New Tools in Chromatic Homotopy Theory

University Of Kentucky Research Foundation, Lexington KY

Investigators

Abstract

Much of modern mathematics is concerned with the intricate interactions between different areas inside of mathematics. One of the primary goals of this project is to use ideas from logic in order to compare topology and algebra. An area of topology called chromatic homotopy theory studies spaces by ``factoring" them into prime parts in the same way that an integer has a prime factorization. Using ideas from logic, we can understand the collection of prime spaces as the prime tends to infinity. It turns out that the resulting collection of spaces can be completely understood using only algebra. This idea is quite new and we hope to develop it into a full theory. This will allow for purely algebraic results to have important topological consequences. Another primary goal of this project is to use topology to build a new bridge between geometry and algebra. This is a familiar story to mathematicians. Classically, certain geometric objects called vector bundles were used to produce an important algebraic invariant of spaces. This algebraic invariant was generalized in the 80's, but in the process of generalization the connection to geometry was somewhat lost. On the other hand, the generalization has a beautiful relationship to an area of algebra called arithmetic geometry. Further developing this relationship with arithmetic geometry should expose part of the geometry that was lost. The PI plans to develop new tools in chromatic homotopy theory that provide both conceptual and computational insight while revealing chromatic homotopy theory as the support for a bridge between geometry and arithmetic geometry. These tools include a geometric construction of Morava E-theory in terms of Stolz--Teichner field theories, a description of the asymptotic behavior of chromatic homotopy theory that introduces Drinfeld elliptic modules into chromatic calculations, and a fusion-system-like combinatorial description of the classifying spaces of finite groups when localized at a chromatic prime leading to the resolution of a conjecture of Ravenel's. These tools are all built on insights gained from the PI's work on and applications of transchromatic homotopy theory. Because of this, the PI will also continue to develop transchromatic homotopy theory. 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.