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This page was the home of the Xtraordinary cohomology theory & K-theory Collective Discussion group* for the fall 2012 semester.
All talks at Berkeley were on Berkeley time (i.e. they started 10 minutes late).
1:00-2:00 -- The EHP spectral sequence
The EHP spectral sequence is a tool used to organize and compute the 2-primary un/stable homotopy groups of spheres. As far as we're concerned, its coolest feature is that it is a maximally interesting example of a spectral sequence; it enjoys basically every imaginable feature that a spectral sequence can. We will construct it, explore its long list of features, compute some homotopy groups of spheres, and conclude with a 2-primary form of Serre's finite generation theorem.
2:30-3:30 -- Functor Calculus and String Topology
I'll discuss a method, motivated by Goodwillie calculus, for approximating the output of a contravariant functor from spaces to spaces. In Goodwillie calculus, the approximation takes the form of a Taylor series, with a constant term, a linear term, a quadratic term, etc. Our approximation is more like polynomial interpolation: we take a function, sample it at (n+1) points, and take the unique degree n polynomial that passes through those points. Given a manifold M, its string topology spectrum LM-TM turns out to be the linear approximation of the based loops of the self-homotopy equivalences of M, which is fairly surprising.
12:00-1:00 -- Geometrizing cohomology
If X is a nice space, there is a natural bijection between H1(X,G), homotopy classes of maps X-->BG, and principal G-bundles on X; if G is abelian, this bijection preserves the natural group structure on each set. In many cases, Chern-Weil theory and Deligne cohomology allow us to "strictify" these bijections. I'll review this theory in the case of line bundles, and discuss its "higher" generalization to gerbes with connection and to bundles of differentiable spaces (due to Pawel Gajer).
2:00-3:00 -- You could've invented tmf
The cohomology theory known as topological modular forms was first introduced as the target of a topological lift of the Witten genus, an invariant of String manifolds taking values in modular forms. However, it also arises quite naturally in the search for a "global height-2 cohomology theory", i.e. a higher analog of rational cohomology (at height 0) and complex K-theory (at height 1). In this talk, I'll explain what all this means, show how it fits into the bigger picture of stable homotopy theory, and give a step-by-step account of how you, too, could've invented tmf.