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This page was the home of the **X**traordinary cohomology theory & **K**-theory **C**ollective **D**iscussion 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.

*Eric Peterson*

** 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

** 12:00-1:00 -- Geometrizing cohomology**

If *X* is a nice space, there is a natural bijection between *H ^{1}(X,G)*, homotopy classes of maps

** 2:00-3:00 -- You could've invented tmf**

The cohomology theory known as

references: Lecture notes from which this talk is assembled.

^{*} the xkcd group -- making xkcd not stand for nothing since 2010!