EXC 2044 - T01: K-Groups and cohomology

Basic data for this project

Description

K-groups and cohomology groups are important invariants in different areas of mathematics, from arithmetic geometry to algebraic and geometric topology to operator algebras. The idea is to associate algebraic invariants to geometric objects, for example to schemes or stacks, C*-algebras, stable ∞-categories or topological spaces. Originating as tools to differentiate topological spaces, these groups have since been generalised to address complex questions in different areas. Cohomology theories such as étale, crystalline and prismatic are crucial in the structural study of schemes and the Langlands programme, while K-theory provides insights into algebraic structures and intersection theory but is also used in algebraic and geometric topology through applications in the study of manifolds and the classification of nuclear C*-algebras.