# Research & writing

### In preparation

- A HKR-type theorem for real Hochschild homology

### Preprints

## On normed $ \mathbb{E}_\infty $-rings in genuine equivariant $ C_p $-spectra (last updated August 2023)

*Abstract:* Genuine equivariant homotopy theory is equipped with a multitude of coherently commutative multiplication structures generalizing the classical notion of an $ \mathbb{E}_\infty $-algebra. In this paper we study the $ C_p $-$ \mathbb{E}_\infty $-algebras of Nardin--Shah with respect to a cyclic group $ C_p $ of prime power order. We show that many of the higher coherences inherent to the definition of parametrized algebras collapse; in particular, they may be described more simply and conceptually in terms of ordinary $ \mathbb{E}_\infty $-algebras as a diagram category which we call \emph{normed algebras}.
Our main result provides a relatively straightforward criterion for identifying $ C_p $-$ \mathbb{E}_\infty $-algebra structures.
We visit some applications of our result to real motivic invariants.

## Categorical dynamics on stable module categories (last updated December 2022)

*Abstract:* Let *A* be a finite connected graded cocommutative Hopf algebra over a field *k*.
There is an endofunctor `tw` on the stable module category StMod_{A} of *A* which twists the grading by 1.
We show the categorical entropy of `tw` is zero.
We provide a lower bound for the categorical polynomial entropy of `tw` in terms of the Krull dimension of the cohomology of *A*, and an upper bound in terms of the existence of finite resolutions of *A*-modules of a particular form.
We employ these tools to compute the categorical polynomial entropy of the twist functor for examples of finite graded Hopf algebras over $\mathbb{F}$_{2}.

## A descent view on Mitchell's theorem *with Elden Elmanto and Denis Nardin*

*Abstract:* In this short note, we given a new proof of Mitchell's theorem that $L_{T(n)}K(\mathbb{Z}) \simeq 0$ for $n \geq 2$. Instead of reducing the problem to delicate representation theory, we use recently established hyperdescent technology for chromatically-localized algebraic K-theory.

### Other

A copy of my minor thesis can be found here.