A.Y. 2020/2021

Spazi di Hilbert di funzioni intere e processi puntuali 

(autunno 2020) 24 ore

Prerequisiti: Spazi di Hilbert, teoria della misura. 

1. Spazi di Hilbert di funzioni intere. Nuclei riproducenti. 

2. Spazio di Paley-Wiener. Nucleo di Bessel. Nucleo di Airy. 

3. Proiettori spettrali degli operatori differenziali di 2 grado. 

4. Spazi di De Branges. 

5. Spazi di Bergman e di Fock. 

6. Processi puntuali. Funzioni di correlazione. 

7. Processi puntuali determinantali. 

8. Teoria di Palm-Khintchine. 

9. Teoremi di convergenza debole. 

10. Quasi-simmetrie: l’analogo del teorema di De Finetti.  

11. Completezza e minimalità.

Introduction to free boundary problems 

Programma: In questa serie di lezioni discuterem in merito a diversi problemi di frontiera libera motivati da diverse applicazioni, tra cui il problema di Bernoulli, il problema dell’ostacolo and il problema a due fasi. 

In particolare, lo scopo principale del corso sarà discutere delle tecniche comunemente utilizzate nel con- testo di problemi a frontiera libera e delle loro applicazioni. In ogni problema trattato, cercheremo di focalizzarci sui principali aspetti della teoria di regolarità della frontiera: 

1. Regolarità ottimale delle soluzioni. 

2. Analisi blow-up della frontiera libera.

3. Regolarità C1,α della frontiera libera. 

Inoltre, verranno presentati alcuni degli aspetti e delle tecniche principali, tra cui 

1. Formulazioni viscose di condizioni sulla frontiera libera. 

2. Formule di monotonia (Alt-Caffarelli-Friedman, Almgren, Weiss e Monneau formula).

3. Improvement of flatness. 

Tempo permettendo, verranno illustrate alcune possibili linee di ricerca e problemi aperti.

Probabilistic methods in Complex and Real Analysis 

We describe the basics of applications of stochastic methods to Complex and Real analysis. 

It turns out that many problems in Complex and Real analysis can be approached from the point of view of stochastic optimal control. This allows us to reduce infinite dimensional problem to finite dimensional (but non-linear) one. 

Many recent achievements in Complex and Real analysis were obtained by this point of view. 

It has an advantage that the path from basic knowledge to real problems is rather short, and at the end of the course the students can try their skills on a real problem. 

One does not need to know Brownian motion theory or anything advanced from Probability, all we need will be covered during the course. 

The basic knowledge of notions from Probability theory is needed: expectation, variance, as well as basic knowledge of what is harmonic and analytic functions.

- representation theory of the symmetric group 

- symmetric function theory 

- plethystic operators 

- diagonal coinvariants 

Introduction to Cartan Connections and Applications

16, 18, 21 and 23 June h 11-13

in presence Aula Seminario VIII piano and On-line: https://unibo.zoom.us/j/3730574442

In this mini-course we will give an introduction to Cartan geometries, which provide a uniform approach to a large variety of differential geometric structures.

We will focus on parabolic geometries which are Cartan geometries infinitesimally modelled on flag varieties. Among the most prominent examples of geometric structures admitting descriptions as parabolic geometries are conformal manifolds (dim>2), projective structures, non-degenerate CR-structures of hypersurface type and various types of bracket-generating distributions.

After having introduced the basic concepts and having studied some examples, we will discuss some applications of Cartan connections to classical problems in differential geometry.

On the one hand, we will see how Cartan connections can be applied to questions of geometric rigidity such as, which Lie groups can act on manifolds preserving a given geometric structure or to which  extent does the group of automorphisms determine the geometric structure.

On the other hand,we will study applications of Cartan connections to compactifications of geometric structures.

Representation theory

Topology of manifolds, varieties, and singular spaces 

(Primavera 2021)

AN INTRODUCTION TO FREE BOUNDARY PROBLEMS

May 24 - June 4 2021

(6 + 6)

Course starts on May 24 at 4 PM.

https://columbiauniversity.zoom.us/j/92676772971?pwd=RWdxOU9vS082THQ1M1NRaHFaMDRrdz09

Tematiche: geometria algebrica e combinatorica con applicazioni in teoria della rappresentazione. Geometria e differenziale e algebrica. 

(Ulteriori dettagli saranno resi disponibili in seguito)