Course - Hodge theory and tame geometry
Prof. Yohan Brunebarbe (CNRS, Bordeaux)
Timetable: 22 and 27 June, 4-6 July (14:00 - 16:00 Sala di Consiglio) ; 8 July ( 10:30-12:30 Sala di Consiglio) ,11July 10:30-12:30 Sala di Consiglio; 13 July 14:00-16:00 Sala di Consiglio.
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Course: 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) classroom 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.
[BH12] H. Berestycki, F. Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math. 65 (2012), 592-648.
[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
[FM77] P.C. Fife, J.B. McLeod, The approach of solutions of non-linear diffusion equations to traveling front solutions, Arch. Ration. Mech. Anal. 65 (1977), 335-361.
[F37] R. A. Fisher, The advance of advantageous genes, Ann. Eugenics 7 (1937), 335-369.
[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.
[KPP37] A. N. Kolmogorov, I. G. Petrovsky, N. S. Piskunov, Etude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bjul. Moskowskogo Gos. Univ. A 1 (1937), 1-26.
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Course - Origini e sviluppi del calcolo differenziale assoluto
Prof. Alberto Cogliati, Università di Pisa
Timetable: 12 - 23 September 2022
Abstract :
- Introduzione alle Disquisitiones circa superficies curvas (1827) di Gauss: l’introduzione della
nozione di curvatura e l’approccio intrinseco alla teoria delle superfici. Il tema della
classificazione delle superfici a curvatura costante: l’apporto di Minding.
- L’Habilitationsvortrag (1854) di Riemann: il concetto di varietà n-estesa, di metrica
riemanniana e la generalizzazione della nozione di curvatura. La Commentatio
mathematica (1861) e l’introduzione del “tensore” di curvatura.
- Il problema dell’equivalenza nei lavori di Christoffel e Lipschitz (1869). Differenti posizioni
storiografiche intorno alla nascita del concetto di tensore.
- Il potere sistematizzatore del calcolo differenziale assoluto: origini, sviluppi e ricezione
dell’opera di Ricci Curbastro
- La geometrizzazione della nozione di derivata covariante: Levi-Civita e la nascita della
teoria delle connessioni.
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Course - Symplectic Geometry
Siye Wu
Timetable: TBA
Abstract:
Symplectic geometry is a branch of differential geometry that studies symplectic manifolds,
which are smooth manifolds equipped with closed non-degenerate two-forms. The course
begins with basic concepts such as Hamiltonian vector fields, Poisson brackets, Lagrangian
submanifolds and the Darboux theorem. Examples include cotangent bundles, K ̈ahler man-
ifolds, coadjoint orbits and fibrations of Lagrangian tori. Comparisons will be made with
contact and Poisson manifolds. The second part is about symmetries of symplectic man-
ifolds. Important notions to be introduced are Hamiltonian group actions, moment maps
and their images, and symplectic quotients. Interesting examples are toric manifolds and
moduli space of flat connections on surfaces. The last part of the course is to be on the
applications of symplectic geometry to classical mechanics (Lagrangian and Hamiltonian
mechanics), solving problems on the motion of rigid bodies and integrable systems.
The course is suitable for students who have already taken an introductory course on man-
ifolds (with calculus of di↵erential forms) and who wish to engage their knowledge in a
constructive and useful setting.
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Reading course
Guido Pezzini (Classical and Geometric Invariant Theory)
Timetable: For days and times please contact Professor Guido Pezzini directly.
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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:
In this course we will consider the problem of constructing optimal locally recoverable codes using number theoretical techniques.
The first part of the course includes an introduction to the theory of locally recoverable codes and to the theory of global function fields (with particular emphasis on Galois extensions of global function fields).
The second part of the course includes an effective method to construct new locally recoverable codes using Galois theory.
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.
Pattern formation is ubiquitous in nature. At micro- and mesoscopic scale physical/chemical systems self-assemble into regular structures, characterized by some periodic alternation of different phases. Among them, the most typical patterns consist of bubbles placed on hexagonal lattices or stripes/lamellae (i.e. one-dimensional structures). Such structures are universally believed to arise from the competition between short range attractive forces (favouring pure phases) and long range repulsive forces (favouring alternation between different phases). Although such a phenomenon is observed in experiments and reproduced by numerical simulations, its mathematical understanding is in most cases a challenging and long-standing open problem. In the zero-temperature approximation, the models are of variational type and the physical states are represented by the minimizers of a free energy functional. The main difficulties from the mathematical point of view lie in the symmetry breaking phenomenon (namely the fact in dimensions larger than 1 that the expected minimizers have less symmetries than the interactions contributing to the energy) and in the nonlocality of the interactions.
The aim of this course is to present a new set of ideas/techniques which have been recently developed by the authors and collaborators and which allowed to give a rigorous proof of symmetry breaking and pattern formation in the form of one-dimensional structures in general dimensions for a large class of functionals. In order to explain such results we will introduce some preliminary notions of Geometric Measure Theory, Calculus of Variations and Reflection Positivity (the latter being the main tool to show periodicity of minimizers in one space dimension).
[DKR19] Sara Daneri, Alicja Kerschbaum, and Eris Runa. “One-dimensionality of the minimizers for a diffuse interface generalized antiferromagnetic model in general dimension”, arXiv preprint arXiv:1907.06419 (2019).
[DR19] Sara Daneri and Eris Runa. “Exact periodic stripes for minimizers of a local/nonlocal interaction functional in general dimension”, Archive for Rational Mechanics and Analysis 231.1 (2019), pp. 519–589.
[DR20] Sara Daneri and Eris Runa. “Pattern Formation for a Local/nonlocal Interaction Functional Arising in Colloidal Systems”, SIAM Journal on Mathematical Analysis 52.3 (2020), pp. 2531–2560.
[DR21a] Sara Daneri and Eris Runa. “Exact periodic stripes for a local/nonlocal minimization problem with volume constraint”, arXiv:2106.08135 (2021).
[DR21b] Sara Daneri and Eris Runa. “One-dimensionality of the minimizers in the large volume limit for a diffuse interface attractive/repulsive model in general dimension”, arXiv:2104.064191676 (2021).
[GR19] Michael Goldman and Eris Runa. “On the Optimality of Stripes in a Variational Model With Non-Local Interactions”, Calculus of Variations and Partial Differential Equations 58.103 (2019).
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.
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
Timetable: TBA
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. 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.
Roberto Pirisi (Sapienza Università di Roma)
First meeting: Friday 11/2/2022
Abstract.
Group cohomology draws a powerful connection between algebra and geometry: a group G naturally determines a topological space BG, the classifying space of G, and we can define the cohomology ring of G as the cohomology of the space BG. The challenge is then to determine how the properties of G are connected to those of the cohomology ring H*(BG).
On the algebro-geometric side, one can construct a different classifying object BG, which can be seen either as an algebraic stack or as a limit of algebraic varieties. A natural analog of the singular cohomology ring in this setting is given by the Chow ring CH*(BG). Over the complex numbers the two are tied by the cycle map CHi(BG) -> H2i(BG), which is an isomorphism after tensoring by the rational numbers.
In this short course we will introduce these two invariants, carry over some basic computations and prove the comparison theorem between the two. Time allowing, we will delve into deeper questions on the depth and regularity of these rings.
The course is based on the book "Group cohomology and algebraic cycles", by Burt Totaro.
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
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 - 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.
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: 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
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)
Analysis
- 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 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