The course is devoted to coarse geometry. The main topic are coarse mappings (in particular quasiisometrics) and invariants of spaces equipped with a coarse structure, first of metric spaces.

In the first half of the course we will define coarse geometry as an approach to the study of spaces from a "large-scale" perspective, in which two spaces that look the same from a large distance are actually equivalent. The plan is to show that many geometric properties of metric spaces are determined by their large-scale coarse structure.

In the middle of the course, we will study some of the coarse invariants of spaces and the useful constructions associated with them. Namely, the ends of metric space and the resulting Freudenthal compactification will be studied.

At the end of the course it is planned to give definitions of notions of growth and amenability of rough spaces. This will allow one to consider the apparatus of rough homological algebra. A coarse version of the fixed point theorem will also be considered.

The course is designed for undergraduate and graduate students.

Prerequisites: group theory, algebraic topology, and functional analysis. Category theory is also desirable.

Lecturer
Arutyunov Andronick Aramovich

Financial support
The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, Agreement no. 075-15-2022-265).

Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center