Type: Seminar

Distribution: World

Expiry: 12 Sep 2011

CalTitle1: Infinite Groups Seminar: Cartwright -- Enumerating the fake projective planes

Calendar2: 12 Sep 2011 1500-1600

CalLoc2: AGR Carslaw 829

CalTitle2: Infinite Groups Seminar: Cowling -- Powers of Random Matrices

Auth: athomas(.pmstaff;2039.2002)@p615.pc.maths.usyd.edu.au

The next Infinite Groups Seminar will be on Monday 12 September at the University of Sydney, with Donald Cartwright speaking at 12 noon and Michael Cowling at 3pm. Details are below. ---------------------------------------------------------------------------- Date: Monday 12 September Time: 12 noon Location: Room 707A, Carslaw Building, University of Sydney Speaker: Donald Cartwright (University of Sydney) Title: Enumerating the fake projective planes Abstract: A fake projective plane is a smooth compact complex surface P which is not biholomorphic to the complex projective plane, but has the same Betti numbers as the complex projective plane, namely 1, 0, 1, 0, 1. A fake projective plane is determined by its fundamental group, In their 2007 Inventiones paper, Gopal Prasad and Sai-Kee Yeung showed that these fundamental groups are the torsion-free subgroups \Pi, with finite abelianization, of index 3/\chi(\bar\Gamma) in a maximal arithmetic subgroup \bar\Gamma of PU(2,1). They show that only a small number of \bar\Gamma can arise, list them explicitly, and found many of the possible subgroups \Pi. Making heavy use of computers, Tim Steger and I have found all the possible groups \Pi, for all of these \bar\Gamma’s, by finding explicit generators and relations for each of these groups \bar\Gamma. We have therefore found all the fake projective planes. It turns out that there are, up to homeomorphism, exactly 50 of them (100 up to biholomorphism). The fundamental group of Mumford’s original fake projective plane will be identified. ---------------------------------------------------------------- Date: Monday 12 September Time: 3pm Location: Room 829 (Access Grid Room), Carslaw Building, University of Sydney Speaker: Michael Cowling (University of NSW) Title: Powers of Random Matrices Abstract: If we select an n by n orthogonal matrix X "at random", using the uniform distribution on the orthogonal group O(n), then the powers of X are not uniformly distributed in O(n). However, as n increases, the distribution of X^n stabilizes. We prove this, consider generalizations to matrices in other compact Lie groups, and make some remarks about random matrices in other Lie groups. ----------------------------------------------------------------