Cursos

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Santander

Lugar de impartición

Santander - Península de la Magdalena

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COLABORACIÓN

ORGANIZADO EN COLABORACIÓN CON

Descripción de la actividad

The theory of operator algebras is a hybrid field initiated in the 1930s by John von Neumann motivated by profound questions in ergodic theory, group representations, and quantum mechanics. It is remarkable that these motivations not only came to spectacular fruition but also continue to provide inspiration for present day research in these areas.

The school will focus on new trends in the structure and approximation theory of C*-algebras. These topics will be presented in four courses. Additionally, there will be contributed talks by participants in the afternoon. The school aims to train national and international Ph.D. students, as well as young researchers with interdisciplinary interests in mathematics.

Course A:            Group stability and approximation problems in C*-algebrasThe theory of operator algebras was developed since its birth in close relationship with the theory of group representations on Hilbert space. One important question here refers to  stability, i.e. when are approximate representations of a group close to actual representations, where closedness is measured by a chosen metric (e.g., the Hilbert-Schmidt distance, the operator norm, etc.). In recent years, there has been a surge of interest in group stability. This is motivated in part by the fact that group stability provides a new angle to attack approximation conjectures for groups. Operator algebraic methods come up naturally in the study of stability for groups.  Moreover, aspects of stability appeared in C*-algebra theory long ago and were proved to be very useful in the study of structure properties of C*-algebras.

Speaker: Tatiana Shulman

Course B:            Model theory and C*-algebrasContinuous model theory, C*-algebras, ultrapowers, EF-games.

This course will delve into the fascinating applications to operator algebras coming from logic and model theory, both of which offer powerful tools for understanding the structure and behavior of mathematical theories, in particular the theory of C*-algebras. The course will address continuous model theory, ultrapowers and EF-games.

Speaker: Andrea Vaccaro (University of Münster, Germany) . Website: https://sites.google.com/view/avaccaro/home

Course C:  Introduction to Cuntz semigroupsThe Cuntz semigroup is a geometric refinement of K-theory that conveniently encodes the comparison theory of operators in a C*-algebra in a partially ordered semigroup. Several of the recent breakthroughs in the structure and classification theory of C*-algebras rely crucially on deep new insights into the fine structure of Cuntz semigroups, and some of the main existing challenges in C*-algebras (such as the famous Toms-Winter conjecture) are intimately connected to Cuntz semigroups.

After an introduction to the modern theory of Cuntz semigroups, based on the fundamental paper by Coward-Elliott-Ivanescu from 2008, this course will cover the recent exciting discoveries about the structure of Cuntz semigroups of stable rank one C*-algebras.

Speaker: Leonel Robert (University of Louisiana at Lafayette). Website: https://userweb.ucs.louisiana.edu/~C00254569/

Course D: Introduction to separated graph C*-algebras.Graph C*-algebras provide models for large families of C*-algebras. In addition, the invariants of these algebras can be computed in terms of the combinatorial information provided by the graphs. In this course, I will cover the basic ingredients of the theory of graph C*-algebras, and then I will also consider some interesting generalizations, including the C*-algebras associated to separated graphs. 

Speaker: Pere Ara (UAB and CRM). Website: https://scholar.google.com/citations?user=7qSDFUMAAAAJ&hl=en