Grants and Contracts per year
Grants and Contracts Details
Description
My recent work is focused on a few open problems in chromatic homotopy theory. First, with Barthel,
Schlank, and Weinstein, I am working on calculating the homotopy groups of the rationalization of the
K(n)local sphere (for all heights and primes). We are making use of the formality machinery created in
``Chromatic homotopy theory is asymptotically algebraic" as well as Scholze and Weinstein''s version of
Falting''s theorem. The goal is to transport the calculation based on the DevinatzHopkins spectral
sequence through Falting''s equivalence to transform it into a calculation about the cohomology of the
general linear group over the padic integers. Second, Barthel and BerwickEvans and I are using our
formulas for power operations on field theories in order to understand the multiplicative structure of
equivariant orientations of complexified topological modular forms. Third, Reeh and Schlank and I are
working on an old question of Ravenel''s that gives a conjectural description of the kernel of the
canonical map from the Burnside ring of a finite group to the K(n)local cohomotopy of the classifying
space. We have made significant progress on this problem and are using it to inform our search for K(n)
local fusion systems. Fourth, with Guillou and several collaborators, I have calculated the homotopy
groups of the KU_Glocal equivariant sphere spectrum for G an odd pgroup. Finally, my students and I
are working on understanding the relationship between multiplicative and additive power operations,
significantly generalizing the exponential relationship between symmetric powers and Hecke operators
described by Ganter. We have produced the kind of exponential relationship that we want and have
discovered that this points in the direction of higher height exterior powers. It will be exciting to
understand these more thoroughly.
Status  Active 

Effective start/end date  9/1/23 → 8/31/28 
Funding
 Simons Foundation: $42,000.00
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Projects
 1 Active

Rational and Equivariant Phenomena in Chromatic Homotopy Theory: Departmental Funds
9/1/23 → 8/31/28
Project: Research project