## Offerta formativa erogata A.A. 2020/2021

**Courses 2020-21**

**Course - **** Cluster algebras and Poisson geometry** (16 hours)

*Alek Vainshtein (University of Haifa)*

**Timetable:**July 5, 7, 9, 12, 14, 16, 19, 21

**Time and room:**2:30-4:30pm, room B

**Abstract:**In this short course I plan to give a self-contained and concise introduction to the connections between cluster algebras and Poisson geometry. I will start from scratch and provide the necessary notions of Poisson geometry as well as several basic examples that lead to the notion of cluster structures. Next, I will proceed to basic definitions, explain the Laurent phenomenon and give an overview of the finite type classification. The central part of the course is built around the notions of compatible Poisson structures, which leads in particular to quantum cluster algebras, and compatible pre-symplectic structures, which leads to cluster algebras defined by triangulated surfaces. Finally, I will present several applications of these ideas to the study of cluster structures in the rings of regular functions on algebraic varieties and to integrable systems.

**Course - ****The topology of positive scalar curvature** (16 hours)

*Thomas Schick* (March 2021)

**First meeting:**Monday 8 March 2021

**Timetable:**Monday 11:15 (sharp) Aula II - Wednesday 15:30 (sharp), Aula Picone. Two (academic) hours each. Also via Meet.

**Course - ****Introduction to GFF, multiplicative chaos and Liouville quantum gravity**

*Nathanael Berestycki*

In recent years rigorous approaches to Liouville quantum gravity have been proposed and this has led to extraordinary progress in many different directions, including our understanding of large random planar maps. These approaches are based on the Gaussian free field and its associated Gaussian multiplicative chaos. I will introduce these notions and discuss several related themes, including (time-permitting): Liouville Brownian motion, the quantum zipper and the mating of trees theorem, and applications to random planar maps.

**Cycle of talks**** - Optimal control and applications**

**Duration**: 30 hours

**Preliminary Zoom meeting**: 2 March 2021 at 2:30pm, contact

*falcone@mat.uniroma1.it*

**Starting date**: 9 March 2021, at 2:30pm

The cycle of talks aims to be an introduction to optimal control theory for systems driven by ordinary differential

equations and to present some recent applications to reinforcement learning and mean field games. We will

discuss the main results related to the dynamic programming approach and the solution of the corresponding

Hamilton-Jacobi equations giving some hints also on the numerical approximation of those problems.

The cycle is organized in three modules of about 10 hours each: Optimal control, Reinforcement learning, Mean field games. Here is a tentative program.

**Optimal control**, M. Falcone (Sapienza)Introduction to some classical problems of deterministic control theory. The direct approach and Pontryagin

principle. Dynamic programming and Hamilton-Jacobi-Bellman equations. Value function and viscosity solutions.

Feedback reconstruction. Numerical approximation and algorithms. Optimal control of diffusion processes .

**Reinforcement learning**, M. Palladino (GSSI, L'Aquila)Introduction to Reinforcement Learning (RL). Model free vs Model based RL. Bayesian RL. Connection between

Optimal control and Reinforcement Learning. Modeling uncertainty in RL.

**Mean field games**, F. Silva (Limoges)Nash equilibria in differential games with infinitely many players and Mean Field Games. Some applications in

economics, finance, social sciences. Characterization of equilibria via a system of nonlinear PDEs of Hamilton-Jacobi-Bellman and Fokker-Planck equations. Existence and uniqueness. Some hints on numerics.

**Course**** - High dimensional probability**

*A. Faggionato, V. Silvestri, L. Taggi (Sapienza Università di Roma)*

**Beginning:** Tuesday, February 9, at 10:00 via zoom.

**Course**** - An introduction to rational homotopy theory**

*Ruggiero Bandiera (Sapienza Università di Roma)*

Roughly, rational homotopy theory is the study of (simply connected, or more in general nilpotent) topological spaces up to rational homotopy equivalences, where a map f:X-->Y is a rational homotopy equivalence if it induces an isomorphism between the torsion-free parts of the homotopy groups. We shall review the two main approaches to rational homotopy theory,

via differential graded Lie algebras (after Quillen) and via differential graded commutative algebras (after Sullivan), as well as the relationship between the two approaches. In both cases, one associates to a space X an algebraic model (a dgla in the work of Quillen, a dgca in the work of Sullivan) which completely encodes the rational homotopy type of X.

