Recent Blog posts

For the spring 2024 edition of our Freudenthal Topology Seminar, I recently gave a talk about parts of Itamar Mor’s thesis on profinite descent methods for computing the Picard group of $K(n)$-local spectra. You can find the notes below. His thesis covers more than what I had time to talk about, and I could only cover the overall ideas, rather than dive into the actual proofs. Nevertheless, I hope that these notes are helpful as an introduction to descent methods for Picard groups in general, as well as a first start on the profinite version.

For our Freudenthal Topology Seminar, I recently gave a talk about algebraicity of chromatic homotopy theory. It was an introduction to the papers by Pstrągowski, Patchkoria-Pstrągowski, and more recently Barkan. You can find the notes below. While the latter two papers discuss much more than ‘just’ the chromatic case, in my talk I decided to focus on the chromatic case only, as this is already a very interesting case, and to keep the length of the talk reasonable.

In this post I’ll motivate and outline the construction of an important invariant of (bicommutative) Hopf algebras in characteristic $p$: its Dieudonné module. This post is a combination of expositions by others; I follow [HL13, §1] for the most part. Another helpful source is the book by Demazure [Dem]. Throughout this post, I’ll assume that all Hopf algebras are bicommutative. Motivation: understanding Hopf algebras Let $k$ be a field. The goal is to understand Hopf algebras over $k$.

For our Junior chromatic homotopy theory seminar, I gave two lectures introducing stacks. The first focused on the basic definitions, while the goal of the second was to introduce quasi-coherent sheaves on a stack. You can find the notes below. I have tried to keep the preliminaries to a minimum, although for the second lecture some familiarity with schemes is very helpful. For both lectures, I have included a number of references that I think are very good for further study.

If $A$ is an abelian group, consider the functor from spectra to graded abelian groups \[ \mathrm{Sp} \to \mathrm{Ab}_{*}, \quad X \mapsto \mathrm{Hom}_{\mathrm{Ab}_*} (\pi_{*}(X) ,\ A), \] where $A$ is considered to be in degree zero. This is generally not a cohomology theory, because it does not satisfy exactness. If $A$ is injective, then this does satisfy exactness, and in fact defines a cohomology theory. By Brown representability it is represented by a spectrum $I_A$.