This project analyses and builds algebraic invariants of geometric objects by simultaneously implementing multiplicative, equivariant, and twisted structures.
The mathematical discipline of algebraic topology studies geometric objects by assigning algebraic invariants to them. One particularly powerful type of these invariants is known as (generalized) cohomology theories, which are widely used in algebraic topology and many other areas of pure mathematics. The aim of this project is to develop cohomology theories that possess multiplicative, equivariant, and twisted properties. This means that these cohomology theories have an additional multiplicative structure (generalizing the multiplication of integers), take symmetries of geometric objects into account (like rotation or flip symmetries), and can be deformed by varying a parameter.
Beyond algebraic topology, the mathematical tools and envisioned results of this project are relevant to fields such as mathematical physics, operator algebras, representation theory, and higher category theory.
