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.
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.
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.
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.
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.
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.
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.
Papers in preparation
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.
Red-shift for Morava K-theory
We prove the red-shift conjecture for Morava K-theory.Joint with Jeremy Hahn, and Dylan Wilson.
Algebraic K-theory of elliptic cohomology
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.
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.
Topological Hochschild homology of the second truncated Brown-Peterson spectrum II
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 Höning.
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.
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.