Rational and Equivariant Phenomena in Chromatic Homotopy Theory (ALN 47.083)

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 Berwick-Evans. 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 well-behaved with respect to the power operations on complexified equivariant elliptic cohomology developed with Barthel and Berwick-Evans. 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 one-day online workshops on the topic during the pan- demic. The conferences brought together experts in the field as well as many young researchers. The one-day 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.