Kempner Colloquium Abstracts

In their 1963 paper, Dixmier-Douady gave a geometric realization of the integral degree 3 cohomology of a space in terms of certain bundles of *-algebras, analogous to the usual interpretation of degree 2 cohomology as the isomorphism classes of complex line bundles. These Dixmier-Douady bundles define twistings of the K-theory of spaces, and in this context have been the subject of much interest in recent years. After reviewing these concepts, I will explain a functorial construction producing important examples, such as the "spin" Dixmier-Douady bundle over a compact Lie group. (Based on joint work with A. Alekseev, arXiv:0907.1257.)

Associated with every Gaussian measure there is a Hilbert space on which it would like to live. However, in an infinite dimensional setting, the measure does not fit on this Hilbert space and one must move to a Banach space in which the Hilbert space is embedded as a dense subspace of measure 0. Nonetheless, many correct predictions can be made by pretending that the measure does live on the Hilbert space. For example, Feynman's representation of solutions to Schrödinger's equation can be viewed as a particularly successful example of such predictions. On the other hand, not all such predictions are correct. Indeed, the centerpiece of this lecture will be an ergodic theorem, proved in a different context by I.M. Segal, which holds only because Gaussian measures do not fit on the Hilbert space.

For a global quotient orbifold Q, that is, the quotient of a manifold by the action of a finite group, Tamanoi introduced for each group Γ a decomposition of Q into singular strata that generalizes the inertia orbifold. This definition introduces a large family of new invariants for orbifolds; indeed, one can apply any orbifold invariant to the decomposition associated to a group Γ, resulting in the so-called "Γ-extension" of this invariant. For example, the stringy orbifold Euler characteristic is the extension of the Euler-Satake characteristic associated to the group Γ = Z².

In this talk, we will discuss generalizations of this construction to arbitrary orbifolds, which are presented by a certain class of groupoids. We will discuss invariants of orbifolds associated to these sector decompositions as well as their applications. We will in particular consider the class of wreath symmetric products of orbifolds, which generalize the symmetric product construction for manifolds.

(This is joint work with Carla Farsi.)

A cubic form with integer coefficients and enough variables has at least one non-trivial 'zero' whose coordinates are all integers.

In order to compute such a point it is desirable to have a bound on the size of the coordinates in terms of a bound on the size of the coefficients of the form.

I shall give a short history of this problem and of my part in it, an overview of the methodology to obtain such a result and, time permitting, estimate the number of zeros of a cubic form defined over a finite field, a consideration vital to the overall approach.

No specialised knowledge of Number Theory , Fourier Analysis or Geometry will be required.

My talk will focus on some operator algebraic ideas motivated by quantum statistical mechanics, in particular the notion of KMS state. We shall see that for a large class of models, namely gauge automorphism groups on approximately proper equivalence relations, KMS states are exactly Gibbs states as defined by Dobrushin, Lanford and Ruelle. Further models will be presented in the C^*-algebraic framework: expansive dynamical systems, which have unique KMS state, and the Bost-Connes system from number theory, which exhibits a phase transition.

In [Bialgebras of type one, 1978], Warren Nichols introduced a family of algebras that initially received little attention. Perhaps MathSciNet gives a clue as to why, calling the paper, "a fairly technical investigation into the structure of certain sorts of Hopf algebras." No surprise that it took more than a decade, and independent discovery by others, for Nichols' algebras to find purchase in the Hopf algebra community.

Today, Nichols algebras play a critical role in the classification of finite dimensional pointed Hopf algebras. They also have a perfectly natural (and non-technical!) definition. I will start here, then give an overview of the ongoing classification program. The ubiquitous Cartan matrices make an appearance.

The "eta invariant" of Atiyah, Patodi and Singer is an example of a 'secondary' index invariant associated to an elliptic operator (such as a Dirac operator on an odd-dimensional Riemannian manifold). In this talk I will describe some recent work with Higson which sets the eta invariant in the context of a certain exact sequence relating the small and large scale structure of manifolds and their universal covers, which we call the "analytic surgery sequence". The presentation will be accessible to graduate students.