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Description
Project Summary
Overview
The PI plans to develop and apply 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. The tools that will be applied include a
powerful test for formality of a cosimplicial E∞ring, developed with Barthel and Schlank, a new
relationship between free loop spaces and transfer maps, developed with Reeh and Schlank, and an
explicit formula, on the level of cocycles, for the power operation for complexified equivariant elliptic
cohomology, developed with Barthel and BerwickEvans. The PI plans to strengthen these tools
while applying them to important problems linking chromatic homotopy theory with arithmetic
geometry, equivariant homotopy theory, and differential geometry.
Intellectual Merit
This project applies newly developed tools in chromatic homotopy theory to address several im
portant open problems regarding the relationship between chromatic homotopy theory and other
mathematical fields. With Barthel and Schlank, the PI will work on an old question of Morava’s.
We will attempt to calculate the homotopy groups of the rationalization of the K(n)local sphere by
applying a novel test for formality to show that the calculation is equivalent to a classical calcula
tion of the continuous cohomology of Gln(Zp) with coefficients in Qp. Secondly, the PI, with Reeh,
and Schlank, will continue the search for the correct notion of a K(n)local fusion system. With
that in hand, we hope to resolve a conjecture of Ravenel’s regarding the kernel of the canonical
map from the Burnside ring of a finite group to the zeroth K(n)local cohomotopy of the classifying
space of the group. Finally, the PI plans to show that the equivariant Witten genus is wellbehaved
with respect to the power operations on complexified equivariant elliptic cohomology developed
with Barthel and BerwickEvans.
Broader Impacts
Throughout his career, the PI has made an effort to mentor graduate students. The PI now has
three graduate students, all of whom are in underrepresented groups in mathematics. With the
help of an NSF grant, the PI plans to support his graduate students.
The PI has worked to increase the awareness of transchromatic homotopy theory by organizing,
with Barthel, Heard, and Naumann, two oneday online workshops on the topic during the pan
demic. The conferences brought together experts in the field as well as many young researchers.
The oneday workshops specifically highlighted the work of young researchers in order to help them
advertise their results and get jobs during the pandemic. The PI is organizing a week long confer
ence on transchromatic homotopy theory in 2023. The PI also plans to work on expository writing
in order to make the field more accessible.
At the University of Kentucky, the PI is one of the founding mentors of the undergraduate
geometry and algebra lab. The lab provides undergraduates with the opportunity to work closely
with faculty on computational research problems. It increases the visibility of the mathematics
department and has had a positive impact on the number of math majors. In the past, the PI’s
lab group wrote Sage software to calculate with formal power series and formal group laws over
large rings. An NSF grant will allow the PI to continue working in the lab and will support one
undergraduate summer research position.
Status  Active 

Effective start/end date  8/15/23 → 7/31/26 
Funding
 National Science Foundation
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Projects
 1 Active

Rational and Equivariant Phenomena in Chromatic Homotopy Theory (ALN 47.049)
8/15/23 → 7/31/26
Project: Research project