Most of my papers are available on the arXiv here. We have an informal seminar here at UNR on topics related to homotopical algebra, broadly construed. The website is here. Please contact me if you'd like to attend.
Research Papers
Higher Groups and Higher Normality, with Landon Fox. Submitted (arXiv).
In this paper we continue Prasma's homotopical group theory program by considering homotopy normal maps in arbitrary ∞-topoi. We show that maps of group objects equipped with normality data, in Prasma's sense, are algebras for a "normal closure" monad in a way which generalizes the standard loops-suspension monad. We generalize a result of Prasma by showing that monoidal functors of ∞-topoi preserve normal maps or, equivalently, that monoidal functors of ∞-topoi preserve the property of "being a fiber" for morphisms between connected objects. We also formulate Noether's Isomorphism Theorems in this setting, prove the first of them, and provide counterexamples to the other two. Accomplishing these goals requires us to spend substantial time synthesizing existing work of Lurie so that we may rigorously talk about group objects in ∞-topoi in the "usual way." One nice result of this labor is the formulation and proof of an Orbit-Stabilizer Theorem for group actions in ∞-topoi.
Projective Geometries and Simple Pointed Matroids as 𝔽₁-modules, with So Nakamura. Submitted (arXiv).
We describe a fully faithful embedding of projective geometries, given in terms of closure operators, into 𝔽₁-modules, in the sense of Connes and Consani. This factors through a faithful functor out of simple pointed matroids. This follows from our construction of a fully faithful embedding of weakly unital, commutative hypermagmas into 𝔽₁-modules. This embedding is of independent interest as it generalizes the classical Eilenberg-MacLane embedding for commutative monoids and recovers Segal's nerve construction for commutative partial monoids. For this reason, we spend some time elaborating its structure.
Brauer Wall Groups and Truncated Picard Spectra of K-theory, with Kiran Luecke and Jack Morava. Submitted (arXiv).
We compute the first two k-invariants of the Picard spectra of KU and KO by analyzing their Picard groupoids and constructing their unit spectra as global sections of sheaves on the category of manifolds. This allows us to determine the E∞-structures of their truncations Pic(KU)[0,3] and Pic(KO)[0,2]. It follows that these truncated Picard spaces represent: the Brauer groups of ℤ/2-graded algebra bundles of Donovan-Karoubi, Moutuou and Maycock; the Brauer groups of super 2-lines; and the K-theory twists of Freed, Hopkins and Teleman. Our results also imply that that these spaces represent twists of String and Spin structures on manifolds and can be used to twist tmf-cohomology. Finally, we are able to identify pic(KU)[0,3] with a cotruncation of the Anderson dual of the sphere spectrum.
Other Writing
Matroids as 𝔽₁-modules Notes from a talk given in the UC-Irvine Algebra Seminar.
Group Theory for ∞-Groups Notes from a talk given in the UIUC Topology Seminar.
Picard Spaces and Orientations: Notes from an expository talk on Picard spaces given at the ``Chromatic Nullstellensatz Seminar."
Toward Higher Algebra over 𝔽₁: Notes from a talk at the conference "Low Dimensional Topology and Number Theory" at Kyushu University.
Interpretations of the Truncated Picard Spectra of KO and KU: Notes from a talk given at the BIRS workshop "Cobordisms, Strings and Thom Spectra," held at the Casa Matématica Oaxaca.
On Braids and Cobordism Theories: Notes from a talk given in University of Glasgow's topology seminar.
A User's Guide: Relative Thom Spectra via Operadic Kan Extensions: An exposition of the main ideas in my paper "Relative Thom Spectra via Operadic Kan Extensions," accessible to graduate students studying homotopy theory.
Notes on Lubin-Tate Cohomology: Some notes about the cohomology of a complex that comes up in deformations of formal groups as well as extensions of n-buds.
THH of X(n): A computation of the Topological Hochschild Homology of Ravenel's X(n) spectra.
The Harmonic Bousfield Lattice: A computation of the Bousfield lattice of the category of p-local harmonic spectra. The main theorem and proof were used by Luke Wolcott here.
Talk Slides
Some Galois Theory for Bordism Homology given in the UNR Mathematics and Statistics Colloquium.
On the PROB of Singular Braids given at The Third Conference on Operad Theory and Related Topics.
Symmetry, Topology and the Nobel Prize, slides for an expository talk on topological phases of matter.
Twisted Forms in Homotopy Theory.
Bialgebras in Spectra.