research
Publications
Isovariant homotopy theory and fixed point invariants (with I. Klang), Advances in Mathematics Vol 433, 2023, 109298, journal page, arXiv:2110.07853
An isovariant map is an equivariant map between G-spaces which strictly preserves isotropy groups. We consider an isovariant analogue of Klein-Williams equivariant intersection theory for finite groups. We prove that under certain reasonable dimension and codimension conditions, the fixed points of self-maps of G-manifolds can be removed isovariantly if and only if the equivariant Reidemeister trace vanishes. In doing so, we build a new homotopy theory on the category of isovariant spaces and prove isovariant analogues of equivariant results, including cellular structures on G-manifolds and a Whitehead theorem.
An isovariant Elmendorf's theorem, Documenta Mathematica Vol. 27, 2022, pp. 613-628, journal page, arXiv:1907.12135
An isovariant map is an equivariant map between G-spaces which strictly preserves isotropy groups. In this paper, we show that for a finite group G, the category of G-spaces with isovariant maps has a Quillen model structure. We prove a Piacenza-style model theoretic proof of an isovariant Elmendorf's theorem, showing that this model structure is Quillen equivalent to a model category of diagrams.
A lax monoidal model for multilinearization, Homology, Homotopy and Applications Vol. 22(1), 2020, pp. 319–331. journal page, arXiv:1706.06915.
Using the category of finite sets and injections, we construct a new model for the multilinearization of multifunctors between spaces that appears in the derivatives of Goodwillie calculus. We show that this model yields a lax monoidal functor from the category of symmetric functor sequences to the category of symmetric sequences of spaces after evaluating at the 0-sphere. We also give a construction which extends the result to symmetric sequences of spectra.
Operads with Homological Stability, (with M. Basterra, I. Bobkova, K. Ponto, and U. Tillmann), Advances in Mathematics 321, 1 December 2017, Pages 391-430. journal page, arXiv:1612.07791
We define operads with homological stability (OHS) and prove that the group completion of algebras over an OHS are infinite loop spaces, generalizing Ulrike Tillmann's work on the surface operad. Using the new definition, we are able to construct maps from geometrically defined Madsen-Tillmann spectra to algebraic K-theory.
Inverting Operations in Operads, (with M. Basterra, I. Bobkova, K. Ponto, and U. Tillmann), Topology and its Applications, Volume 235, 15 February 2018, Pages 130-145. journal page, arXiv:1611.00715
Using a modified hammock localization, we construct an operad localization for one-ary operations which preserves operads with homological stability.
Goodwillie calculus and injections, UIUC thesis.
We give a model for Goodwillie calculus using the indexing category of finite sets and injective maps. This yields a classification of n-excisive functors from spaces to spectra, and a monoidal model for the derivatives of an endofunctor of spaces multilinearization of symmetric functor sequences. There are errors here about properties of cross-effects.
Unbased calculus for functors to chain complexes, (with M. Basterra, K. Bauer, A. Beaudry, R. Eldred, B. Johnson and M. Merling), Women in Topology: Collaborations in Homotopy Theory, AMS Contemporary Mathematics Series, Vol. 641 (2015). journal page, arXiv:1409.1553v2
The Johnson-McCarthy discrete calculus for homotopy functors was extended to include functors from an unbased simplicial model category to spectra. This paper completes the constructions needed to ensure that there exists a discrete calculus tower for functors from an unbased simplicial model category to chain complexes over a fixed commutative ring.
Expository articles
Mathematics user's guide, 2017, Enchiridion.
Provides meta data and context for the results from my thesis organized into four sections: key insights, metaphors and mental imagery, development timeline, and a colloquial summary of results. You can also find user's guides from other mathematicians at this link.
