Nathaniel Stapleton

We compute the algebraic Picard group of the category of $K(n)$-local spectra, for all heights $n$ and all primes $p$. In particular, we show that it is always finitely generated over $\mathbb{Z}_p$ and, whenever $n \geq 2$, is of rank $2$,…

We compute the rational homotopy groups of the $K(n)$-local sphere for all heights $n$ and all primes $p$, verifying a prediction that goes back to the pioneering work of Morava in the early 1970s. More precisely, we show that the inclusion…

We prove that the image of the total power operation for Burnside rings $A(G) \to A(G\wr\Sigma_n)$ lies inside a relatively small, combinatorial subring $\mathring A(G,n) \subseteq A(G \wr \Sigma_n)$. As $n$ varies, the subrings $\mathring…

For a finite abelian group $A$, we determine the Balmer spectrum of $\mathrm{Sp}_A^{\omega}$, the compact objects in genuine $A$-spectra. This generalizes the case $A=\mathbb{Z}/p\mathbb{Z}$ due to Balmer and Sanders \cite{Balmer-Sanders},…

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…

The Segal conjecture describes stable maps between classifying spaces in terms of (virtual) bisets for the finite groups in question. Along these lines, we give an algebraic formula for the p-completion functor applied to stable maps…

In our previous paper, we constructed and studied a functorial extension of the evaluation map $S^1 \times \mathcal{L}X \to X$ to transfers along finite covers. In this paper, we show that this induces a natural evaluation map on the full…

We construct and study a functorial extension of the evaluation map $S^1 \times \mathcal{L} X \to X$ to transfers along finite covers. For finite covers of classifying spaces of finite groups, we provide algebraic formulas for this…

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…

The close relationship between the scheme of level structures on the universal deformation of a formal group and the Morava $E$-cohomology of finite abelian groups has played an important role in the study of power operations for Morava…

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…

Inspired by the Ax--Kochen isomorphism theorem, we develop a notion of categorical ultraproducts to capture the generic behavior of an infinite collection of mathematical objects. We employ this theory to give an asymptotic solution to the…

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…

We prove that the $p$-completed Brown-Peterson spectrum is a retract of a product of Morava $E$-theory spectra. As a consequence, we generalize results of Ravenel-Wilson-Yagita and Kashiwabara from spaces to spectra and deduce that the…

This expository paper introduces several ideas in chromatic homotopy theory around Morava's extraordinary E-theories. In particular, we construct various moduli problems closely related to Lubin-Tate deformation theory and study their…

The purpose of this note is to verify that several basic rings appearing in transchromatic homotopy theory are Noetherian excellent normal domains and thus amenable to standard techniques from commutative algebra. In particular, we show…

We show that Sullivan's model of rational differential forms on a simplicial set $X$ may be interpreted as a (kind of) $0|1$-dimensional supersymmetric quantum field theory over $X$, and, as a consequence, concordance classes of such…

In this paper we compute the total power operation for the Morava $E$-theory of any finite group up to torsion. Our formula is stated in terms of the $GL_n(Q_p)$-action on the Drinfeld ring of full level structures on the formal group…

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…

We prove that the $p$th Hecke operator on the Morava $E$-cohomology of a space is congruent to the Frobenius mod $p$. This is a generalization of the fact that the $p$th Adams operation on the complex $K$-theory of a space is congruent to…