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This page was the home of the Xtraordinary cohomology theory & K-theory Collective Discussion group* for the spring 2014 semester.
All talks at Berkeley were on Berkeley time (i.e. they started 10 minutes late).
11:00-12:00 -- 380F Sloan Hall -- Power operations and Commutative Ring Spectra
Highly structured ring spectra play a distinguished role in modern stable homotopy theory. Such operations help to differentiate between different "commutative" structures. We will compute these operations on relative smash products using the Kunneth spectral sequence. Then, we will interpret the homotopy of these relative smash products and the algebra of operations in terms of different realizations of highly structured DGAs. We will also discuss the relation to the relevant notion of cotangent complexes. Be forewarned, there is a wide turn in this road.
1:30-2:30 -- 383N Sloan Hall -- A Spectrum-Level Hodge Decomposition on Topological Hochschild Homology
We'll give a geometrically-flavoured introduction to Hochschild homology and topological Hochschild homology of commutative rings, before describing a fragmentation of the THH spectrum into Hodge-graded pieces. We'll then speculate about interpretations of our work in derived algebraic geometry.
3:15-4:15 -- 380X Sloan Hall -- Loop Spaces and Friends
This will be a sort of casual overview talk about loop spaces of (derived) schemes and stacks, and the role they play in (derived) algebraic geometry. It's much too large of a topic to adequately cover in an hour, so I'll try to touch on different facets of the theory based on audience interest.
Sam Nolen will not talk about ∞-categories at 4:15 in 380X Sloan Hall.
10:00-11:00 -- n-categories as sheaves on n-manifolds
I'll explain how a (suitably finite) n-category gives invariants of (suitably finite) n-manifolds. This construction captures loads of classical and recent constructions, such as factorization homology and various knot invariants. This talk will be mostly definitions and constructions -- I'll assume familiarity with homotopical category theory.
2:30-3:30 -- Localization sequences in and around algebraic K-theory
Algebraic K-theory is a spectral invariant of categories of modules often reputed to be difficult to compute. Localization sequences can relate different algebraic K-theory spectra to each other and provide computational power. I will provide a survey of some localization sequences arising in algebraic K-theory (and the closely-related theory of topological Hochschild homology) and provide some indication as to how such results might be proven.
4:00-5:00 -- K-Theory of Endomorphisms
The K-theory of endomorphisms (or "parametrized K-theory") is the study of the "additive invariants" of endomorphisms over a discrete ring. Almkvist, Grayson, Kelley, Spanier, et al. studied these in the 60s and 70s and discovered that these additive invariants lived as a dense λ-subring of the ring of (big) Witt vectors. Lindenstrauss and McCarthy have more recently given a complete description of the higher invariants (the higher K-groups) of these endomorphisms as a "topological Witt vector construction". We discuss some of the historical motivation for the study of the subject, the results of Lindenstrauss and McCarthy, and possible generalizations of their work.
references: Lecture notes.
* the xkcd group -- making xkcd not stand for nothing since 2010!