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, Lewis Dominguez, 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 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, 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.