Pure Mathematics Seminars Abstracts 1999
Martin Otto
January 6th 1999
The robust decidability properties of modal logics have recently been contrasted with a number of suprisingly strong undecidability results concerning their counterparts for logics with two variables. These two-variable logics form natural extensions of the underlying modal logics, with some desirable but still very limited additional first-order closure properties. The proofs of the undecidability results rest on logical reductions of domino problems, thereby illustrating the versatility of domino problems for undecidability proofs in logic.
Arthur Apter
January 11th 1999
The Axiom of Determinacy (AD), which states that certain types of two person games of infinite length played with sets of reals always have a winning strategy for either player, has profound implications in measure theory and set theory. Some of AD's consequences include that all sets of reals are Lebesgue measurable, all sets of reals satisfy the property of Baire, every uncountable set of reals contains a perfect subset, and in set theory, the existence of many measurable cardinals. These consequences, along with the consistency of AD, will be discussed.
Chris Chalk
March 15th 1999
In 1975 Dave Johnson gave a talk at UEA which posed the problem of determining the finiteness or infiniteness of certain cyclically presented groups. This is a follow-up to this talk! and describes how well known techniques used in the proof of the 4-colour theorem can be adapted to answer Dave's problem in certain cases (and without requiring use of a computer). The subject matter should be accessible to non-group theorists/undergraduates etc.
Dan Velleman
May 3rd 1999
The concepts of continuity and differentiability of functions of two variables are designed to tell us about the local behavior of a function near a point, where "local" is defined by reference to the standard topology on R2. Partial derivatives, in contrast, only tell us about the behavior of the function at nearby points in the coordinate directions. This suggests that, although the standard topology is useful for studying continuity and differentiability, it may not be the best topology to use to study partial derivatives. In this talk I will propose a different topology that may be more useful for studying partial derivatives, and investigate some of its properties. In some respects this topology turns out to be strikingly different from the standard topology.
William Zwicker
May 3rd 1999
(This work is joint with A. Taylor).
We consider the type of "yes-no" voting system that is used to pass legislation and that can be modelled by a simple game G? What does it mean to say that one voter, or one coalition of voters, has less influence than another? For individual voters, the standard desirability relation provides the only reasonable answer. Additionally, this relation is transitive and sometimes detects that the underlying voting system is not "weighted voting" by producing a cycle.
Comparing the influence of coalitions is more subtle. Lapidot and Winder proposed different coalitional relations. Each has intransitive strict and equivalence forms, and is better at detecting nonweightedness (but not perfect). The recently introduced trading versions of these relations do detect every instance of nonweightedness, and also have transitive strict forms. Transitivity of "equivalent influence" proves, however, to be an impossibility for any relation that one might reasonably call a coalitional desirability. The proof uses the Rudin-Keisler ordering -- a notion borrowed from the theory of ultrafilters.
Nikolai Gordeev
June 7th 1999
In 1951 O.Ore conjectured that every element in a finite simple group is a single commutator. This talk is devoted to some constructions in linear groups, which allowed to solve this problem for all finite groups of Lie type over fields containing more than 8 elements. In particular, the analogue of the well known Gauss decomposition of matrices is considered in the case of Steinberg-Chevalley groups.