I am the Ralph E. and Norma L. Edwards Research Professor in the mathematics department at the University of Kentucky.

From 2016-2017 I was a postdoc at the University of Regensburg. From 2014-2016 I was a postdoc at the Max Planck Institute for Mathematics. From 2011-2014 I was a CLE Moore Instructor at MIT. My Ph.D. advisor was Charles Rezk at the University of Illinois at Urbana-Champaign.

I work on interactions between algebraic topology, algebraic geometry, representation theory, and arithmetic geometry. Specifically, I am interested in chromatic homotopy theory and the Morava E-theories.

This semester I am primarily working with Tobias Barthel, Dan Berwick-Evans, Bert Guillou, David Mehrle, Sune Precht Reeh, Tomer Schlank, and Jared Weinstein on a variety of projects. My current PhD students are Nate Cornelius, Lakshay Modi, and Millie Rose.

I am a 2021 Sloan Research Fellow. My work is partially supported by the National Science Foundation grant DMS-2304781 and was previously supported by DMS-1906236. I am also supported by a Simons travel support for mathematicians grant. My collaboration with Tomer Schlank is partially supported by the U.S. Israel Binational Science Foundation grant 2018389. I may be contacted at nat.j.stapleton at gmail.com. My office is POT 765.

Here is my CV.

## Papers

*On Hopkins' Picard group*, with Tobias Barthel, Tomer Schlank, and Jared Weinstein. *On the image of the total power operation for Burnside rings*, with Nate Cornelius, Lewis Dominguez, David Merhle, Lakshay Modi, and Millie Rose. *On the rationalization of the K(n)-local sphere*, with Tobias Barthel, Tomer Schlank, and Jared Weinstein. *The homotopy of the KUG-local equivariant sphere spectrum*, with Tanner Carawan, Rebecca Field, Bert Guillou, and David Mehrle. *On the KUG-local equivariant sphere*, with Peter Bonventre and Bert Guillou. *Evaluation maps and transfers for free loop spaces II*, with Sune Precht Reeh and Tomer Schlank. *Evaluation maps and transfers for free loop spaces I*, with Sune Precht Reeh and Tomer Schlank. *Power operations in the Stolz-Teichner program*, with Tobias Barthel and Dan Berwick-Evans*accepted for publication in Geom. Topol.,*2021 .*Transfer ideals and torsion in the Morava E-theory of abelian groups*, with Tobias Barthel,*J. Homotopy Relat. Str.,*2020 .*Additive power operations in equivariant cohomology*, with Peter Bonventre and Bert Guillou. *Level structures on p-divisible groups from the Morava E-theory of abelian groups*, with Zhen Huan*accepted for publication in Math. Z.,*2022 .*Monochromatic homotopy theory is asymptotically algebraic*, with Tobias Barthel and Tomer M. Schlank*accepted for publication in Adv. Math.,*2020 .*Lubin-Tate theory, character theory, and power operations*, Handbook of Homotopy Theory,2020 .*Chromatic homotopy theory is asymptotically algebraic*, with Tobias Barthel and Tomer M. Schlank,*Invent. Math.,*2020 .*The Balmer spectrum of the equivariant homotopy category of a finite abelian group*, with Tobias Barthel, Markus Hausmann, Niko Naumann, Thomas Nikolaus, and Justin Noel,*Invent. Math.,*2019 .*Excellent rings in transchromatic homotopy theory*, with Tobias Barthel,*Homology Homotopy Appl.,*2018 .*A formula for p-completion by way of the Segal conjecture*, with Sune Precht Reeh and Tomer M. Schlank*accepted for publication in Topol. Appl.,*2022 .*A canonical lift of Frobenius in Morava E-theory*,*Homology Homotopy Appl.,*2018 .*Brown-Peterson cohomology from Morava E-theory*, with Tobias Barthel and an appendix by Jeremy Hahn,*Compos. Math.,*2017 .*The character of the total power operation*, with Tobias Barthel,*Geom. Topol.,*2017 .*Centralizers in good groups are good*, with Tobias Barthel,*Algebr. Geom. Topol.,*2016 .*On the ring of cooperations for 2-primary connective topological modular forms*, with Mark Behrens, Kyle Ormsby, and Vesna Stojanoska,*J. Topol.,*2019 .*A transchromatic proof of Strickland's theorem*, with Tomer M. Schlank,*Adv. Math.,*2015 .*Singular cohomology from supersymmetric field theories*, with Chris Schommer-Pries,*accepted for publication in Adv. Math.,*2020 .*A relative Lubin-Tate theorem via meromorphic formal geometry*, with Aaron Mazel-Gee and Eric Peterson,*Algebr. Geom. Topol.,*2015 .*Subgroups of p-divisible groups and centralizers in symmetric groups*,*Trans. Amer. Math. Soc.,*2015 .*Transchromatic twisted character maps*,*J. Homotopy Relat. Str.,*2015 .*Transchromatic generalized character maps*,*Algebr. Geom. Topol.,*2013 .

## Expository and Notes

*A note on Strickland's MO argument for the zeroth homotopy of the K(1)-local sphere*.*Notes from a talk on the character of the total power operation*.*Notes from an introductory talk on etale homotopy theory*.*An introduction to HKR character theory*, Appendix in Formal geometry and bordism operations, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2018 by Eric Peterson.*The E-theory Seminar Notes*. These are notes from the talks, they were written primarily by Eric Peterson.

## Programming

I have contributed code to Macaulay 2. In particular, I wrote the package RationalPoints.m2 and coauthored the packages GenericInitialIdeal.m2, NoetherNormalization.m2, and Regularity.m2. Currently, I am working with an undergraduate lab group at Kentucky to add code to Sage to work with formal group laws. Here is a calculation of the universal formal group law.

## Art

In the spring of 2012 I took a print-making class at MIT. Some of my art is below:

## People

Here are some of my collaborators:

Tobias Barthel, Mark Behrens, Dan Berwick-Evans, Peter Bonventre, Lewis Dominguez Martin Frankland, Bert Guillou, Aaron Mazel-Gee, Zhen Huan, Niko Naumann, Justin Noel, Kyle Ormsby, Eric Peterson, Sune Precht Reeh, Tomer Schlank, Chris Schommer-Pries, David Spivak, Vesna Stojanoska.