Bordism of symmetries: From global groups to derived orbifolds

Introduction

Bordism theory connects many central areas of mathematics. Defined via smooth manifolds, it has proved to have deep connections to formal groups from algebraic geometry and stable homotopy theory from algebraic topology. In particular, complex bordism serves as an effective organizational tool for studying the stable homotopy category, splitting up the homotopy groups of spheres into components of different wavelengths.

Project Goal

The goal of BorSym is to develop a similarly powerful correspondence that classifies the symmetry of spaces through novel connections to equivariant algebraic geometry and smooth actions.

Recent Developments

While such a relationship for symmetries has long been conjectured, it was only recently that foundational results have been established for the actions of abelian groups. Key achievements include:

1. The tensor-triangular classification of actions on finite complexes.
2. My proof of the equivariant Quillen theorem.

At the same time, bordism of symmetries has become a central object of interest also in the seemingly unrelated field of symplectic topology. Groundbreaking work of Abouzaid and collaborators has identified derived orbifold bordism as the universal target for Floer homology, thereby solving long-standing open problems in the field.

Emerging Picture

In view of these events, a new picture is emerging which paints a universal role of equivariant bordism, going substantially beyond even its classical role described above.

Objectives

The goal of BorSym is to unravel the full potential of this emerging picture:

• To thoroughly develop a chromatic homotopy theory of group actions.
• To employ the newly established set of tools to tackle major open problems in transformation groups and obtain new information about the derived orbifold bordism ring.
• To extend the connections between formal groups, thick subcategories, and bordism rings beyond the abelian case.