## COURSES 2021-22

__Travelling fronts and spreading properties for reaction-diffusion equations__*François Hamel (Aix-Marseille Université)*

**Timetable: every wednesday from 2pm to 4pm, from 18/05/2022 to 8/06/2022, (except Wednesday 1st June) aula C**

**Abstract:**

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, since the pioneering works of Fisher [F37] and Kolmogorov, Petrovsky and Piskunov [KPP37]. Much progress was made in the 70’s from the founding papers of Fife and McLeod [FM77], and Aronson and Weinberger [AW78], based on the use of powerful tools from elliptic and parabolic partial differential equations. The theory has been booming again in the past 20 years, from 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 Hamel and Nadirashvili [HN01], Berestycki, Hamel and Nadirashvili [BHN10], Berestycki and Hamel [BH12], 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. In the course, some essential aspects of travelling fronts and propagation phenomena for reaction-diffusion equations in the whole space will be discussed, especially from references [AW78], [HN01] and [HR21]. If time permits, the general notions introduced in [BH12] will be mentioned.

[AW78] D. G. Aronson, H.F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math. 30 (1978), 33-76.

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__Origini e sviluppi del calcolo differenziale assoluto__*Prof. Alberto Cogliati, Università di Pisa*

**Timetable: 12 - 23 September 2022**

**Abstract :**

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**Symplectic Geometry***Prof. Siye Wu*

**Timetable:**TBA

**Abstract**:

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__Corso di lettura__*Guido Pezzini (Classical and Geometric Invariant Theory)*

**Timetable:**

*Per i giorni e l'orario si prega di contattare direttamente il Professor Guido Pezzini.*

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**Course -**

**Constructing Locally Recoverable Codes using Galois Theory over global function fields.***Giacomo Micheli (University of South Florida)*

**Timetable:**

*1-8-15 June 2022 (classroom B, 15:30 -17:30)*

**Abstract**:

**Course - **__Energy-driven pattern formation: emergence of one-dimensional periodic structures__

*Eris Runa - 12 hours*

**Place and time:**Sala di Consiglio, 11:00-13:00

**Timetable:**17-19-21-24-26-28 January 2022

**Abstract.**

**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

**Abstract.**

**Course - Travelling fronts and spreading properties for reaction-diffusion equations**

**Period:**May-June 2022

**Timetable:**TBA

**Abstract.**

**Course - **__Classifying spaces, group cohomology and Chow rings__* - (about 20 hours)*

*Roberto Pirisi (Sapienza Università di Roma)*

**First meeting:**Friday 11/2/2022

**Timetable:**Friday 11:00-14:00, Room B, link: meet.google.com/gve-otgc-hoh

**Abstract.**

^{i}(BG) -> H

^{2i}(BG), which is an isomorphism after tensoring by the rational numbers.

**Reading course - Introduction to the theory of schemes**

*Simone Diverio (Sapienza Università di Roma)*

**First meeting:**Wednesday 13 October 2021

**Timetable:**Monday, 14:30 - 16:30, Aula B

**Main reference:**Q. Liu, “Algebraic Geometry and Arithmetic Curves”

**Course - **__Topics on Fano varieties__* - about 20 hours*

*Enrico Fatighenti (Sapienza Università di Roma) *

**First lecture:** Wednesday 27 April 2022

**Timetable:** Wednesday 15:30-17:30, Friday 14:30-16:30.

**Abstract.**

**Course - Quiver representations**

*Giovanni Cerulli Irelli (Sapienza Università di Roma)*

**Time:**mid March till mid May 2022 (about 8 weeks)

**Timetable:**first lesson on Wednesday 30 March, second lesson on Friday 1 April, 10-12, Aula B

**Abstract.**

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**

__Reading courses 2021-22__

**Algebra and geometry**

**- An introduction to the theory of schemes (S. Diverio) - ACTIVE**

**Analysis**

**Probability, mathematical physics and numerical analysis**

More PhD courses:

**PhD courses at SBAI "Sapienza" University of Roma
PhD courses at University of Roma "Tor Vergata"**

PhD courses at University Roma Tre

PhD courses at University of Pisa

PhD courses at University Roma Tre

PhD courses at University of Pisa