Pure Mathematics Seminars Abstracts 1998
Ilijas Farah
September 16th 1998
In his 1959 book of problems, S.Ulam has suggested study of the "stability" of mathematical objects in a rather broad sense. In particular, he asked for which compact metric groups the "approximate automorphisms" are near to strict automorphisms. I will give some answers to this, and discuss the related questions. In particular, I will relate "instability" with the existence of pathological submeasures.
Mirna Dzamonja
October 12th 1998
The notion of universality appears in various branches of mathematics, and is a very natural one. Studying this notion from the abstract point of view, inside of model theory, we arrive to the problem of classifying first order theories according to how well they admit the existence of a universal model.
By some classical results in model theory, the question does not make much sense in the presence of GCH (a set theoretic axiom claiming that the cardinal arithmetic is as simple as it can be), so one is forced to work with independence results. After explaining the general methods, we shall see how all this can be applied to a problem from functional analysis, namely that of the existence of a universal uniform Eberlein compactum.
Peter Collins
November 2nd 1998
The talk will concern a central problem in topology and will be partly historical. Some theorems encompassing much of what has gone before will be presented and some hard unsolved problems discussed. The talk is designed to be accessible to the non-specialist mathematician.
Stephen Power
November 6th 1998
Inclusions of complex matrix algebras (Mn → Mm) are determined up to similarity by one statistic - multiplicity. It follows that locally matricial algebras are determined up to isomorphism by the generalised integer coming from the multiplicities of a subalgebra chain. This data (which is essentially K0 data) is only a partial invariant for locally finite algebras and it is of interest to determine additional classifying invariants, which of course have their origins in finite dimensional inclusions. I shall talk about the general problem area, which is motivated in part by the analysis of approximately finite operator algebras, and I shall focus in particular on certain sparse matrix algebras, and the classification of their inclusions.
Dan Segal
December 7th 1998
I'll try to explain what profinite groups are and why one wants to know about their closed subgroups. I'll talk about recent progress on an old problem about subgroups of finite index in finitely generated profinite groups: are they open?
Vladimir Bavula
December 14th 1998
Let K be an algebraically closed field of characteristic zero. Let Λ be the ring of (K-linear) differential operators with coefficients from a regular commutative Noetherian affine domain of Krull dimension 2 which is the tensor product of two regular commutative Noetherian affine domains of Krull dimension 1. A typical example is the second Weyl algebra A2 ≅ K[X1, X2, ∂/∂X1, ∂/∂X2]. The Gelfand-Kirillov dimension of a simple Λ-module is either 2 or 3. In the first case such a Λ-module is called holonomic.
The aim of the talk is to give a description (a classification) of simple holonomic Λ-modules. Basic results on differential operators, Gelfand-Kirillov dimension and localizations will be explained (recalled). As a consequence a classification of simple modules over the ring of differential operators with coefficients from a regular commutative Noetherian affine domain of Krull dimension 1 will be given.