One model for the braid group is: the fundamental group of a
complex vector space, minus some hyperplanes, modulo a group generated
by reflections across them. We call any group arising from this construction
"braid-like". We are mostly interested in infinite arrangements in (for example)
the complex ball, with the ultimate goal of proving the "monstrous proposal".
That is: a particular braid-like group (coming from the complex 13-ball), modulo
the squares of the braid-like generators, is (almost the same as) the monster
finite simple group. This will require finding generators and relations for this
braid-like group, and the subject of the talk will be how we found generators.
(We don't know the relations.) The method we used can be applied to other
hyperplane arrangements, giving a general tool for finding generating sets for
Higher dimensional analogues of fake projective planes
July 17, 2014 (Thurs.), 15:30 am -
Room 204, RIMS, Kyoto University
Gopal Prasad (Univ. of Michigan)
A fake projective plane is a smooth projective complex algebraic surface which is not isomorphic to the complex projective plane but whose Betti numbers are that of the complex projective plane. The fake projective planes are algebraic surfaces of general type and have smallest possible Euler-Poincare characteristic among them. The first fake projective plane was constructed by D.Mumford using p-adic uniformization, and it was known that there can only be finitely many of them. A complete classification of the fake projective planes was obtained by Gopal Prasad and Sai-Kee Yeung. They showed that there are 28 classes of them,and gave at least one explicit example in each class. Later, using long computer assisted computations, Cartwright and Steger found that the 28 families altogether contain precisely 100 fake projective planes. Using the work of Prasad and Yeung, they also found a very interesting smooth projective complex algebraic surface whose Euler-Poincare characteristic is 3 but whose first Betti number is 2. Prasad and Yeung have developed a notion of higher dimensional analogues of fake projective planes and to a large extent determined. The talk will be devoted to an exposition of this work.