In particular, given such an algebraic model one can easily extract (rational) homotopical information on X, such as the rational homotopy groups, rational (co)homology groups, rational Whitehead brackets, rational Postnikov towers, et cet.. Moreover, these algebraic models can be often determined explicitly, especially in Sullivan's approach, making rationalhomotopy theory much more amenable to explicit computations than ordinary homotopy theory. Some classical applications to geometry will be discussed, such as the formality of compact Kaehler manifolds (after Deligne, Griffiths, Morgan and Sullivan), or the existence of infinitely many (geometrically distinct) closed geodesics on compact simply connected Riemannian manifolds

whose rational cohomology algebra requires at least two generators (after Sullivan and Vigué). Some elements of model category theory and strong homotopy algebras shall also be reviewed along the way, when needed. Additional topics might vary depending on the interests of the participants: possible topics include

- the rational dichotomy between rationally elliptic and rationally hyperbolic spaces;

- rational Lusternik-Schnirelmann category;

- algebraic models of function spaces and disconnected rational homotopy theory.

**Course**** - Combinatorics of diagonal coinvariants**

*Michele D'Adderio (Libre Université de Bruxelles)*

**Duration**: 10 hours

**Period**: January-February 2021

Contact: for more information please write to bravi@mat.uniroma1.it

Since the pioneering works of Frobenius, Young and Schur, more than a century ago, the interplay between representations of the symmetric group, symmetric functions and the endless combinatorics arising in this context, has been a central paradigm in algebraic combinatorics. In the last three decades, since Macdonald introduced his famous symmetric functions, there has been a plethora of new developments, all apparently related to so-called "diagonal coinvariants".

The goal of this mini-course (five 2-hour lectures) is to discuss some recent and surprising breakthroughs and conjectures, which gave rise to quite a bit of excitement in the community. The mini-course has very little prerequisites, and should be accessible to any student with a basic background in algebra (bachelor level), mathematical curiosity and open-mindedness.

**Corso** (8 crediti) - **Seminari di ricerca in didattica e storia della matematica**

Il corso, che si articola in seminari in parte indipendenti fra loro, si propone di offrire un panorama di alcune tematiche di ricerca in didattica della matematica e in storia della matematica. Gli argomenti affrontati saranno diversi dagli argomenti dell’analogo corso svolto per il dottorato nel 2019-20, ma in continuità con quelli.

Per il calendario della prima parte del corso cliccare qui.

Il link per partecipare a tutti i seminari del ciclo è qui (è richiesta la registrazione al primo accesso).

Il corso è coordinato da Claudio Bernardi e Annalisa Cusi; intervengono come docenti anche Alessandro Gambini, Nicoletta Lanciano, Marta Menghini, Enrico Rogora. Saranno inoltre occasionalmente invitati docenti di altre università italiane e straniere.

**Reading courses**

**- Mathematical methods in Quantum Mechanics (A. Teta, G. Panati, Monaco)**

**- Nonlinear elliptic equations (F. De Marchis, F. Pacella) - 4 credits**

- **Variational methods in material sciences (A. Garroni, E. Spadaro) **

Timetable: 1-2 introductory lectures (end of January-beginning of February), students' talks (end of February-beginning of March), working group (from mid March on)

The reading course will cover some of the main mathematical tools needed to study, in the framework of Calculus of Variations, models related to the mechanics of materials (in particular models in elasticity and plasticity). Some examples: rigidity theory, Gamma convergence, linear and non linear elasticity, analysis of topological defects.

The main goal will be to provide the essential background in order to attend a series of seminars (in the form of a working group) delivered by A. Garroni and E. Spadaro and devoted to an overview of recent results in the study of models for low angle grain boundaries due to Lauteri and Luckhaus (with the detailed description of the main analytical new ideas).

