Geometric Phenomena in Homotopy Theory

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

DMS-0072300 Igor Kriz The aim of the present project is to study certain connections between mathematics and mathematical physics. The area of mathematics concerned is Algebraic topology, which employs methods of algebra to investigate properties of topological spaces (shapes) which remain unchanged (invariant) under continuous deformation. Among such properties or invariants, a distinguished role is played by certain invariants of linear nature, known as generalized cohomology theories. The main goal of this project is studying interpretations and interactions of such cohomology theories with certain areas of physics, geometry and algebraic geometry (spaces of solutions of algebraic equations). Concretely, the investigator focuses on two areas. The first object is finding a geometric (or physical) model of elliptic cohomology. It is conjectured that elliptic cohomology can be recovered by infinite loop space theory from a category of holomorphic conformal field theories. The other area concerns the study of Morava K-theories in a homotopy category of algebraic varieties introduced by Voevodsky.