ABSTRACT: The two fundamental ``machines'' of 
non-commutative geometry are cyclic homology and 
(bivariant) topological K-theory. We describe these two 
theories and their connections. Cyclic theory can be viewed as a far
reaching generalization of the classical de Rham cohomology, while
bivariant K-theory includes the topological K-theory of
Atiyah-Hirzebruch as a very special case.

The classical commutative theories can be extended to a degree of
generality which is quite striking. It is important to note however
that this extension is by no means simply based on generalizations of
the existing classical methods. The constructions are quite different
and give, in the commutative case, a new approach and an unexpected
interpretation of the well-known classical theories. One aspect is
that some of the properties of the two theories become visible only in
the non-commutative category. For instance, both theories have certain
universality properties in this setting. There is a multiplicative
transformation (the bivariant Chern-Connes character) from bivariant
K-theory to bivariant periodic cyclic theory. This transformation is
a vast generalization of the classical Chern character in differential
geometry.

The modern approach to both theories is based on extensions of
algebras. We describe this approach and the construction of the
Chern-Connes character. The natural framework where all this works
smoothly is the category of topological algebras which are projective
limits of Banach algebra. Both theories can however be extended to
arbitrary complex algebras (without specifying a topology) provided
that they have a countable basis. An intriguing and fundamental
testing case is the Weyl algebra, i.e. the unital algebra with two
generators x and y satisfying the Heisenberg commutation relation
xy - yx = 1. Its K-homology is generated by a two-dimensional
class which is represented by an extension of length 2.

* THESE SPEAKERS ARE 
CANDIDATES FOR A POSITION 
IN THE DEPARTMENT.

ABSTRACT: In this talk, we present a new idea of establishing the isoperimetric and Sobolev inequalities 
with a consideration of the sharp constant. 
More precisely, we not only split the isoperimetric 
inequality into two geometric inequalities in terms of Hausdorff content, but also partition the Sobolev 
embedding inequality into two analytic inequalities
by using Choquet integral with respect to Hausdorff 
content.

ABSTRACT: The octonions are the largest of the four normed division algebras.  While somewhat unloved 
due to their nonassociativity, they
are important for understanding many "exotic structures"
in mathematics, such as the exceptional simple Lie groups.  The largest of the exceptional simple Lie groups is E_8, whose root lattice gives the densest lattice 
packing of spheres in 8 dimensions.  In this expository talk we show how to multiply octonions, describe the E_8 root lattice as a set of octonions closed under addition and multiplication, and show how this illuminates 
some features of E_8. 

ABSTRACT: I will be describing the main assumptions and 
results of an approach toquantum gravity called loop 
quantum gravity.  Recent proposals for using
planned astrophysical observations for experimental tests of quantum theories of gravity will be discussed.

ABSTRACT:   We survey some recent developments in 
the study of PDEs in which  differential geometry has 
played a fundamental role. First, we recall some properties widely considered as indicators of  integrability: existence 
of linearizing and/or Backlund transformations, 
existence of an infinite number of symmetries and/or conservation laws,  existence of linear problems for which 
the equation at hand is the  compatibility condition.
  Second, we introduce the class of equations of pseudo-spherical type, and show that these properties can 
be understood in geometric terms. 
  Third, we summarize four different applications: 
(a) We show that there exist transformations between 
arbitrary solutions of any two equations in the class of equations of pseudo-spherical type.
(b) We define hierarchies of equations in this class, and 
show that the transformations of (a) extend to this general framework. 
(c) We present comparison theorems relating some of the integrability notions mentioned at the beginning of the talk. 
(d) We consider three water wave equations, Camassa-Holm, Hunter-Saxton, and KdV. We show how to find their associated linear problems, construct conservation laws and "nonlocal symmetries", and also how these symmetries 
can be used to generate nontrivial solutions. 
  Fourth, we summarize some current research (involving symplectic geometry  and infinite-dimensional Lie groups) 
and state some open problems.

 ABSTRACT: The classical Center-Focus problem posed 
by H. Poincare in 1880's asks about the characterization of planar polynomial vector fields such that all their integral trajectories contain a fixed point (which is called a center). 
In my talk I describe a new general approach to the
Center Problem.

ABSTRACT: In this talk I will give an introduction to and overview of Higher-Dimensional Category Theory. 
The talk will be at a general level and aimed at 
mathematicians and scientists at any level from 
undergraduate upwards, and possibly others. 
In particular, no knowledge of Category
Theory will be assumed.

