In Balmer's framework of tensor triangular geometry, the prime thick tensor ideals in a tensor triangulated category C form a space which admits a continuous map to the Zariski spectrum Spec^h(End_u(1)) of homogeneous prime ideals in the graded endomorphism ring of the unit object. (Here the grading is induced by an element u of the Picard group of C.) If C is the stable motivic homotopy category and u is the punctured affine line, then this endomorphism ring is the Milnor-Witt K-theory ring of the base field. I will describe work by my student, Riley Thornton, which completely determines the homogeneous Zariski spectrum of Milnor-Witt K-theory in terms of the orderings on the base field. I will then comment on work in progress which uses the structure of this spectrum to study the thick subcategories of the stable motivic homotopy category.

Finite resolutions of spectra constructed by Goerss, Henn, Mahowald and Rezk provide a powerful tool for K(2)-local computations at the prime 3 because of the associated tower spectral sequences. I will talk about the methods used in constructing these resolutions and extending them to the prime 2. I will present some old and new results at the prime 2 and construct the duality resolution of the spectrum closely related to the K(2)-local sphere.

I will review the construction of the Adams spectral sequence in a triangulated category equipped with a projective or injective class. Then, following Cohen and Shipley, I'll describe higher Toda brackets in a triangulated category and present a result which gives a relationship between these Toda brackets and the differentials in the Adams spectral sequence. I will also mention a result due to Heller and Muro that shows that the 3-fold Toda brackets determine the triangulation and hence the higher Toda brackets. This is joint work with Martin Frankland.

I plan to talk about recent joint work with Lennart Meier on RO(Q)-graded homotopical Gorenstein duality for kR, tmf_1(3) and BPR

For any finite subgroup G of the Morava stabilizer group of height n at a prime p, there is an associated ring spectrum EO = (E_n)^{hG} of homotopy fixed points of Morava E-theory E_n under its G-action. I will discuss joint work with Drew Heard and Akhil Mathew, in which we show that, when n=p-1, Pic(EO) is always cyclic, so that every invertible EO module is a suspension of EO.

Multiplicative structure and power operations have been used to great effect in many familiar spectral sequences. One main application is an easy proof of the collapse of a spectral sequence or a computation of the multiplicative structure or power operations on the target of a spectral sequence. In the case of the Adams spectral sequence one can do more. In his thesis, Bruner gave definitive formulas for differentials in the Adams spectral sequence of an H_oo-ring spectrum. In particular, this gives a nice intuitive explanation of the Hopf invariant one differential

In explaining this differential, we will expose the moving parts of such a result. We will also present a C_2-equivariant form of some of Bruner's results.