SEMINAR:
Lectures on Growth in Groups and Graphs
Tue, 04 May 2010 12:00 - Maths Lecture Room 2 and Maths Lecture Room 3
Nick Gill
Nick Gill (University of Bristol) will be giving a series of three talks
on the topic of Growth in Groups and Graphs as follows:
* Tuesday 4 May, 12 noon, MLR2, I: Sum-Product
* Friday 7 May, 11am, MLR3, II: Growth in Groups of Lie Type
* Friday 14th May 11am MLR3, III: Escape
These are part of a UWA research collaboration award funding visits to
Perth by Nick Gill and Harald Helfgott. The aim of the talks is to give
the necessary background and introduction to research in the area. Nick
will give a further three talks later in the year. Nick has prepared a
page of supporting material at
https://www.maths.bris.ac.uk/~manpg/austlit.html
The titles and abstracts for the first three talks are as follows.
I: SUM-PRODUCT We introduce the idea of growth in groups, before
focussing on the abelian setting. We take a first look at the sum-
product principle, with a brief foray into the connection between sum-
product results and incidence theorems.
We then focus on Helfgott’s restatement of the sum-product principle in
terms of groups acting on groups.
II: GROWTH IN GROUPS OF LIE TYPE Since Helfgott first proved that
“generating sets grow” in SL_2(p) and SL_3(p), our understanding of how
to prove such results has developed a great deal. It is now possible to
prove that generating sets grow in any finite group of Lie type; what is
more the most recent proofs are very direct – they have no recourse to
the incidence theorems of Helfgott’s original approach.
We give an overview of this new approach, which has come to be known as
a ”pivotting argument”. There are five parts to this approach, and we
outline how these fit together.
III: ESCAPE The principle of “escape from subvarieties” is the first
step in proving growth in groups of Lie type. We give a proof of this
result, and its most important application (for us) – the construction
of regular semisimple elements.
We then examine other related ideas from algebraic geometry, in
particular the idea of non-singularity.
For more information:
Michael Giudici
giudici@maths.uwa.edu.au
Starts : Tue, 04 May 2010 12:00
Ends : Fri, 14 May 2010 12:00
Last Updated : Thu, 29 Apr 2010 11:28