**Courses 2019-20**

**List of courses held by invited professors ** (details below)

T. Schick - The topology of positive scalar curvature

M. Hauray - M. Pulvirenti - Scaling limits and effective equation in kinetic theory

C. De Concini - Hodge theory and matroids

D. Noja - Nonlinear Schroedinger equation on graphs

C. Bernardi, A. Cusi - Seminari di ricerca in didattica della matematica

P. Pragacz - Gysin formula for homogeneous spaces

G. Besson - Finiteness and compactness in negative curvature

**List of graduate courses (in common with Laurea Magistrale)**

Details here and here

**Fall term**

ISTITUZIONI DI ALGEBRA SUPERIORE - MAT/02

ALGEBRA SUPERIORE - MAT/02

ISTITUZIONI DI GEOMETRIA SUPERIORE - MAT/03

TOPOLOGIA ALGEBRICA - MAT/03

GEOMETRIA RIEMANNIANA - MAT/03

CORSO MONOGRAFICO DI STORIA DELLA MATEMATICA - MAT/04

ISTITUZIONI DI ANALISI SUPERIORE - MAT/05

EQUAZIONI ALLE DERIVATE PARZIALI - MAT/05

EQUAZIONI DIFFERENZIALI NON LINEARI - MAT/05

PROCESSI STOCASTICI - MAT/06

STATISTICA MATEMATICA - MAT/06

CALCOLO STOCASTICO E APPLICAZIONI - MAT/06

FISICA MATEMATICA SUPERIORE - MAT/07

ANALISI DI SEQUENZE DI DATI - MAT/07

MECCANICA DEI FLUIDI - MAT/07

MODELLI DI RETI NEURALI - MAT/07

ISTITUZIONI DI ANALISI NUMERICA - MAT/08

FISICA MODERNA - FIS/08

TEORIA DEGLI AUTOMI - INF/01

TEORIA DEI CODICI - INF/01

METODI NUMERICI PER LE EQUAZIONI ALLE DERIVATE PARZIALI NON LINEARI - ING-IND/06

__Spring term__

MATEMATICA DISCRETA - MAT/02

GEOMETRIA ALGEBRICA - MAT/03

GEOMETRIA SUPERIORE - MAT/03

MATEMATICHE ELEMENTARI DA UN PUNTO DI VISTA SUPERIORE - MAT/03

FONDAMENTI DELLA MATEMATICA - MAT/04

DIDATTICA DELLA MATEMATICA - MAT/04

SPAZIO E FORMA - MAT/04

ANALISI FUNZIONALE - MAT/05

ANALISI SUPERIORE - MAT/05

MODELLI ANALITICI PER LE APPLICAZIONI - MAT/05

ISTITUZIONI DI PROBABILITA' - MAT/06

ISTITUZIONI DI FISICA MATEMATICA - MAT/07

SISTEMI DINAMICI - MAT/07

METODI NUMERICI PER LE EQUAZIONI ALLE DERIVATE PARZIALI - MAT/08

ELEMENTI DI FISICA TEORICA - FIS/02

TEORIA DEGLI ALGORITMI - INF/01

**Courses held at the department of Mathematics at the school of engineering (SBAI)**

Details here

**Reading courses**

Here is a list of potential reading courses by the faculty and of some reading courses held in the previous years. Graduate students are encouraged to contact faculty members and request the activation of reading courses of their interest.

**Abstracts and details**

Dates: April 2020 (to be determined)

**COURSE (6 credits) - 24 hours**

Maxime Hauray - Mario Pulvirenti

__Scaling limits and effective equation in kinetic theory__Monday 15.30-17.30, Wednesday 14.00-16.00

First class: Monday 10 February 2020

Abstract.

*In this course I will discuss the scaling limits necessary to outline the physical regimes one wants to discuss, starting from large (classical) particle systems. The goal is to derive rigorously the effective equations which are largely used in kinetic theory, as the Boltzmann, Vlasov and Landau equations. From a mathematical side we have very few results and many open challenging problems.*

**COURSE (3 credits) - 12 hours**

First lecture: Tue 4 February, Aula B 15:00-17:00

**COURSE (3 credits) - 10 hours**

Diego Noja

__Nonlinear Schroedinger equation on graphs__Room B

Mon 27 Jan - at 16:00-18:00

Tue 28-Wed 29-Thu 30 Jan - at 14:00-16:00

Fri 31 Jan - at 10:00-12:00

**CORSO (8 crediti)**

__Seminari di ricerca in didattica della matematica__Giovedì, 15:00-17:30 in Aula B - Calendario degli incontri

Il corso, che si articola in seminari in parte indipendenti fra loro, si propone di offrire un panorama di alcune tematiche di ricerca in didattica della matematica. Fra i temi che saranno affrontati, citiamo: aspetti legati alla dimostrazione (sviluppo di argomentazioni, vari stili di dimostrazione, ecc.), l'evoluzione del concetto di matematiche elementari da un punto di vista superiore, le concezioni iniziali degli studenti, il passaggio dalla scuola secondaria all’università.

