Rational and Equivariant Phenomena in Chromatic Homotopy Theory

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 Devinatz--Hopkins spectral sequence through Falting''s equivalence to transform it into a calculation about the cohomology of the general linear group over the p-adic integers. Second, Barthel and Berwick-Evans 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_G-local equivariant sphere spectrum for G an odd p-group. 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.