Pure Mathematics Seminars Abstracts 1997
P. R. Hewitt
January 27th 1997
We will look at some ways cohomology can be used to attack problems in group theory. These problems concern issues that classically fall under the cohomological aegis - group extensions, module extensions, and their ilk - but the novelty lies in trying to determine properties of a group from information about small sections of the group. We will look at three different ways a group can be built up from small sections - groups that are residually finite, finitary, or profinite - and explore what happens to the cohomological information when we ``pass to the limit''.
N. Vavilov
February 3rd 1997
In the talk we describe a remarkable class of representations of Lie algebras and Chevalley groups and corresponding combinatorial structures which appear more or less everywhere, from combinatorics and Chevalley groups, to differential and algebraic geometry.
E. Kolev
February 17th 1997
Although it is a coding theory problem it could be considered as a pure combinatorial problem. K2(n,R) stands for minimal cardinality of a set C of binary vectors of length n with the property: for any binary vector z of length n there exists a vector x from C such that z and x differ in at most R coordinates.
P. Voutier
May 19th 1997
The Pell equation x2−Dy2=1, where D>1 is a square-free integer, is a simple example of a norm form equation. Despite its simplicity, this example illustrates some important phenomena which can occur with the integer solutions of norm form equations. There are infinitely many integer solutions, but these solutions all belong to a single family which arises in a very natural way.
This notion of a family of solutions can be generalized to all norm form equations and Schmidt has shown that the number of integer solutions must lie in only finitely many such families. It is even possible to get bounds for the number of these families and in some cases these bounds have a nice ``minimal'' dependence on the particular norm form. After an introduction which will hopefully be motivational, such bounds will be the subject of this lecture.