Il corso è coordinato da Claudio Bernardi e Annalisa Cusi; intervengono come docenti anche Alessandro Gambini, Marta Menghini, Nicoletta Lanciano, Enrico Rogora. Saranno inoltre occasionalmente invitati docenti di altre università italiane.

**COURSE**

**Gysin formula for homogeneous spaces**

Piotr Pragacz (Polish Academy of Sciences)

**Lectures**

1. Flag bundles, Segre polynomials, and push-forwards

Wed 13 November, 14:00-15:00, Sala di Consiglio

*We give Gysin formulas for all flag bundles of types A, B, C, D. The formulas (and also the proofs) involve only the Segre classes of the original vector bundles and characteristic classes of universal bundles. As an application we provide new determinantal formulas.*

This is a joint work with Lionel Darondeau.

This is a joint work with Lionel Darondeau.

2.-3. Gysin maps, duality, and Schubert classes I-II

Thu 14 and Mon 18 November, 15:00-16:00, Room B

*We establish a Gysin formula for Kempf-Laksov flag bundles and we prove a duality theorem for Grassmann*

*bundles. We then combine them to study Schubert bundles, their push-forwards and fundamental classes.*

This is a joint work with Lionel Darondeau.

This is a joint work with Lionel Darondeau.

4. A Gysin formula for Hall-Littlewood polynomials

Thu 21 November, 15:00-16:00, Room B

*We give a formula for pushing forward the classes of Hall-Littlewood polynomials in Grassmann bundles, generalizing Gysin formulas for Schur S- and P -functions.*

**COURSE**

**Finiteness and Compactness in Negative Curvature**

Gérard Besson (Grenoble)

**Lectures:** 1, 3, 8, 10 October 2019 - Room B, 14:00-17:00

**Abstract**

*Finiteness and compactness problems in Riemannian geometry date back to the pioneering works of Cheeger, Gromov, Grove-Petersen, Anderson et al. ([C],[G],[GP],[A]) which started the ``theory of convergence of Riemannian manifolds'', one of the main trends in Riemannian geometry and topology as of today. *

*The aim of the course is to generalize, in negative curvature, the classical works of Cheeger and Gromov under much weaker assumptions.*

*Actually, the classical bounds on sectional/Ricci curvature and on the injectivity radius are replaced, in our setting, only by a bound on the entropy, which is a much more flexible (and global) invariant. Morover, by "negative curvature" we mean Gromov-hyperbolicity, which is a substitute of classical Riemannian negative sectional curvature on large-scale. This allows us to consider larger classes of spaces (not only Riemannian manifolds) and to better characterize the limit spaces. At present, the more refined results of convergence theory apply only to classes of manifolds with a lower bound on the Ricci curvature, without any control of the regularity (and dimension) of the arising limit spaces.*

*The main topics touched in the course will be:*

*-a Margulis' lemma for groups acting on Gromov-hyperbolic spaces;*

*-a Bishop-Gromov inequality for Gromov-hyperbolic spaces;*

*-estimates of the first Betti numbers for quotients of Gromov-hyperbolic spaces;*

*-finiteness and compactness theorems.*

*The course will be based on the joint works (partly in progress) with G.Courtois, S.Gallot and A.Sambusetti:*

*[BCGS] G.Besson, G.Courtois, S.Gallot, A.Sambusetti, Curvature-free Margulis lemma for Gromov-Hyperbolic spaces, preprint arxiv: 1712.08386 (2017)*

*[BCGS2] G.Besson, G.Courtois, S.Gallot, A.Sambusetti, Finiteness and compactness for Gromov-Hyperbolic spaces, in preparation.*

PhD courses at Roma Tor Vergata and Roma Tre:

**PhD courses at Roma "Tor Vergata"**

PhD courses at Roma Tre