Category Theory is the study of mathematical structures in general.  It starts from the observation that a collection of mathematical structures, together with structure-preserving maps between them, is itself a mathematical structure that 
can be studied in its own right, yielding
greater insight into the structures we started with.  Category Theory provides a language and framework for studying structures in this way. What is Higher-Dimensional Category Theory?  This is a good question; in fact most research in this field is still concerned with answering this apparently basic question.  We try to generalise the framework above to
enable greater expressive possibilities as demanded by studying the rest of mathematics.  However, this is not a straightforward process; at least 15 different theories are proposed in the literature, and the relationship
between them is far from clear.  I will discuss the issues involved with moving into higher dimensions, as well as where the subject currently is, where it is going, and why it is 
a good idea to go there.

Abstract: The problem of quantizing general relativity is a 
focus for research into the problem of quantum geometry. 
We will survey some directions, then present results of our own research. 

Abstract: The rational homotopy groups of a simply-connected space Y assemble
into a Lie algebra, with Lie bracket given by the Samelson product.
Suppose there is a connected space X such that the rational cohomology
algebra of Y is a k-rescaling of the rational cohomology algebra of X,
and, moreover, the latter algebra is Koszul.  Then, the homotopy Lie
algebra of Y equals, up to k-rescaling, the rational Lie algebra
associated to the lower central series of pi_1(X).  If Y is a formal
space, this equality is actually equivalent to the Koszulness of
H^*(X,Q). The "Rescaling Formula" can be upgraded to a "Malcev
Formula," at the level of filtered groups, precisely when X is formal.

Among spaces that admit naturally defined homological rescalings are
complements of complex hyperplane arrangements, and complements of
classical links. The Rescaling Formula holds for Koszul (in
particular, for fiber-type) arrangements, as well as for links with
connected linking graph.  On the other hand, for generic arrangements
the Rescaling Formula fails, while for links, the Malcev Formula can 
fail even when the Rescaling Formula does hold.

This is joint work with Stefan Papadima (Institute of Mathematics of the Romanian Academy). 

Abstract:I will describe loop quantum gravity in 3 + 1 dimensions in the
case  of a positive cosmological constant and will present both old and 
new results which support the case that this theory provides a 
satisfactory quantum theory of gravity.  These include the existence of 
a ground state which both is an exact solution to the constraints of 
quantum gravity and has a semiclassical limit which is deSitter 
spacetime.The long wavelength excitations of this state are studied 
and are shown to reproduce both gravitons and, when matter is 
included, quantum field theory on deSitter spacetime.  Furthermore, 
one may derive Planck scale, computable corrections to the 
energy-momentum relations for matter fields.  This may lead in the next 
few years to experimental tests of the theory.

To study the excitations of the Kodama state requires the use of 
spin networks involving representations of a *quantum* group, with 
the quantum deformation arising because of the nonzero cosmological
constant.

The talk is designed to be an introduction to loop quantum gravity,
requiring no prior knowledge of the subject.  The deep relationship
between quantum gravity and topological field theory is stressed
throughout.
 

Abstract: In the calculus of real variables, one can use a Taylor series to approximate a function about a point within the radius of convergence using only the derivatives of the function at that point.  Goodwillie calculus is a way of using this idea to approximate FUNCTORS rather than functions.  For many functors F, Goodwillie calculus provides a notion of
the degree n approximation to F (analogous to the n-th partial sum of the Taylor series) and the derivatives of F.  However, in contrast with the Taylor series analogy, the Taylor tower of F may not always be computed by simply knowing the values of each of the derivatives of F.  We will describe a theorem which allows us to compute the Taylor tower from the derivatives of F, and show how this theorem recovers some well known
results such as the Poincare-Birkhoff-Witt theorem.

In this talk, we will give an introduction to derived categories of rings as they arise in the representation theory of finite groups and finite-dimensional algebras. A central role is played by the notion of
derived equivalence [1]. Following [5], [3], [4] we will then define the derived Picard group of an algebra A. It should be thought of as the `group scheme' of automorphisms of the derived category.  The Lie
algebra of this `group scheme' will turn out to be the Hochschild cohomology of the algebra A, where the Lie bracket is the Gerstenhaber bracket [2].  It will follow that derived equivalences preserve the
Gerstenhaber bracket. More refined techniques allow one to extend this result to all `reasonable' actions of an operad on the Hochschild complex.

[1] J. Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), 436-456.

[2] M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. (2) 79 (1964), 59-103.

[3] R. Rouquier, A. Zimmermann, Picard groups for derived module categories, preprint, 1998, to appear in Proc. London Math. Soc.

[4] A. Yekutieli, Dualizing complexes, Morita equivalence and the derived Picard group of a ring, Preprint, 1998, math/9810134.

[5] A. Zimmermann, Derived equivalences of orders, Canadian Math. Soc. Conference Proceedings 18 (1996), 721-749.