Chromatic homotopy theory is an attempt to divide and conquer algebraic topology by studying a sequence of what were first assumed to be “easier” categories. These categories turn out to be very strangely behaved — and furthermore come equipped with intriguing and exciting connections to number theory. To give an appreciation for the subject, I’ll describe the most basic of these strange behaviors, then I’ll describe an ongoing project which addresses a small part of the “chromatic splitting conjecture”.

In recent work, Hess and Shipley have defined topological coHochschild homology (coTHH) for coalgebra spectra. In this talk, I will discuss coTHH for coalgebra spectra and dg-coalgebras, and give some computational results in about cofree coalgebras in both contexts. In the spectral world, these follow from an analogue of the Bokstedt spectral sequence for computing topological Hochschild homology, which I will also discuss. This is joint work with Gerhardt, Hogenhaven, Shipley, and Ziegenhagen.

The Hill-Hopkins-Ravenel solution to the Kervaire invariant one problem not only resolved a classical geometric question, it also began the serious investigation of (more rigid) equivariant methods in studying chromatic homotopy theory. In this talk we discuss ongoing work to carry out this idea at odd primes. The primary hurdle is to find an analog of the spectrum of Real cobordism where there is no obvious replacement for the role played by complex conjugation. We will revisit the prime 2 with an eye towards generalization, and then share our preliminary results at odd primes.

An equivariant infinite loop space machine is a functor that constructs genuine equivariant spectra out of simpler categorical or space level data. In the late 80's Lewis-May-Steinberger and Shimakawa developed generalizations of the operadic approach and the Gamma-space approach respectively. In this talk I will describe work in progress that aims to understand these machines conceptually, relate them to each other, and develop new machines that are more suitable for certain kinds of input. This work is joint with Anna Marie Bohmann, Bert Guillou, Peter May and Mona Merling.

I will report recent progress of computing stable stems, joint with Dan Isaksen and Guozhen Wang. I will review known methods and explain two new methods that we recently developed. The first one uses the Kahn-Priddy theorem and has applications on smooth structures on spheres. The second one uses motivic homotopy theory in a fundamental way, and gives a connection to chromatic homotopy theory. Combining with known methods, I will describe a project towards solving the open case of the Kervaire invariant problem in dimension 126.