# Research

My work focuses on invariants of ring spectra that shed light on arithmetic and geometry.

## Published and Accepted papers

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

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.

To appear in Geometry & Topology. Joint with Christian Ausoni, Dominic Leon Culver, Eva Höning, and John Rognes.#### Topological Hochschild homology of the second truncated Brown-Peterson spectrum I arXiv published

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.

Published in Algebraic & Geometric Topology. Joint with Dominic Culver and Eva Höning.#### Complex orientations of TP of complete DVRs arXiv published

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.

Published in Homology, Homotopy and Applications.#### Detecting beta elements in iterated algebraic K-theory of finite fields arXiv published

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 a conjecture of Lichtenbaum. This also suggests a new approach to studying the redshift conjectures of Ausoni-Rognes.

Published in Transactions of the American Mathematical Society.#### 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

#### Topological \Delta G-homology of twisted G-rings arXiv

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

Joint with Mona Merling and Maximilien Peroux.#### Algebraic K-theory of real topological K-theory arXiv

We compute algebraic K-theory of real topological K-theory after smashing with a finite complex at the prime 2 using syntomic cohomology.

Joint with Christian Ausoni and John Rognes.#### A deformation of Borel equivariant homotopy arXiv

We construct a deformation of the Borel G-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 Jia Kong.#### 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 rings with anti-involution.

Joint with Teena Gerhardt and Mike Hill.#### 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.#### 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.

## In progress

#### Syntomic cohomology of Morava K-theory

We compute the syntomic cohomology of any MU-algebra form of the connective Morava K-theory. As qualitative consequences, we prove redshift, the telescope conjecture, and the Ausoni-Rognes-Lichtenbaum-Quillen conjecture holds for algebraic K-theory of Morava K-theory.

Joint with Jeremy Hahn and Dylan Wilson.#### Syntomic cohomology of truncated Brown-Peterson spectra

We compute the syntomic cohomology of any MU-algebra form of truncated Brown--Peterson spectra. As qualitative consequences, we prove redshift, the telescope conjecture, and the Ausoni-Rognes-Lichtenbaum-Quillen conjecture holds for algebraic K-theory of arbitrary MU-algebra forms of truncated Brown-Peterson spectra.

Joint with Jeremy Hahn and Dylan Wilson.#### Real syntomic cohomology

We study even filtrations in equivariant homotopy theory with applications to Real topological cyclic homology. In particular, we provide constructions of Real syntomic cohomology and Real prismatic cohomology. As an application, we study Real topological cyclic homology of Real topological K-theory.

Joint withHana Jia Kong and J.D. Quigley.#### Syntomic cohomology of 2-primary topological modular forms

We compute the syntomic cohomology of 2-primary topological modular forms. Notably topological modular forms is most interesting at the primes 2 and 3 and the primes 2 and 3 require different techniques.

Joint with Christian Ausoni, Robert R. Bruner, Jack Morgan Davies, John Rognes, and Tristan Yang.#### Syntomic cohomology of 3-primary topological modular forms

We compute the syntomic cohomology of 3-primary topological modular forms. Notably topological modular forms is most interesting at the primes 2 and 3 and the primes 2 and 3 require different techniques.

Joint with Jack Morgan Davies, and Tristan Yang.#### The odd primary even filtration

We study a version of the even filtration in equivariant homotopy theory for a cyclic group of odd primary order. We also present a Mayer-Vietoris approach to slice spectral sequence computations and combine these two ideas to provide a computational tool for studying equivariant homotopy theory for a cyclic group of prime order.

Joint with Mark Behrens, Eva Belmont, Maxwell Johnson, and Hana Jia Kong.

## 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 unstable modules 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.