New Tools in Chromatic Homotopy Theory

Grants and Contracts Details


Transchromatic homotopy theory is concerned with the relation between the chromatic layers of the stable homotopy category. After developing tools in transchromatic homotopy theory between 2010 and 2013, I began looking for interesting applications. During the last five years, I have found several applications together with several collaborators. The highlight is a recent result on the Balmer spectrum of the equivariant homotopy category with Barthel, Hausmann, Naumann, and Noel that was accepted for publication in Inventiones Mathematicae. Chromatic homotopy theory studies the stable homotopy category by localizing at "chromatic primes" that depend on the choice of a prime p and a natural number n. Recently, Barthel, Schlank, and I applied some ideas from mathematical logic to study the asymptotic behavior of these localized categories when the natural number n is fixed and as the prime goes to infinity. We have already produced an interesting application to local Smith-Toda complexes. We hope to extend our results in order to connect stable homotopy theory to the arithmetic of function fields, a whole area that has historically seen little application to stable homotopy theory. The Stolz-Teichner program aims to relate the geometry of certain n-dimensional field theories to chromatic homotopy theory. Schommer-Pries and I have provided an algebraic notion of a field theory that we hope will provide a place where field theories and chromatic homotopy theory can interact. Barthel, Berwick-Evans, and I have been working on the multiplicative part of this relation. We have shown that the power operation applied to a field theory has a close connection to certain calculations in chromatic homotopy theory due to Barthel and I. Reeh, Schlank, and I have been pursuing a conjecture of Ravenel's regarding the relationship between the Burnside ring of a finite group and chromatic homotopy theory. We have a program for proving this conjecture that is in progress. By understanding the relationship between fusion systems and chromatic homotopy theory we hope to uncover the notion of a "height n" fusion system and use it to produce a "height n" Burnside ring. We see a path to proving Ravenel's conjecture with such a ring in hand.
Effective start/end date9/1/199/1/19


  • Simons Foundation


Explore the research topics touched on by this project. These labels are generated based on the underlying awards/grants. Together they form a unique fingerprint.