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
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
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
On spaces and spectra associated with blocks of finite groups.
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
has a natural compactification (as a space of equivalence classes of
which is also (by a theorem of Davis, Januszkiewicz, and Scott)
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
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'
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
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
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.