Progetto formativo per il XLII ciclo del Dottorato in Scienze Matematiche

Progetto formativo per il XLII ciclo del Dottorato in Scienze Matematiche

n. TITOLO ORE DESCRIZIONE DEL CORSO
1. The mathematical physics of lattice models 15 Lattice models are a cornerstone in the theoretical description of solids and gases, providing a unifying framework for understanding complex physical phenomena. Their importance is reflected in major scientific recognitions, including the Nobel Prizes awarded to Philip Anderson (1977) and Giorgio Parisi (2021), as well as the Fields Medal to Hugo Duminil-Copin (2022). These models offer powerful tools to tackle fundamental open problems and to interpret a wide range of experimental observations.

Over the past three decades, lattice models have also become a central topic in Mathematical Physics, where rigorous results have improved our understanding of non-perturbative behaviors in physical systems and have attracted significant interest in the mathematical community.

This course aims to provide an accessible introduction to the mathematical theory of lattice models, while highlighting some of the most recent developments in the field.
2. Argomenti avanzati di geometria algebrica 15 Il corso propone alcuni argomenti avanzati nell’ambito della geometria algebrica e che siano propedeutici alla ricerca in tale ambito. Gli argomenti spaziano dalla geometria delle varietà algebriche, all’applicazione di metodi omologici per il loro studio alla loro teoria delle deformazioni.
3. Logica e Categorie 15 Il corso propone una selezione di argomenti di ricerca tratti dalle aree dell’Algebra Categoriale, della Teoria dei Topos, della Logica Algebrica, della Logica Categoriale, della Logica Computazionale e della Teoria dei Modelli. Gli argomenti specifici saranno scelti tenendo conto degli interessi dei potenziali frequentanti, avendo cura di discutere le necessarie nozioni preliminari.
4. Approssimazione con reti neurali 15 L’uso delle reti neurali è diventato molto popolare negli algoritmi di apprendimento. Nonostante il loro grande successo nella pratica, non è stata ancora trovata una spiegazione teorica convincente. Poiché nelle applicazioni le reti neurali forniscono approssimazioni di una funzione, parte del loro successo risiede nell’accuratezza di tali approssimazioni.

Il corso si propone di introdurre all’approssimazione di una funzione nota mediante reti neurali, sottolineando le differenze rispetto ai metodi più classici
5. Advanced topics on stochastic control and games 15 This 15-hour PhD course offers an introduction to some modern topics in stochastic control and game theory, focusing on strategic interactions under uncertainty, large-population models, and informational asymmetries. After a brief review of the classical framework of stochastic control, the course will discuss mean field control and mean field games, highlighting the contrast between centralized optimization and decentralized equilibrium. It will then turn to stochastic differential games with asymmetric information, where players act strategically on the basis of different observations or beliefs, and finally introduce Bayesian persuasion as a framework for information design and strategic disclosure. The course is intended to provide participants with a coherent mathematical overview of these themes, emphasizing both the underlying probabilistic structures and their connections with current research
6. Wick calculus and Wick renormalization 16 Wick calculus is central in modern and current physics and probability. It is the first renormalization tool in Quantum Field Theory and Statistical Physics. It underlies chaos decompositions and is essential in the foundations of stochastic calculus. Besides the most classical one (related to algebras of Gaussian variables and free quantum fields), it takes various less familiar forms, including for example in finite or noncommutative probability. Ultimately, it has been the subject of various articles recently, dedicated to it or using it intensively. At our best knowledge, no systematic account of it, covering all its facets, is available in the literature.

The series of lectures will provide a systematic, complete and self-contained account of Wick and Wick renormalization, explaining why and how it has been used, classically and recently, in its various application domains. Furthermore, they will present Wick calculus featuring the recent understanding provided by an algebraic approach (in the vein of the Connes-Kreimer and Hairer approaches to renormalization).
7. Geometric and Functional Inequalities and their Applications 18 In recent years, there has been growing interest in geometric and functional inequalities—such as isoperimetric inequalities, the Brunn–Minkowski inequality, and Sobolev and Gagliardo–Nirenberg inequalities—along two main research directions.

The first direction focuses on stability problems, aiming to understand whether a function that almost attains equality in such inequalities must be close (and in what precise sense) to the optimal functions, which are typically known. This line of work seeks quantitative refinements of classical inequalities.

The second direction concerns the extension of these inequalities beyond Euclidean spaces, investigating their validity in more general settings such as Riemannian manifolds and even non-smooth spaces with controlled geometric properties. The minicourse will investigate these lines of research and will describe the recent results on these topics.

The minicourses will investigate these lines of research and will describe the recent results on this topics.
8. Local and Nonlocal Minimal Surfaces and Geometric Flows 18 This school explores recent advancements and foundational techniques in the study of minimal surfaces and geometric evolution equations. The lectures will cover a broad spectrum of topics in modern geometric analysis, from both an elliptic and parabolic perspective, highlighting analogies and differences in the study of local and nonlocal minimal surfaces as well as extrinsic (e.g. mean curvature) and intrinsic (e.g. Ricci) curvature flows.

Participants will be introduced to state-of-the-art analytical methods, with a focus on regularity theory, the structure of singularities, and long-time asymptotics. Designed primarily for PhD students and early-career researchers, the school aims to provide both the necessary background and an overview of cutting-edge research. Attendees will have the opportunity to engage with open problems and acquire advanced mathematical tools in geometric measure theory and nonlinear partial differential equations.
9. Analyitic de Rham stacks  15 Applying the new theory of analytic stacks of Clausen and Scholze we introduce a general notion of derived Tate adic spaces. We use this formalism to define the analytic de Rham stack in rigid geometry, extending the theory of D-cap-modules of Ardakov and Wadsley to the theory of analytic D-modules. We prove some foundational results such as the existence of a six functor formalism and Poincaré duality for analytic D-modules, generalizing previous work of Bode. Finally, we relate the theory of analytic D-modules to previous work of the author with Rodrigues Jacinto on solid locally analytic representations of p-adic Lie groups.

 
10. Étale and tame-étale cohomology  15 This PhD course explores étale and tame cohomology, tracing their development from classical algebraic geometry to modern rigid analytic geometry. We begin with the foundational properties of the étale site, covering classical results. Addressing the pathological behavior of étale cohomology for $p$-torsion sheaves in positive characteristic, we then introduce the tame site, which resolves these structural deficiencies by suppressing wild ramification. Finally, we apply these tools to rigid analytic varieties and adic spaces.