Offerta formativa programmata 2020/2021


Elenco dei corsi/attività primo anno

titolocrediti
T. Schick - The topology of positive scalar curvature 6
N. Berestycki - Introduction to GFF, multiplicative chaos and Liouville quantum gravity 8
M. Falcone - Optimal control 3
M. Palladino - Reinforcement learning 3
F. Silva - Mean field games 3
A. Faggionato, V. Silvestri, L. Taggi - High dimensional probability 8
R. Bandiera - An introduction to rational homotopy theory 6
M. D'Adderio - Combinatorics of diagonal coinvariants 6
C. Bernardi - Seminari di ricerca in didattica e storia della matematica 8
F. De Marchis, F. Pacella - Nonlinear elliptic equations (reading course) 4
A. Garroni, E. Spadaro - Variational methods in material sciences (reading course) 4
A. Teta, G. Panati, D. Monaco - Mathematical methods in Quantum Mechanics (reading course) 4

Eventuali maggiori informazioni per le voci sopra elencate

Courses 2020/21
 

The list of available seminars is at the following link.


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 (April-May 2021)

 

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 falcone@mat.uniroma1.it
Starting date: 9 March 2021, at 2:30pm
Contact: for more information write to falcone@mat.uniroma1.it

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.
 

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

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)


 

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)
 
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)

 

 


PhD courses at Roma Tor Vergata and Roma Tre:

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


 



Modalità di scelta del soggetto della tesi

I dottorandi sono scelti in base ad una procedura che mette in rilievo i loro interessi scientifici. Quando entrano a far parte del dottorato in Matematica vengono assegnati ad un tutor che li guida nella scelta dei corsi e nella scelta di un direttore di tesi, figura molto importante per i dottorati in Matematica. La scelta della tesi viene fatta insieme al direttore di tesi; spesso è un problema che quest'ultimo propone al dottorando.

Modalità delle verifiche per l'ammissione all'anno successivo

Alla fine del primo anno di dottorato i dottorandi devono aver superato con successo gli esami relativi alle 3 attività formative previste e approvate dal collegio. Inoltre devono aver individuato in maniera chiara un'area di ricerca e un soggetto di tesi.
Verso la metà di ottobre ogni dottorando che abbia completato il primo anno di studi deve affrontare la prova di passaggio d'anno. Tale prova consiste di una dettagliata relazione di 3-6 pagine attorno al problema di ricerca che lo studente intende affrontare nella sua futura tesi di dottorato, e di una presentazione davanti ad una commissione scelta dal collegio, nel quale dovrà dimostrare in particolare di aver acquisito gli strumenti necessari per lo svolgimento della tesi.

Unitamente alle attività formative, il superamento di tale prova è condizione necessaria per l'ammissione all'anno successivo. In alcuni casi dubbi, il collegio può ammettere il dottorando all'anno successivo con riserva e riservarsi di sottoporlo a un'ulteriore verifica dopo alcuni mesi.



Elenco dei corsi/attività secondo anno


Eventuali maggiori informazioni per le voci sopra elencate

Optional courses 2020/21
 

The list of available seminars is at the following link.


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 (April-May 2021)

 

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 falcone@mat.uniroma1.it
Starting date: 9 March 2021, at 2:30pm
Contact: for more information write to falcone@mat.uniroma1.it

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.
 

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

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)


 

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)
 
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)

 

 


PhD courses at Roma Tor Vergata and Roma Tre:

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


 



Modalità di preparazione della tesi

Ricerca individuale, con una forte interazione con il direttore di tesi.

Modalità delle verifiche per l'ammissione all'anno successivo

A metà ottobre il dottorando deve affrontare la prova di passaggio al terzo anno.
Tale prova consiste di una dettagliata relazione scritta sullo suo lavoro di tesi in corso da presentare al collegio dei docenti.
L'ammissione dello studente al terzo anno è subordinata all'approvazione di tale relazione da parte del collegio.
Se ritenuto necessario, allo studente sarà anche richiesto di sostenere un colloquio davanti a una commissione scelta dal collegio.



Elenco dei corsi/attività terzo anno


Eventuali maggiori informazioni per le voci sopra elencate

Optional courses 2020/21
 

The list of available seminars is at the following link.


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 (April-May 2021)

 

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 falcone@mat.uniroma1.it
Starting date: 9 March 2021, at 2:30pm
Contact: for more information write to falcone@mat.uniroma1.it

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.
 

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

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)


 

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)
 
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, D. 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)

 

 


PhD courses at Roma Tor Vergata and Roma Tre:

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


 



Modalità di ammissione all'esame finale

Le modalità sono quelle previste dal regolamento di Ateneo.
Entro il 30 ottobre del terzo anno i dottorandi devono consegnare la versione finale della tesi che viene mandata a due valutatori esperti del campo scelti dal collegio.
Gli studenti che hanno depositato la versione finale della tesi sono anche invitati a presentarne il contenuto in una conferenza pubblica, usualmente all'interno di uno dei seminari permanenti del dipartimento.
Il collegio, avvalendosi anche dei pareri dei due valutatori esterni e tenendo conto dell'attività svolta durante il triennio, dispone l'ammissione all'esame finale.
Gli studenti ammessi all'esame finale possono depositare la versione definitiva della tesi.


Modalità di svolgimento dell'esame finale

Nel mese di ottobre vengono formate le commissioni per gli esami finali degli studenti del terzo anno.
Le commissioni per l'esame finale vengono nominate seguendo il regolamento di Ateneo e scegliendo membri particolarmente esperti negli ambiti di ricerca coperti dalle tesi presentate. Questa scelta garantisce un giudizio molto approfondito del lavoro di tesi e permette ai candidati e ai risultati da essi ottenuti di avere un'importante visibilità all'interno della loro comunità scientifica di riferimento.
La tesi è inviata ai membri della commissione che la leggono e ne verificano il contenuto. Viene quindi fissata una data per l'esame orale finale, il quale è pubblico e viene sempre pubblicizzato nel notiziario dei seminari del nostro dipartimento.

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