding open problem (at least, in some of the conjectured ambient dimensions). The aim of the course is to discuss the original Bernstein problem and its proofs, and to some of the recent approaches to the more general problem of characterizing complete, stable minimal hypersurfaces (the “stable Bernstein problem”). |
| Dott,ssa Laura Pertusi, Unimi | Condizioni di stabilità su categorie derivate e componenti di Kuznetsov (Stability conditions on derived categories and Kuznetsov components) |
Aprile 15 ore (3 crediti) Anno accademico 2024-25 |
The construction of stability conditions on the bounded derived category of higher dimensional varieties is a difficult task starting from dimension three. Nevertheless, once a stability condition is given, many interesting applications are known, in particular concerning the study of the geometry of the associated moduli spaces of semistable objects. In this course, we will introduce the notion of derived category and semiorthogonal decompositions. Then we will define stability conditions and the known construction technique via tilt stability in lower dimension. Then we will focus on the case of Kuznetsov components in semiorthogonal decompositions, explaining a criterion to induce stability conditions, and applications to the geometry of certain Fano threefolds and fourfolds. |
| Prof. Marco Seveso, Unimi | Hecke characters and their associated L-functions |
15 ore (3 crediti) Anno accademico 2025-26 |
We will discuss Hecke charactes of number fields and their associated L-functions, following the adelic approach introduced by J. Tate, with the subsequent reformulation by P. Deligne recognizing the L and epsilon factors. Time permitting, we will mention p-adic analogues of these results for the field of rational numbers. |
| Prof. Federico Binda, Unimi | Derived geometry | Febbraio
15 ore (3 crediti) Anno accademico 2024-25 |
Learning the basic tools used in derived geometry, with applications ranging from deformation theory to recent breakthroughs in K-theory and (topological) Hochschild homology due to Antieu, Bhatt, Mathew, Morrow, Nikolaus, Scholze and others . We will build the theory from scratch, starting from Toen-Vezzosi’s “classical” approach. |
| Prof. Niels Benedikter, Unimi | Mathematical Methods for Many-Body Quantum Systems |
15 ore (3 crediti) Anno accademico 2025-26 |
After a very brief review of the mathematical framework of quantum mechanics, I will discuss second quantization methods for the analysis of many-body quantum systems. Topics include Fock space, creation and annihilation operators, coherent states, Bogoliubov transformations, quasifree states, Bose-Einstein condensation, and if time permits touch also upon aspects of variational approximations such as Hartree-Fock theory and the BCS theory of superconductivity. |