Pure Maths Seminar Archive - Abstracts 2002
An algebraic Zd-action is an action of Zd on a compact abelian group X by continuous automorphisms of X. We describe various rigidity properties of algebraic Zd-actions on zero dimensional groups.
In this talk I will introduce some of the groups in which I am interested, namely the classical groups over a p-adic field, and try to give some idea of how to go about constructing (complex) representations of them, particularly the supercuspidal ones.
A problem in model theory leads one to consider extensions of infinite permutation groups by profinite modules. As with the extension problem for finite groups, these are classified by certain cohomology groups. I'll try to explain the problem, what some of the words mean and sketch part of a proof that, for some rather natural permutation groups and modules, these cohomology groups are finite.
In this talk (which is based on joint work with B.Kuelshammer and J.B.Olsson), we discuss how to define (from a fairly elementary approach) a "block theory" of the symmetric group for every positive integer l. As demonstrated by S.Donkin, this is related to the representation theory of quantum groups related to Hecke algebras of type A (though our approach is character-theoretic and self-contained).
Vector spaces, say over the rationals, algebraically closed fields of a given characteristic, and free groups are three familiar examples of classes of mathematical structures which are categorical in all uncountable cardinals, i.e. any two uncountable such structures of the same size are isomorphic. In each of these instances, there is a notion of dimension (linear dimension, transcendence degree, the number of generators) which captures the isomorphism-type of the structure. This is a general model-theoretic phenomenon:
Theorem: (Lessmann) If K is a reasonable class of mathematical structures which is categorical in some uncountable cardinal, then inside each mathematical structure there is a definable pregeometry whose dimension determines its isomorphism-type, and furthermore, the class is categorical in all uncountable cardinals.
By reasonable, we mean (1) axiomatized using at most countably many first order axioms (the first two examples above, in this case this is the classical Baldwin-Lachlan theorem), or, more generally, (2) axiomatized using not necessarily first order axioms but in such a way that there is a good notion of universal domain (a homogeneous model as in the example of free groups, or a full model, as in Shelah's excellent classes; we will illustrate this latter case with Zilber's pseudo-analytic structures). The difficulty in (2) is that the compactness theorem fails.
Time permitting, we will also discuss the use of dimension theory beyond categoricity: Inside any mathematical structure, we can define what we mean by "A is independent from B (over C)" using the automorphism group of the structure. Buechler and I showed that this independence relation has good properties under very general, not necessarily first order, model-theoretic circumstances, called simplicity and stability. Examples of stable and simple mathematical structures are those described above (in each case the independence relation becomes the usual one: in vector spaces it becomes linear independence, and algebraic independence in an algebraically closed field). Examples also include non-categorical structures such as Hilbert spaces, where the independence relation coincides with orthogonality.
There exists a large body of results about the arithmetic of linear recurrence sequences. These can be used to pose questions about sequences defined by higher order recurrences. I will survey recent results about elliptic divisibility sequences showing how some results seem harder to obtain whilst others are provably easier.
Let V be a finite dimensional vector space over a field K and let G be a finite linear group in GL(V). Then G acts naturally on the algebra S = K[V] of polynomial functions on V. Further, let R = SG be the algebra of invariants. The branch locus X in Spec S of the extension S/R is the set of ramified points. We consider the connection between codim X and such characteristics of S as homological dimension, defect and depth.
Let (W,S) be a Coxeter system for which W is finite and let H be the associated Hecke algebra. For each subset J of S, the subgroup WJ of W generated by J has a Coxeter system (WJ ,J ) and an associated Hecke algebra HJ . We investigate the use of the cell structure of W to determine various homomorphic images of the H-modules obtained by inducing appropriate HJ -modules, leading to an understanding of the representation theory of H.
If one needed to choose such a space, it would seem to be the closed unit interval. This is the compact, metrizable topology to which all topologies are compared. That is, most topology is done on completely regular spaces - those with the property that whenever a point lies in an open set, there is a continuous function into the unit interval taking the value 1 at the point, and 0 off the set.
But we are unintimidated, and present a challenger for the title.