Course - Polynomial identities and combinatorial methods
Antonio Giambruno (Università di Palermo) - 6 hours
Place and time: Room B, 11-13
Timetable: 11, 15, 16 November 2021
A polynomial identity of an algebra A is a polynomial in noncommutative variables vanishing under any evaluation in A. The set of polynomial identities of A is an ideal of the free algebra, called T-ideal, invariant under endomorphisms. The purpose of the minicourse is to give some highlights of the study of T-ideals through the representation theory of the symmetric or general linear group.
Course - Travelling fronts and spreading properties for reaction-diffusion equations
Period: May-June 2022
Reaction-diffusion equations are involved in many fields of physics and life sciences, and they are also mathematically extremely rich. These parabolic partial differential equations are the most frequently used in many models of population dynamics and play a central role in the description of biological invasions. They can be written in the simplest case as
u_t=Delta u+f(u), where u_t is the partial derivative with respect to the time variable, Delta u is the Laplacian with respect to the space variables, accounting for the diffusion process, f(u) stands for the nonlinear reaction terms involving birth, death, cooperation and/or competition mechanisms. The existence of traveling fronts and spreading solutions is an essential feature of the reaction-diffusion equations and has greatly contributed to their popularity. The mathematical theory of these equations goes back to more than 80 years ago. But it has been booming again in the past 20 years, based on the introduction of new general notions of propagation and also both fed by some important applications such as the modelling of epidemics, just to mention one.
The founding works of Berestycki and Hamel '[BH12], Berestycki, Hamel and Nadirashvili [BHN05] [BHN10], Hamel and Nadirashvili [HN01], on the notions of transition fronts and propagation speeds for general evolution equations have shed a completely new and unexpected light on the theory and the description of the solutions. The notion of transition fronts, which involves families of moving hypersurfaces and the convergence to some limit states far away from these interfaces, uniformly in time, extend all the previously known cases of traveling or pulsating fronts in homogeneous or periodic environments. These works, together with the recent paper [HR21] on spreading and asymptotic one-dimensional symmetry for the solutions of the Cauchy problem with initial conditions having general unbounded support, open fascinating prospects for a better understanding of the long-time dynamics and qualitative properties of the solutions for a large class of equations in various geometrical configurations.
[BH12] H. Berestycki, F. Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math. 65 (2012), 592-648.
[BHN05] H. Berestycki, F. Hamel, N. Nadirashvili, The speed of propagation for KPP type problems I - Periodic framework, J. Europ. Math. Soc. 7 (2005), 173-213.
[BHN10] H. Berestycki, F. Hamel, N. Nadirashvili, The speed of propagation for KPP type problems II - General domains, J. Amer. Math. Soc. 23 (2010), 1-34.
[HN01] F. Hamel, N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in RN, Arch. Ration. Mech. Anal. 157 (2001), 91-163.
[HR21] F. Hamel, L. Rossi, Spreading speeds and one-dimensional symmetry for reaction-diffusion equations, https://arxiv.org/abs/2105.08344.
Course - Group cohomology and algebraic cycles - (about 20 hours)
Roberto Pirisi (Sapienza Università di Roma)
First meeting: TBA
Reading course - Introduction to the theory of schemes
Simone Diverio (Sapienza Università di Roma)
First meeting: Wednesday 13 October 2021
Timetable: every Wednesday, 15:30 - 17:30, Aula B
Content: Presheaves and sheaves, spectrum of a ring, ringed spaces, schemes, reduced and integral scheems, dimension. Base change, algebraic varieties, finite morphisms, separate morphisms, proper morphisms. Normal, regular, flat schemes. Flat and étale morphisms. Zariski main theorem. Coherent sheaves, Čech cohomology, higher direct images.
Note: it is a monographic reading course, aimed at getting acquainted with basic notions in the theory of algebraic schemes. Classes will be held by the participants.
Some results in commutative algebra will be recalled as they will be needed.
Main reference: Q. Liu, “Algebraic Geometry and Arithmetic Curves”
Course - Topics on Fano varieties
Enrico Fatighenti (Sapienza Università di Roma)
Time: Winter-Spring 2022 (about 20 hours)
Fano varieties are one of the most studied classes of varieties in algebraic geometry, for example in the subfields of birational geometry. In this course we will survey some results from a more representation-theoretical angle, starting from Mukai's classification of prime Fano threefolds and their models inside homogeneous varieties. We will then show how to complete the classification of threefolds using only "Mukai-style" biregular tools, and we will give some partial results on the fourfold case. Time permitting, we will also survey some higher-dimensional Fano varieties with special Hodge-theoretical features, such as Fano varieties of K3 and Calabi-Yau type, and their link with hyperkaehler geometry.
Course - Quiver representations
Giovanni Cerulli Irelli (Sapienza Università di Roma)
Time: mid March till mid May 2022 (about 8 weeks)
Timetable: every Tuesday and Thursday, 10-12, Aula B
This is a first course on quiver representations and representation theory of finite dimensional associative algebras from the point of view of cluster algebras. We will start by exploring in full detail the representation theory of quivers of type A. We will then introduce Auslander-Reiten theory for acyclic quivers and prove Gabriel's theorem. Meanwhile we introduce classical BGP reflection functors and compare it with the Auslander-Reiten translation. After that we will focus on derived categories of quiver representations following Happel's work in 1988. At this point I will introduce the cluster category introduced by Buan-Marsh-Reiten-Reineke-Todorov to categorify cluster algebras. I want to finish the course by showing the general version of the reflection functors given by Derksen-Weyman-Zelevninsky and their theory of quivers with potential.
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 2020-21, ma in continuità con quelli.
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 2021-22
Algebra and geometry
- Algebraic combinatorics (C. Malvenuto)
- Symmetric and quasi-symmetric functions (C. Malvenuto)
- Combinatorial Hopf algebras (C. Malvenuto)
- Actions and representations of algebraic groups (G. Pezzini)
- Toric and spherical varieties (G. Pezzini)
- Classical and geometric invariant theory (G. Pezzini)
- Symplectic reflection algebras (P. Bravi, G. Pezzini)
- An introduction to the theory of schemes (S. Diverio) - ACTIVE
- 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)
- Gamma-convergence (A. Garroni)
- Geometric measure theory (E. Spadaro)
- Linear elliptic equations with singular drift term (L. Boccardo)
- Variational methods in material sciences (A. Garroni, E. Spadaro)
Probability, mathematical physics and numerical analysis
- Mathematical methods in quantum mechanics (D. Monaco, G. Panati, A. Teta)
- 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)
- 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