Research & writing

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.

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.

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.