Abstract
A simple permutation is one which does not map any proper non-trivial interval into an interval. We will discuss the enumeration of simple permutations (in terms of their generating function, and asymptotically) as well as their importance in analysing the structure of classes of permutations.
Abstract
The theory of linear recurrences has a rich history, and has a close connection to more applied topics such as primality testing, integer factorization and public-key cryptography. In this talk we focus on the problem of determining tendencies in the factorization of terms which appear in a particular class of such sequences; the so-called Lucas sequences. We discuss some of the more pertinent results on this matter, some related open problems, and recent progress being made.
Abstract
We will give a necessary and sufficient condition for the Gilbert HNN extension of inverse semigroups to be finitely generated and presented.
Abstract
A variational argument will be presented to establish the existence of periodic solutions of a system of Hamilton's equations for almost all values to the energy. (The analogous result for all energy values is easily seen to be false.) The energy levels for which existence is established are characterised as those at which the critical value (in the sense of linking in the calculus of variations) which is a monotone functions of the energy, is differentiable. (This is an idea of Michael Struwe adapted to our situation.)
Abstract
The multiplicative group of a finite field is cyclic. Additively, a finite field extension can also be viewed as a cyclic module. There are many questions regarding the existence of generators with further specified properties: some may be answered affirmatively provided the field cardinality, or degree of the extension, or characteristic is sufficiently large. The focus of the talk will be questions for which a complete unconditional answer either is known or may be feasible to obtain.
Abstract
For a sequence u, defined over a finite set, the block-complexity p(n) is the number of distinct blocks that appear in u. It is conjectured that if p(n)=O(n) for the sequence u then the corresponding b-ary expansion is either rational or transcendental. There is a similar conjecture for the continued fraction expansion except in this case the conclusion is:it is either a quadratic irrational or it is transcendental. We discuss recent results towards resolving these conjectures.
Abstract
We review the definition and fundamental properties of (word) hyperbolic groups. In particular, a finitely generated group is hyperbolic if and only if it has a Dehn algorithm. This implies that the word problem can be solved in linear time. A stronger result is that any word in the generators of a hyperbolic group can be reduced to a normal form (the so-called shortlex normal form) in linear time. Finally we shall report on a recent result of Holt and Epstein that the conjugacy problem in a hyperbolic group is solvable in linear time.
Abstract
The groups of Lie type are among the most important structures in modern mathematics. Examples of such groups include reductive Lie groups, reductive algebraic groups, and finite groups of Lie type (which include most of the finite simple groups). Many problems in the representation theory of groups of Lie type have been solved using computers (for example, by the CHEVIE group). In this talk, I discuss descriptions of these groups (via the Steinberg presentation or highest weight representations) that are useful for computation. I also give methods for dealing with the twisted groups using Galois cohomology and Lang's theorem.
This talk describes joint work with Arjeh Cohen, Sergei Haller, and Don Taylor.
Abstract
We discuss an analogue for curved arcs of the Kakeya problem for straight lines, which arises from Hörmander's conjecture about oscillatory integrals in the same way as the straight line case comes from the restriction and Bochner-Riesz problems. We first review the negative results due to Bourgain, then give an idea of some positive results achieved by geometric and combinatorial methods.