Power operations in the Stolz--Teichner program

The Stolz--Teichner program proposes a deep connection between geometric field theories and certain cohomology theories. In this paper, we extend this connection by developing a theory of geometric power operations for geometric field theories restricted to closed bordisms. These operations satisfy relations analogous to the ones exhibited by their homotopical counterparts. We also provide computational tools to identify the geometrically defined operations with the usual power operations on complexified equivariant $K$-theory. Further, we use the geometric approach to construct power operations for complexified equivariant elliptic cohomology.

Transfer ideals and torsion in the Morava $E$-theory of abelian groups

Let $A$ be a finite abelian $p$ group of rank at least $2$. We show that $E^0(BA)/I_{tr}$, the quotient of the Morava $E$-cohomology of $A$ by the ideal generated by the image of the transfers along all proper subgroups, contains $p$-torsion. The proof makes use of transchromatic character theory.

Monochromatic homotopy theory is asymptotically algebraic

In previous work, we used an $infty$-categorical version of ultraproducts to show that, for a fixed height $n$, the symmetric monoidal $infty$-categories of $E_{n,p}$-local spectra are asymptotically algebraic in the prime $p$. In this paper, we prove the analogous result for the symmetric monoidal $infty$-categories of $K_{p}(n)$-local spectra, where $K_{p}(n)$ is Morava $K$-theory at height $n$ and the prime $p$. This requires $infty$-categorical tools suitable for working with compactly generated symmetric monoidal $infty$-categories with non-compact unit. The equivalences that we produce here are compatible with the equivalences for the $E_{n,p}$-local $infty$-categories.

On the ring of cooperations for 2-primary connective topological modular forms

We analyze the ring tmf_*tmf of cooperations for the connective spectrum of topological modular forms (at the prime 2) through a variety of perspectives: (1) the E_2-term of the Adams spectral sequence for tmf ^ tmf admits a decomposition in terms of Ext groups for bo-Brown-Gitler modules, (2) the image of tmf_*tmf in the rationalization of TMF_*TMF admits a description in terms of 2-variable modular forms, and (3) modulo v_2-torsion, tmf_*tmf injects into a certain product of copies of TMF_0(N)_*, for various values of N. We explain how these different perspectives are related, and leverage these relationships to give complete information on tmf_*tmf in low degrees. We reprove a result of Davis-Mahowald-Rezk, that a piece of tmf ^ tmf gives a connective cover of TMF_0(3), and show that another piece gives a connective cover of TMF_0(5). To help motivate our methods, we also review the existing work on bo_*bo, the ring of cooperations for (2-primary) connective K-theory, and in the process give some new perspectives on this classical subject matter.

Centralizers in good groups are good

We modify the transchromatic character maps to land in a faithfully flat extension of Morava E-theory. Our construction makes use of the interaction between topological and algebraic localization and completion. As an application we prove that centralizers of tuples of commuting prime-power order elements in good groups are good and we compute a new example.