Pure Mathematics Summer 2012
All seminars will take place in the McCrea Building, Room 219, on Tuesdays at 2 pm, unless stated otherwise. Tea will be served after the seminar at 3pm in Room 237 of the McCrea Building. All are welcome!
1st May Wojciech Samotij (University of Cambridge): Independent sets in hypergraphs (abstract)
8th May Tim Burness (University of Southampton): Extremely primitive groups (abstract)
15th May Tony Skyner (University of Bristol): Classifying quasi-monomial droups (abstract)
29th May Riddhi Shah (Jawaharlal Nehru University, New Delhi): Some properties of distal groups (abstract)
Wojciech Samotij: Many important theorems and conjectures in combinatorics, such as the theorem of Szemeredi on arithmetic progressions and the Erdos-Stone Theorem in extremal graph theory, can be phrased as statements about families of independent sets in certain uniform hypergraphs. In recent years, an important trend in the area has been to extend such classical results to the so-called `sparse random setting'. This line of research has recently culminated in the breakthroughs of Conlon and Gowers and of Schacht, who developed general tools for solving problems of this type.
Tim Burness: A primitive permutation group is said to be extremely primitive if a point stabilizer acts primitively on each of its nontrivial orbits. By a theorem of Mann, Praeger and Seress, every finite extremely primitive group is either almost simple or of affine type, and the affine examples have essentially been classified. I will explain some of the main ideas in the proof of this theorem, and I will report on recent joint work with Cheryl Praeger and Akos Seress towards a complete classification.
Tony Skyner: Let rho in Irr(Gal(K/Q)) be a non-trivial irreducible complex representations of Gal(K/Q) and associate to this the Artin L-function L(s,rho). Artin conjectured that this function is analytic and proved this in the case when Gal(K/Q) is a monomial group. In 2006 Andrew Booker presented an algorithm to test Artin's conjecture, but only when Gal(K/Q) satisfies a certain condition, which can be thought of as being close to a monomial group. He called these groups almost monomial groups. The same algorithm can be used to test the Riemann hypothesis, though our group can satisfy a weaker condition. We shall call such groups quasi-monomial groups.
In this talk I shall present some properties of these groups and families of groups known to satisfy the conditions needed.
Riddhi Shah: A locally compact group is said to be distal if the closure of the orbit of each non-trivial element under the conjugacy action stays away from the identity of the group. We will discuss some properties of distal and pointwise-distal groups. We characterise pointwise distal groups in terms of behaviour of convolution powers of probability measure on it. We also relate distal groups with groups with polynomial growth and unimodularity of the group. We also discuss a necessary and sufficient condition for the group to be pointwise distal.