research

Publications

   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 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.


    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.


   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.


   Using a modified hammock localization, we construct an operad localization for one-ary operations which preserves operads with homological stability.


   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.


   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

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

Notes

I've scanned handwritten notes from some conferences I've attended. All mistakes are my own.

Seminar talks

Other notes