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This page was the home of the Xtraordinary cohomology theory & K-theory Collective Discussion group* for the fall 2014 semester.
All talks at Berkeley were on Berkeley time (i.e. they started 10 minutes late).
12:00-1:00 -- 380F Sloan Hall -- Clifford algebras and K-theory
The goal of this talk is to explain some aspects of the relationship between Clifford algebras and K-theory. We'll focus in particular on the relationship between Bott periodicity for Clifford algebras and for K-theory and on the relationship between bundles of Clifford algebras and twists of K-theory.
4:00-5:00 -- 383N Sloan Hall -- Some formal geometry around K-theory [joint with the Stanford topology seminar]
Classical K-theory ranks among the most well-studied objects in homotopy theory, and its many rich structures enjoy interpretations which are simultaneously geometric and algebraic. The "chromatic" perspective on homotopy theory promotes the use of algebraic geometry to organize such algebraic information and uses it to highlight useful patterns and generalizations. I'll explain how such organizational techniques apply to K-theory, including a sketch of how they show the existence of the complex sigma-orientation, and then speculate about how a computation joint with Hughes and Lau suggests the presence of an interesting new infinite loopspace over BU.
1:00-2:00 -- 891 Evans -- The cotangent complex of an operad
I will describe the cotangent complex formalism for operads rather than algebras over operads and a few applications of such a theory. If there is time, I would like to discuss how this theory relates to the Costello-Gwilliam formalism of observables in Quantum Field Theory.
2:30-3:30 -- 939 Evans -- Complex-oriented Quillen stratification
A classical theorem of Quillen states that if G is a finite group, then the mod p cohomology of G can be recovered, up to an equivalence relation called F-isomorphism (a strong form of nilpotence), as the ring of compatible families of cohomology classes on the elementary abelian p-subgroups of G. I'll explain how Quillen's theorem can be reinterpreted (following Carlson) as a form of "descent up to nilpotence" in the category of Fp-modules with G-action. If time permits, I'll explain a generalization of this to complex-oriented theories and equivariant spectra, which is joint work with Niko Naumann and Justin Noel.
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