Arithmetic and Homotopy Theory

There will be three lecture series consisting of 3-4 lectures each.

• Jeremy Hahn (Cambridge)

The Galois cohomology of a p-adic number field satisfies Tate duality, which is a form of Poincare duality where the fundamental class is determined by the Hasse invariant.  I will describe an emerging picture of how this generalizes to higher chromatic fields, such as p-completed KU.  In these higher chromatic cases, Galois cohomology is replaced by the associated graded of a filtration on localized algebraic K-theory.  Time permitting, I may discuss how to define a higher chromatic analog of the category of (crystalline) Galois representations, or relations to the theory of affine stacks.  All of this is based on works in progress with many people, including Devalapurkar, Raksit, Rognes, Senger, Wilson, and Yuan.

• Matthew Morrow (CNRS)

This mini-course will overview recent progress in motivic cohomology.In the A^{^1}-invariant context, i.e. related to Weibel's A^{^1}-invariant K-theory, we will explain how Voevodsky's conjectural programme to define motivic cohomology using motivic homotopy theory has now been realised. In the non-A^{^1}-invariant context, i.e., related to actual algebraic K-theory, we will present a definition of motivic cohomology which is known to satisfy almost all desired properties and which agrees with the classical theory in the case of smooth algebraic varieties; depending on time, we may also discuss its relation to algebraic cycles, and its conjectural relation to the motivic spectra of Annala, Hoyois, and Iwasa. The new results in the mini-course are joint work with Bachmann--Elmanto and Elmanto, and work of Bouis.

• Jared Weinstein (BU)

The field of p-adic geometry is opening up new frontiers in chromatic homotopy theory. We’ll report on new results, joint with many collaborators, on the vanishing and splitting conjectures. The main hammer in these investigations is Faltings’ “two-tower isomorphism”, along with some interesting and novel generalizations.