## 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*

**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

**Optimal control**, M. Falcone (Sapienza)

**Reinforcement learning**, M. Palladino (GSSI, L'Aquila)

**Mean field games**, F. Silva (Limoges)
**PROPOSAL OF COURSES AND ACTIVITIES RELATED TO PROBABILITY**

**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 bandiera@mat.uniroma1.it

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

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**

__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)

**Analysis**

- 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)

**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 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