Exploring equivariant homotopy theory

Samenvatting

Mathematicians have always been intrigued by shapes and geometric objects. One of the recent themes in algebraic topology aims at capturing geometry of objects with symmetries by finding simpler models for them. This proposed project, which consists of two parts, uses new developments of model categories by Quillen together with good models for stable homotopy theory and motivic homotopy theory to build our understanding of spaces with symmetries in algebraic topology and in algebraic geometry.

In the first part I will study algebraic invariants of spaces with symmetries called rational G-cohomology theories. By extracting essential structural information from the category of rational G-cohomology theories we will be able to provide a much easier, algebraic description of it. The ultimate aim is to do it for any compact Lie group G. This part of the project will add to our understanding of spaces with symmetries, their invariants and possibly result in constructing new invariants starting from purely algebraic data, as it was done by Greenlees in the case of the circle group action.

The second part of the project aims at understanding homotopic Galois and Hopf-Galois theory of objects with finite group action (a generalisation of symmetries) in algebraic geometry. The aim of this project is to provide a good model-categorical framework for motivic homotopic (Hopf-)Galois extensions and work out interesting examples (like KO to KU in standard homotopy theory). This will help understand complicated objects with group actions by linking them to simpler ones without group action via descent theory.

Both parts of this research project will use new methods developed in my thesis and my recent work with collaborators on induced model structures and abstract foundations for Galois extensions in a general model category.

Kenmerken

Projectnummer

639.031.757

Hoofdaanvrager

Dr. M. Kedziorek

Verbonden aan

École Polytechnique Fédérale de Lausanne

Looptijd

01/09/2017 tot 31/08/2020