Elliptic cohomology in the last ten years.

In the 1980's Witten discovered the remarkable connection between string theory and elliptic cohomology. Since then there has been great progress in the study of elliptic cohomology in stable homotopy theory and also in the mathematics of string theory. It is now possible to tell a story in stable homotopy theory which complements and significantly refines the physical results described by Witten. In addition, many things we don't know about elliptic cohomology seem to correspond closely to issues of current interest in the study of open strings.

Spaces of embeddings and the Fulton-McPherson compactification.

We will discuss certain filtrations of the space of embeddings of manifolds. To describe the filtration we will need certain construction closely related to the Fulton-Mcpherson compactification. We will discuss the properties of the filtration and relate it to some classical constructions in geometric topology.

Representations of braid groups.

Braid groups have many equivalent definitions, and an even larger number of applications. The Jones representation of braid groups by matrices was originally defined using the Hecke algebra, and can be used to obtain the famous Jones polynomial of a knot or link.

I will give two definitions of the braid group - one algebraic and one topological. I will then describe a new definition of the Jones representation from this more topological point of view. It is to be hoped that this will shed new light on the topological meaning'' of the invariants obtained from it.

Tate cohomology of finite groups, cohomology of differential graded algebras, and obstruction theory.

After a brief introduction to Tate cohomology, I shall discuss the following question. Given a module X over Tate cohomology of a finite group G, how do we tell whether it is the Tate cohomology of a kG-module? I shall describe a single obstruction which determines whether it is a direct summand of the Tate cohomology of a module. The obstruction lies in Ext of the X with itself, in degree (3,1). This obstruction is the image of a universal predecessor, which lies in the Hochschild cohomology of the Tate cohomology of G, in the same degree.

The analogous question for differential graded algebras is this. Given a module over the cohomology of a DGA, how do we tell whether it is the cohomology of a module? The answer is essentially the same: there is a single obstruction to realizability as a direct summand of the cohomology of a module, and there is a universal predecessor in degree (3,1) Hochschild cohomology of the DGA.

The obstruction in each case can be viewed as a universal Massey triple product. In the language of $A_\infty$-algebras, it is the element of Hochschild cohomology determined by the structure map m3.

For an arbitrary triangulated category with direct sums, the obstruction in Ext in degree (3,1) can still be formulated, but it seems that more structure is necessary for the predecessor in Hochschild cohomology to work.

It is a pleasure to acknowledge that the research described in this talk is joint work with Henning Krause and Stefan Schwede.

Periodic homology of infinite loop spaces.

Consider the Goodwillie tower of the functor that assigns to a spectrum X, the suspension spectrum of its 0th space. This tower has rth fiber equal to the rth extended power of X. Our theorem is that, after localizing at any Morava K-theory K(n) (n>0), the tower naturally splits (for ALL spectra X). This has implications for computing the E-homology or cohomology of infinite loopspaces, where E is Bousfield equivalent to K(n). It also suggests some general conjectures regarding how Goodwillie calculus and K(n)-localization interact.

New analogs of the braid groups.

The space of unordered configurations of distinct points in the plane is aspherical, with Artin's braid group as its fundamental group. Remarkably enough, the space of ordered configurations of distinct points on the real projective line, modulo projective equivalence, has a natural compactification (as a space of equivalence classes of trees) which is also (by a theorem of Davis, Januszkiewicz, and Scott) aspherical.

The classical braid groups are ubiquitous in modern mathematics, with applications from the theory of operads to the study of the Galois group of the rationals. The (rather mysterious) fundamental groups of these new configuration spaces are not braid groups, but they have many similar formal properties. This talk will be an introduction to their study, and their (possible) applications.

On functor homology.

Homological algebra in the category of functors from finite pointed sets to abelian groups plays an important role in algebra and topology. We will explain the main recent applications and particularly the applications to the homology theory of commutative algebras.

Varieties on the field with one element.

Tits, and later Manin, suggested the existence of an algebraic geometry over "the field with one element". I will describe their arguments in favor of such a theory. Then I will propose a provisional definition of these varieties. Those speculations will also lead us to a motivic interpretation of the image of Adams' J-homomorphism.

Algebraic Morse Theory and factorization of birational maps.

We develop a Morse-like theory for complex algebraic varieties. In this theory the Morse function f is replaced by a K*-action. The critical points of the Morse function correspond to fixed points of the action. The homotopy type changes when we pass through the critical points. Analogously, in the algebraic setting passing through'' the fixed points of the K*-action induces some simple birational transformations called blowups, blowdowns and flips. They are analogous to spherical modifications.

In classical Morse Theory by means of the Morse function we can decompose the manifold into elementary pieces -"handles". In the algebraic Morse Theory we decompose a birational map between two smooth complex algebraic varieties into a sequence of blowups and blowdowns with smooth centers. This provides an affirmative solution to a long standing factorization conjecture of birational maps.