HOMOTOPY METHODS IN ALGEBRAIC TOPOLOGY

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Topological Andre-Quillen cohomology (TAQ), introduced by Hunter & McClure, provides the natural cohomology theory in the category of E-infinity ring spectra (Robinson & Whitehouse independently introduced a similar, perhaps identical, theory). Hunter-McClure defined Steenrod operations for TAQ with coefficients in HZ/p and formulated a "Miller spectral sequence" to calculate it. Kriz observed that the k-invariants for the Postnikov towers of E-infinity ring spectra should refine to classes in TAQ with Eilenberg-MacLane spectrum coefficients, and so obstructions to existence of E-infinity ring structures and of E-infinity ring maps lie in these groups. Basterra reformulated the foundations of this theory in a convenient form and wrote the first complete arguments in a preprint to appear soon in JPAA.

With coefficients in an Eilenberg-MacLane spectrum, the Topological
Andre-Quillen cohomology of an E-infinity ring spectrum can be
calculated as the Quillen cohomology (as defined in the monograph
* Homotopical Algebra *) of a related E-infinity differential graded
algebra. In joint work with Basterra, we use this to interpret the
Hunter-McClure Steenrod operations in terms of the Ginzburg-Kapranov
Getzler-Jones operadic bar duality (operadic "Koszul" duality). In
addition, we calculate the Andre-Quillen cohomology of the
representing objects, i.e. the abelian objects in the category of
E-infinity differential graded Z/p-algebras. This provides a
calculation of the algebra of (unstable) operations on TAQ with
coefficients in HZ/p. I plan to talk about these calculations.

While looking for functors of rings to spectra which admit natural transformations to algebraic K-theory we were led to develop a "calculus" which is formally dual to that of T. Goodwillie for homotopy functors. In this talk we briefly discuss the problem in algebraic K-theory we wish to study and why the dual calculus is a natural first approximation to consider. We then provide the general construction and universal properties for this dual calculus and finish by discussing some specifics about the theory as it applies to the algebraic K-theory of rings.

In this talk I will introduce some basic concepts, results and conjectures in the homotopy theory of algebraic varieties. I will emphasize on the analogy with standard algebraic topology, showing how analogues of some classical results, for instance dealing with rational homotopy theory or Adams spectral sequences, are related to highly non-trivial results or conjectures on motives.

(joint with Mark Hovey) We give a classification of the thick subcategories of finitely generated modules over certain kinds of finite-dimensional Hopf algebras, as well as a classification of the Bousfield classes of all modules over those Hopf algebras. One class of examples consists of the finite sub-Hopf algebras of the mod 2 Steenrod algebra, after extending scalars to the algebraic closure of the field with 2 elements.

This talk will have four parts. First, we will discuss Thom's original theorem. Second, we will discuss improved methods of computation. Third, we will discuss the results for structural groups other than O. Fourth, we will discuss applications of cobordism theory. Along the way, we will mention where mistakes were made.

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