# 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.

## Current projects

#### Homology of twisted G-rings

We construct a variant of topological Hochschild homology associated to any crossed simplicial group. The input of this construction is a twisted G-ring, which generalizes the notion of a ring with anti-involution.

Joint with Mona Merling and Maximilien Peroux.#### A deformation of Borel C_p-equivariant homotopy

We construct a deformation of the Borel C_p-equivariant stable homotopy category that recovers the a-complete Artin-Tate real motivic stable homotopy category of Burkland-Hahn-Senger.

Joint with Mark Behrens, Eva Belmont, and Hana Kong.#### Syntomic cohomology of Morava K-theory

We compute the syntomic cohomology of connective Morava K-theory and use this to study algebraic K-theory of Morava K-theory.

Joint with Jeremy Hahnand Dylan Wilson.#### G-theory of ring spectra with many objects

We give a new construction of G-theory in the context of ring spectra with many objects and study the corresponding localization sequences.

Joint with Christian Ausoni, Eva Höning, and John Rognes.

## Published and Accepted Papers

#### Complex orientations of TP of complete DVRs arXiv

I give an explicit description of the height one formal group law associated to TP of a ring of integers in a local field over a certain base ring.

Accepted in Homotopy, Homology, and Applications.#### 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

#### Algebraic K-theory of elliptic cohomology arXiv

We compute mod p,v_1,v_2 homotopy of algebraic K-theory of the second truncated Brown-Peterson spectrum at primes p greator or equal to seven.

Joint with Christian Ausoni, Dominic Leon Culver, Eva Höning, and John Rognes.#### Real Topological Hochschild homology via the norm and Real Witt vectors arXiv

We interpret Real topological Hochschild homology as the norm for the orthogonal group of two-by-two-matrices. We then prove a multiplicative double coset formula. Using this we define Real Hochschild homology and Witt vectors for rins with anti-involution.

Joint with Teena Gerhardt and Mike Hill.#### Topological Hochschild homology of the second truncated Brown-Peterson spectrum I arXiv

We compute topological Hochschild homology of forms of the second truncated Brown-Peterson spectrum with coefficients in connective Morava K-theory. We also compute topological Hochschild homology of forms of arbitrary truncated Brown-Peterson spectra with coefficients in the p-local integers.

Joint with Dominic Culver and Eva Höning.#### Chromatic complexity of algebraic K-theory of y(n) arXiv

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

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 an application, we prove a version of Mitchell's theorem for truncated Brown-Peterson spectra.

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

I prove that a certain family of beta elements 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.

## Notes

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

This note provides user friendly conditions for checking when a map of simplicial spectra induces a cofibration on geometric realizations. 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 describe 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 a construction of the topological Hochschild-May spectral sequences, use it to compute topological Hochschild homology of the image of j after smashing with the Smith Toda complex of type one, and then I detect certain beta elements in iterated algebraic K-theory of finite fields. This 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.