THE MAHLER LECTURES (AUSTMS) Speaker: Prof Peter Sarnak, School of Mathematics, Institute for Advanced Study Title: Thin integer matrix groups and the affine sieve Location: Room V206, Mathematics Building (Callaghan Campus) The University of Newcastle Access Grid Venue: Other Time and Date: 3:30 pm, Fri, 12th Aug 2011 Access grid talk from the University of Western Australia Abstract: Infinite index subgroups of integer matrix groups like $SL(n,Z)$ which are Zariski dense in $SL(n)$ arise in geometric diophantine problems (e.g., integral Apollonian packings) as well as monodromy groups associated with families of varieties. One of the key features needed when applying such groups to number theoretic problems is that the congruence graphs associated with these groups are "expanders". We will introduce and explain these ideas and review some recent developments especially those connected with the affine sieve. [Permanent link]