Matroidi e spazi di configurazioni
Luca Moci – Luca Migliorini – Federico Ardila – Emanuele Delucchi
30 h
Some nonlocal problems in PDEs and Geometric Measure Theory
Eleonora Cinti
16 h
ANNULLATO -> Introduction to free boundary problems <- ANNULLATO
Giorgio Tortone
16 h
Gruppi topologici e misura di Haar
Alessio Savini
30 h - february - march 2020
- Spazi omogenei e misure indotte
- Gruppi di Lie (definizioni, esempi, algebra di Lie associata, mappa esponenziale)
- Cenni di geometria Riemanniana e spazi simmetrici
- Reticoli in gruppi di Lie
Probabilistic methods in Complex and Real Analysis
Alexander Volberg
16 h
April 2020
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.
Optimal transportation and applications
Cristian Gutierrez
16 h
May 2020
Optimal mass transportation concerns the optimal allocation of resources. For example, allocation of persons to jobs, shipping goods from warehouses to shops, transforming one image into another. In each of these cases, a function is given representing the cost of mapping or transporting a unit of one item into another item. A question is then to find a way to allocate all resources simultaneously so that the total cost is minimum. The question originates with the work of Monge in the 18th century to solve a military problem (linear cost) and was dormant until 1940 when Kantorovich, motivated from problems in economics, discovered a probabilis- tic formulation. In addition and as a very important part in the development of these ideas, linear programming (LP) was invented during WWII to solve planning problems in wartime operations. In the postwar period, many industries found these developments very useful for optimizing their businesses, and these tools are continued to be used until today. Major players and founders in LP include G. Dantzig, who discovered the simplex method, John von Neumann who discovered the duality theory, and T. Koopmans for the applications to economics. In 1975, Kantorovich and Koopmans were awarded the Nobel Prize in Economics for their contributions to the optimal allocation of resources. The subject has flourished in the last three decades having mathematical connections with convex analysis, optimization, probability, and pdes; and it was found to have applications to fields such as optics, image processing, and machine learning.
The purpose of this course is to develop some of this theory and show examples of applications. Topics include:
(1) Monge and Kantorovich problems; existence of optimal maps/plans; use of convex anal- ysis; dual problems.
(2) Introduction to Monge-Ampe`re equations in this context.
(3) Brenier’s polar factorization theorem.
(4) Monge-Kantorovich distance.
(5) Applications to geometric optics.
The proposed length of the course is four weeks between May 14th and June 5th, 2020. The course will be of interest for students in analysis, probability, applied mathematics and economics. Prerequisites are knowledge of real analysis, basic pdes, abstract measure theory and basic functional analysis.
Parabolic Geometry
Katharina Neusser
8 h
May 2020
This mini-course will give an introduction to parabolic geometries, which provide a uniform approach to a large variety of differential manifolds infinitesimally modelled on flag manifolds.
The most prominent examples of geometric structures admitting descriptions as parabolic geometries are conformal manifolds (dim>2), projective structures, almost quaternionic manifolds, and certain types of CR-structures. We shall also give some applications of Cartan connections to symmetries and to the construction of invariant differential operators for parabolic geometries.
Topics in Mathematics 2019/2020
organizzatori: Carolina Beccari e Luca Moci
24 h