# Research

My work focuses on invariants of ring spectra that shed light on interactions with arithmetic and geometry. For more information about my research, see my research statement.

## Published Papers

#### On topological Hochschild homology of the K(1)-local sphere arXiv published

I compute mod (p,v_1) topological Hochschild homology of the connective cover of the K(1)-local sphere spectrum using the topological Hochschild-May spectral sequence.

Published in Journal of Topology.#### The Segal Conjecture for Topological Hochschild Homology of Ravenel spectra arxiv published

We solve the homotopy limit problem for topological Hochschild homology of Ravenel's spectra X(n) with respect to all cyclic groups of order a power of p. This implies that, after p-completion, topological negative cyclic homology and topological periodic cyclic homology of X(n) are homotopy equivalent.

Published in Journal of Homotopy and Related Structures. Joint with J.D. Quigley.#### A May-type spectral sequence for topological Hochschild homology arXiv published

We construct a spectral sequence for higher topological Hochschild homology associated to a multiplicative filtration of a commutative ring spectrum. In particular, we show that the Whitehead tower of a commutative ring spectrum can be built as a multiplicative filtered commutative ring spectrum. We use this spectral sequence to give a bound on topological Hochshcild homology of a connective commutative ring spectrum.

Published in Algebraic & Geometric Topology. Joint with Andrew Salch.

## Submitted Papers

#### Chromatic complexity of algebraic K-theory of y(n) arXiv pdf

We compute Morava K-theory of topological periodic cyclic homology and topological negative cyclic homology of the Thom spectra y(n). This gives evidence for a version of the red-shift conjecture for topological periodic cyclic homology at all chromatic heights.

Joint with J.D. Quigley.#### Commuting unbounded homotopy limits with Morava K-theory arXiv pdf

We give conditions for Morava K-theory to commute with certain homotopy limits with an eye towards applications to topoligical periodic cyclic homology and algebraic K-theory. As our application, we a version of Mitchell's theorem for truncated Brown-Peterson spectra.

Joint with Andrew Salch.#### Detecting the beta family in iterated algebraic K-theory of finite fields arXiv pdf

I prove that the beta family is detected in iterated algebraic K-theory of finite fields and consequently iterated algebraic K-theory of the integers. This gives evidence for a higher chromatic height version of the Lichtenbaum conjecture, which I call the Greek-letter-family red-shift conjecture, after the red-shift conjecture of Ausoni-Rognes.

## Papers in preparation

#### Topological Hochschild homology of the second truncated Brown-Peterson spectrum

We compute topological Hochschild homology of the second truncated Brown-Peterson spectrum with coefficients in the first truncated Brown-Peterson spectrum.

Joint with Dominic Culver and Eva HÃ¶ning.#### Real Topological Hochschild homology, Witt vectors, and norms

We give a description of the Mackey functor homotopy groups of Real topological Hochschild homology of noncommutative rings with anti-involution in terms of Witt vectors for rings with anti-involution. We also provide a Hill-Hopkins-Ravenel norm model for Real topological Hochshild homology.

Joint with Teena Gerhardt and Mike Hill.#### Topological crossed simplicial group homology

We give a new construction of topological crossed simplicial group homology, which gives a combinatorial approach to certain Hill-Hopkins-Ravenel norms for compact Lie groups.

Joint with Mona Merling and Maximilien Peroux.

## Notes

#### Maps of simplicial spectra whose realizations are cofibrations arXiv pdf

This note provides user friendly conditions for checking when a map of simplicial spectra induces a cofibration on geometric realizations. The main result of this paper shows that the category of symmetric spectra satisfies the hypotheses of our joint paper A May-type spectral sequence for topological Hochschild homology. The results are proven in a completely elementary way.

Joint with Andrew Salch.#### K(n)-local homotopy groups via Lie algebra cohomology pdf

I compute the homotopy groups of the K(1)-local mod p Moore spectrum at odd primes in order to illustrate a method that can be used to compute K(n) local homotopy of more general type n complexes. This an expository article based on work of Doug Ravenel.

#### Auslander-Reiten quiver of the category of unstable modules over a sub-Hopf algebra of the Steenrod algebra pdf

I study the Auslander-Reiten quiver associated to the category of unstable E(1) modules where E(1) is the sub-Hopf algebra of the Steenrod algebra generated by the first two Milnor primitives.

## Theses

#### Periodicity in iterated algebraic K-theory of finite fields link

I give an exposition of joint results with Andrew Salch on the topological Hochschild-May spectral sequence, compute topological Hochschild homology of algebraic K-theory of a large class of fields after smashing with a four cell complex. I then detect some classes in the Hurewicz image of iterated algebraic K-theory of finite fields. These results are improved upon in the preprints above, but this thesis contains additional exposition, which may is also of value. This project is my PhD thesis completed under the direction of Andrew Salch.

#### Galois cohomology and algebraic K-theory of finite fields pdf

I compute algebraic K-theory of finite fields using Galois cohomology and an Atiyah-Hirzebruch type spectral sequence for algebraic K-theory. This project is my master's thesis completed under the direction of Andrew Salch.

#### Social Justice Through Mathematics: An Exploration of Social Justice Pedagogy and a Reflection on Mathematics in Summer Programs for Youth link

I reflect on my experiences working in the KCIS's Think Summer! program and developing mathematics curriculum for the program. I also design my own summer program for teaching mathematics through a social justice lens, which I call Just Math, based upon reflections on work of Robert Moses, Paulo Freire, and Eric Gutstein. This is my undergraduate thesis completed under the direction of John Fink.