Chiara Franceschini

Frédéric van Wijland

Summary: High dimensional data are everywhere, from physical systems to the spreading of epidemics, the world of economic agents or that of neural networks and machine learning. The course aims at stressing the basic techniques underpinning our understanding of the emergence of collective behaviors in very large assemblies of interacting agents.

Gabriela Alexandra Estevez Jacinto

Summary: We study the space of compact operators defined in Hilbert spaces. We saw some examples and the relation of this space with the other spaces such as the space of operators of finite rank, the set of self-adjoint operators, the set of operators with non-empty eigenvalues. We also studied some of the main properties of compact operators such as the characterization of self-adjoint and compact operators (Spectral Theorem) and some applications.

Anna Benini

Summary: Anna Miriam Benini introduced topics such as conformal mappings, Riemann surfaces, Riemann mapping theorem, classification of Riemann surfaces, and meromorphic functions on the Riemann sphere.

Gabriele Benedetti

Charlene Kalle

Summary: A comprehensive review on Banach and Hilbert spaces, covering foundational concepts and structural properties essential to understanding these spaces. In addition, we presented major theorems in functional analysis, including the Hahn-Banach Theorem, Banach-Steinhaus Theorem, and Riesz Representation Theorem, explaining their implications and applications. All lectures were accompanied by tutorial sessions, a homework was given to be sent by email by students.
 

Peter Stevenhagen

Summary: This is part of the regular International Master in Mathematics program.
 

Krerley Oliveira

Summary:  I covered the section of the course related to methods for computing and estimating eigenvalues and eigenvectors. My focus was on the foundational aspects of this theory, including classic theorems and methodologies. Several examples were thoroughly discussed with the students. I placed particular emphasis on class discussions and practical exercises, which included hands-on activities and a set of homework assignments.
 

Etienne Pardoux

Summary:  I will treat conditional expectation, uniform integrability tightness, and convergence in law. The exact content will depend upon how reactive the students are. I will give them exercises along the way.

 

Frank Neumann