## Midwest Topology Seminar

### Northwestern University

### April 24, 2010

- Dan Isaksen (Wayne State)

Motivic stable homotopy groups

Abstract:
I will survey several different approaches to explicit
computations of motivic stable homotopy groups: geometric
constructions, Toda brackets, and Adams spectral sequences. The
motivic Adams spectral sequence over C presents some curious exotic
phenomena starting in the 25-stem. Z/2-equivariant stable homotopy
groups will play a role in the discussion, since they are related to
motivic stable homotopy groups over R.

- Jack Morava (Johns Hopkins)

Base motives and homotopic descent

Abstract:
This a report on work in progress with K. Hess.
Blumberg, Gepner, and Tabuada, in
have constructed a category which seems to
accomodate both the motives of algebraic geometry
(defined via derived categories of qcoh sheaves)
and the objects of differential topology (defined
in terms of categories of modules over the functional
duals of finite complexes). Their category is naturally
enriched over the K-theory of the sphere
spectrum S and it is plausible that
some version of it is homotopically Tannakian over
the category of S-modules, with an appropriate Hopf ring-spectrum
as `motivic group'. Over the rationals, this can be
identified with (the dual to) the universal enveloping
algebra of a free Lie algebra on odd-degree generators,
called the `cosmic Galois group' by arithmetic geometers