Talks at conferences/workshops
Isovariant homotopy theory, at Algebraic K-Theory of Spaces, Regensburg, Germany, July 2023
An isovariant Whitehead theorem, at AMS Fall Western sectional, online, October 2021
An isovariant Elmendorf's theorem, at Equivariant Topology and Derived Algebra (gong show), in Trondheim, Norway, July 2019
Chain rules and operads in abelian functor calculus, at Functor Calculus Workshop, Ohio State University, March 2019
Isovariant cell complexes, at Women's Intellectual Network Research Symposium, University of Virginia, September 2018
Social Choice and Functoriality, at MAA MathFest, Denver, August 2018
Isovariant homotopy theory, at Higher structures in homotopy theory, INI, Cambridge, July 2018
Chain rules and operads in abelian functor calculus, at CMS summer meeting Fredericton, New Brunswick, June 2018
The Dundas-McCarthy Theorem, joint with Aaron Royer, at Midwest Topology Summer School on Trace Methods in Algebraic K-Theory at Indiana University, August 2017
Inverting Operations in Operads, at AWM Research Symposium at UCLA, April 2017
Derivatives in Goodwillie Calculus, at AMS Special Session on Women in Topology, JMM, Atlanta, January 2017
Operads with Homological Stability, at Midwest Topology Seminar at Purdue University, Indiana, September 2016
Derivatives in Goodwillie calculus, at 31st Summer Topology Conference in Leicester, England, August 2016
Intro to K-theory, at European Talbot in Klosters, Switzerland, July 2015
A chain rule for Goodwillie calculus, at Young Topologists Meeting in Lausanne, Switzerland, July 2015
Poster presentation at JMM 2015 AWM special session in homotopy theory, San Antonio, January 2015
Classifying n-excisive functors, at Young Topologists Meeting in Copenhagen, Denmark, July 2014
Classifying n-excisive functors, at Manifolds, K-theory, and related topics in Dubrovnik, Croatia, June 2014
Notes
I've scanned handwritten notes from some conferences I've attended. All mistakes are my own.
Michael Ching minicourse March 2015 at UIUC
MSRI Summer Workshop 2013
Midwest F2013
Seminar talks
UCR topology seminar, May 2022
UNAM algebraic topology seminar, April 2022
Ohio State student homotopy seminar, February 2021
UCR topology seminar, February 2021
UCLA topology seminar, February 2021
UCR category theory seminar, November 2020
MSRI newcomers seminar, February 2020
University of Louisiana Lafayette topology seminar, November 2019
Maryland geometry and topology seminar, April 2019
North Carolina State University colloquium, March 2019
Northwestern University student seminar, February 2019
Iowa State colloquium, January 2019
University of Minnesota Topology seminar, November 2018
Union College student seminar, April 2018
University at Albany algebra topology seminar, April 2018
University of Calgary peripatetic seminar, February 2018
Howard University math colloquium, October 2017
Wayne State topology seminar, April 2017
Johns Hopkins topology seminar, April 2017
George Mason topology and dynamics seminar, December 2016
Maryland geometry and topology seminar, November 2016
Boulder topology seminar, October 2016
Virginia topology seminar, April 2016
Purdue topology seminar, February 2016
KTH/Stockholm topology seminar, December 2015
University of Kentucky topology seminar, November 2015
Johns Hopkins topology seminar, November 2015
Ohio State topology seminar, October 2015
Indiana University topology seminar, October 2015
Notre Dame topology seminar, October 2015
UIC topology seminar, October 2015
University of Chicago topology seminar, October 2015
Northwestern topology seminar, October 2015
WWU-Münster topology seminar, July 2015
Other notes
Prelim notes: A write-up trying to understand definitions of calculus and the category of finite sets and injective maps. This helped prepare me for my prelim, spring 2013.
Analyticity: Thinking about what it means for a functor to be analytic and the relationship with n-excision. circa 2014
Diagrams: Thinking about when diagrams in symmetric monoidal categories are symmetric monoidal. incomplete, summer 2012.
A User's Guide: for my thesis on Goodwillie calculus. Beware of errors about cross effects. Also at the link: user's guides by other people on different papers!