COURSES 2020/21

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
Starting date: 9 March 2021, at 2:30pm
Contact: for more information write to

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. 
Further details (including a link to  the meeting room and possibility to follow the lectures also at the department) will appear on the webpage of Prof. A. Faggionato soon.

Course - An introduction to rational homotopy theory

Ruggiero Bandiera (Sapienza Università di Roma)

Duration: TBA
Period: TBA
Contact: for more information please write to

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

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 2020-21


Algebra and geometry


- Algebraic combinatorics (C. Malvenuto)

- Symmetric and quasi-symmetric functions (C. Malvenuto)

- Combinatiorial Hopf algebras (C. Malvenuto)

- Actions and representations of algebraic groups (G. Pezzini)

- Toric and spherical varietes (G. Pezzini)

- Classical and geometric invariant theory (G. Pezzini)

- Kobayashi hyperbolicity and relations to arithmetic and algebraic geometry (S. Diverio)

- Positivity in analytic and algebraic geometry, and its counterpart in complex differential geometry (S. Diverio)

- Atiyah-Singer index theorem (P. Piazza)




- Vectorial calculus of variations (A. Garroni, E. Spadaro)

- Gamma-convergence (A. Braides, A. Garroni, A. Malusa)

- Geometric measure theory (E. Spadaro)
- Linear elliptic equations with singular drift term (L. Boccardo)
- Some integral functionals with easy minimisation, but not so easy Euler-Lagrange equation (L. Boccardo)

- Evolution of Harmonic maps and liquid crystals (A.Pisante)
- Nonlinear elliptic equations (F. De Marchis, F. Pacella) - 4 credits - ACTIVE

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

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).



Probability, mathematical physics and numerical analysis


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

- Dynamics of infinitely many particles and models of viscous friction (P. Buttà, G. Cavallaro). 

- Gradient flow and applications to discrete spaces (G. Basile, L. Bertini)

- High dimensional probability and statistics (A. Faggionato) 

- Stochastic systems of interacting particles (G. Posta)

- Mixing time in Markov chains (G. Posta)

- Numerical methods in linear algebra (S. Noschese) 
- Implicit methods for hyperbolic problems (G. Puppo)

- Numerical methods and modelling for vehicular traffic (G. Puppo)

- Numerical methods for optimal controls and Mean Field Games (E. Carlini)



More PhD courses:

PhD courses at University of Roma "Tor Vergata"
PhD courses at University Roma Tre

PhD courses at University of Pisa


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