Applications of homotopy theory to algebraic geometry and physics

Harvard University, Cambridge MA

Investigators

Abstract

The field of homotopy theory is the study of mathematical structures that are insensitive to deformations. It is applicable whenever one is interested in studying qualitative aspects of a system, emergent properties of a statistical ensemble, or whenever there might be imprecision in the specification of the state of a system. In recent years the methods of homotopy theory have found use in fields as diverse as the physics of materials and the study of algebraic equations. This project aims to bolster these relationships with new tools from homotopy theory, and is focused on applications to classical algebraic geometry and the theory of manifolds. Broader impacts of this project include work with students and postdoctoral researchers. In more specialized terms, the work in this proposal involves two main themes of study. One describes advances in motivic homotopy theory that achieve the goals of a longstanding vision concerning an interface between complex analysis, algebraic geometry and algebraic topology. It exhibits a very close alignment between the topological construction of vector bundles over smooth algebraic varieties and the much more challenging construction of algebraic ones. Another aspect of motivic homotopy theory studied is the structure of cohomology rings of Eilenberg-MacLane spaces in that setting. The other theme offers a new approach to the "Immersion Conjecture," which is one of the most famous theorems about general manifolds, and promises to simplify and clarify the notoriously complicated existing proof. Part of this line on the Immersion Conjecture involves spaces which are potentially models for spaces BO/I_n constructed by Brown and Peterson that are understood only weakly. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.