SEMINAR:
Lectures on Growth in Groups and Graphs Part II
Wed, 21 Jul 2010 11:00 - Maths Lecture Room 2
Nick Gill
Nick Gill (Bristol)
will continue his series of lectures on growth in groups and graphs
at the following times
Wednesday 21/7 at 11am
Friday 23/7 at 11am
and Tuesday 27/7 at 1pm
All seminars will be held in MLR2
I: Growth in soluble subgroups of GL_r(p)
Abstract: We show how to reduce the study of exponential growth in
soluble subgroups of GL_r(p) to the nilpotent setting. We make use of
ideas based on the sum-product phenomenon, as well as some machinery
from linear algebraic groups. We will not assume any background from
these areas.
II: An introduction to expanders
Abstract: This is a background seminar preparing the way for Seminar
III, where we connect results on growth in simple groups to the explicit
construction of families of expanders. In this seminar we will define
what we mean by a family of expanders, stating (and sometimes even
proving!!) background results that will be important later.
We will also try to explain why expanders are of such interest to so
many different groups of people.
III: Using growth results to explicitly construct expanders
Abstract: We outline the method of Bourgain and Gamburd. They were the
first to use results concerning growth in simple groups to explicitly
construct expander graphs. Let S be a set in SL_2(Z) and define S_p to
be the natural projection of S modulo p. Now let G_p be the Cayley graph
of SL_2(F_p) with respect to the set S_p. Bourgain and Gamburd give
precise results as to when the set of graphs {G_p : p a prime} forms a
family of expanders. They make crucial use of the result of Helfgott
(encountered in an earlier seminar) which states that "all generating
sets in SL_2(p) grow".
All welcome
For more information:
Michael Giudici
giudici@maths.uwa.edu.au
Starts : Wed, 21 Jul 2010 11:00
Ends : Tue, 27 Jul 2010 13:45
Last Updated : Wed, 08 Sep 2010 12:42