HOMOTOPY METHODS IN ALGEBRAIC TOPOLOGY

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In joint work with C. Rezk and G. Arone, we give a simple classification of functors of finite degree in the sense of Goodwillie (i.e. polynomial functors) from spaces to spectra. Passing to a limit gives some information about analytic functors.

Our approach is to work with a construction which extracts a small amount of data from a functor F and builds the largest polynomial subfunctor of F of a given degree. (This is pretty much dual to the usual Goodwillie tower construction, which assigns to F its maximal polynomial quotients.) Our construction is explicit and functorial. If F itself is polynomial, the extracted data gives a complete set of invariants for F.

Some years ago, Jeff Smith noticed that certain spectra, such as the representing spectra for Morava K-theory or Brown-Peterson homology, could be given a non-unique structure as an A-infinity spectrum under MU, with its canonical A-infinity structure. Using the obstruction theory developed by Mike Hopkins and myself, one can recapture this result for all spectra E with coefficient rings of the form E_* = S^{-1}MU_*/I where I is an ideal in MU_* generated by a regular sequence and S is a multiplicatively closed subset of MU_*/I. This covers many of the standard examples of spectra under MU. Furthermore, one can calculate the moduli space of all A-infinity structures on such a spectrum, and the space of A-infinity maps between two such spectra; in particular, taken together, they form a centric diagram in the sense of Dwyer and Kan.

In this talk, I will explain and prove the Lichtenbaum-Quillen conjecture for such fields.

The topological Theta function is at this point a fantasy object derived from contemplating Theta functions in the context of elliptic cohomology. This point of view has recently led me to a new proof of Borcherds' mod 24 congruences for the Theta functions of even unimodular lattices. I plan to explain this proof, the idea of the topological theta functions, and I'll try and indicate some the good things having the real deal would do.

Benson, Carlson, and Rickard classified thick subcategories of finite stable k[G]-modules, where G is a finite p-group and k is an algebraically closed field of characteristic p. Hovey and Palmieri have recently extended these results to apply to certain graded connected Hopf algebras, including finite subHopf algebras of the mod 2 Steenrod algebra. But we still had to work over the algebraic closure of F_2. In this talk, we use Galois theory to show that both the Benson-Carlson-Rickard theorem and the Hovey-Palmieri theorem apply over arbitrary fields of the appropriate characteristic.

I will talk about certain questions of Real-oriented and algebraic stable homotopy theory. In particular, I will discuss a "Rost spectrum" and its connection with an algebraic version of the Hopf invariant 1 problem. I will also talk about a related question of exotic spheres in Voevodsky's algebraic stable category.

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