Infinite Groups Seminar: Cartwright, Cowling

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.

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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.

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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.

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