Pure Maths Seminar Archive - Abstracts 2011
We use Schlage-Puchta's concept of p-deficiency and Lackenby's property of p-largeness to show that a group having a finite presentation with p-deficiency greater than 1 is large. What about when p-deficiency is exactly one? We also generalise a result of Grigorchuk on Coxeter groups to odd primes.
The group of permutions of the natural numbers can be written as an increasing chain of proper subgroups. The minimal length of such a chain is called the cofinality of the symmetric group. Its investigation is connected to the search for interesting types of subgroups, comparisons with other common cardinal characteristics of the real numbers and the computation of possible lengths in various models of set theory. There are some customised forcing extensions for increasing these lengths.
In the centre of this talk is a geometric characterisation of critical points of eigenvalue functions of the transition probability matrix of random walks on finite, vertex transitive graphs in the case of higher multiplicity. We will explain applications to Archimedean solids and to the Cayley groups of finite Coxeter groups. The results are joint work with Ioannis Ivrissimtzis.
(Joint with Jamshid Derakshan.) We give a detailed analysis of the topological structure of sets definable in the ring of adeles over a number field K, paying attention to difficult issues of uniformity in K. All definable sets are measurable (but not in general locally closed), and our method allows calculation of the measures in many cases. The general problem here is related to motivic integration.
I will describe the genesis and development of a first-year, compulsory course in problem solving for undergraduate mathematics students at Durham University. I will outline the course aims, its structure, how it runs and what contribution we think it makes to helping students make the transition from school mathematics to university mathematics. I'll talk about some of the problems (both difficulties encountered with the course, and mathematical puzzles tackled). I will put the course in a pedagogical context, but will avoid educational theorising.
A topological space X is said to be κ-resolvable (for κ a finite or infinite cardinal number) if X contains κ disjoint dense subsets. Most "nice" spaces (e.g. metric, or compact, or linearly ordered ones) are maximally resolvable in a precise sense. But there are countable regular (hence nice) spaces that are irresolvable, i.e. not 2-resolvable.
The aim of this talk is to review several recent results concerning this concept. We describe joint work with L. Soukup and Z. Szentmiklossy about a method that enables us to construct spaces with a large variety of resolvability properties. Then we present very recent joint work with M. Magidor about a purely set theoretic characterization of maximal resolvability of monotonically normal spaces, a natural class of spaces that includes both metric and linearly ordered ones.