University of Hawaii Mathematics Department

Distinguished Lecture Series

February 21, 22, 23, 2007


The Distinguished Lecture Series is a series of three lectures. The first lecture is intended for a general audience. This year the lectures will be given by:


Martin R. Bridson

Professor of Pure Mathematics
Imperial College, London


A poster advertising the event created by Ryan Smith.


The language of symmetry and the grammar of space

Wednesday, February 21, 2007 - 4:30 pm - Crawford 105
Reception 3:30 pm - Campus Center Room 203E

Beauty all around us is connected with the presence or subtle absence/breaking of symmetry. *Groups* are the mathematical objects that afford a precise language in which to explore symmetry. I shall explain some elements of the modern theory of groups, relying heavily on visual examples. In this context, I shall address the fundamental question:

How can one quantify complexity? How can one quantify the "hardness" of a problem, or the "subtlety of a space".

I shall explain some of the mathematics associated to answering such question, focussing in particular on the problem of navigating in 2- and 3-dimensional spaces of varying geometry.

Curvature, complexity, and the universe of finitely presented groups

Thursday, February 22, 2007 - 4:30 pm - Keller 403
Refreshments 3:00 pm - Keller 418

This is a version of the Invited Lecture that I gave at the ICM in Madrid this summer.

A universe of finitely presented groups is sketched and explained, leading to a discussion of the fundamental role that manifestations of non-positive curvature play in group theory. The geometry of the word problem and associated filling invariants are discussed. The subgroup structure of direct products of hyperbolic groups is analysed and a process for encoding diverse phenomena into finitely presented subdirect products is explained. Such an encoding is used to solve problems of Grothendieck concerning profinite completions and representations of groups. In each context, explicit groups are crafted to solve problems of a geometric nature.

Automorphism groups of free, free-abelian and surface groups, and actions on CAT(0) spaces
Friday, February 23, 2007 - 4:30 pm - Keller 403
Refreshments 3:00 pm - Keller 418

In this talk I shall present the content of several papers that are in preparation and which concern the actions of the above groups on spaces of geometric interest.

There will be an initial focus on CAT(0) spaces, but the discussion is largely of a topological nature, involving dimension theory and Smith theory. Sample theorems include: SL_n(Z) cannot act without a fixed point on any complete CAT(0) space of dimension less than n-1, and SAut(F_n) has no non-trivial action whatsoever on any Z_2-homology sphere of dimension less than n-1. A less difficult but rather lovely fact is that among the wide diversity of semisimple actions of Out(F_3) on CAT(0) spaces one has the following common feature: if the Nielsen generators do not have fixed points, then the maximal-rank abelian subgroups must act discretely, each stabilizing an isometrically embedded copy of Euclidean 3-space where the action has as fundamental domain the rhombic dodecahedron.


Location:

For the location of the relevant buildings (Keller Hall, Campus Center, Crawford Hall) consult the campus map.


The Sponsors:

The lectures are sponsored by the National Science Foundation, and the Department of Mathematics and the College of Natural Sciences of the University of Hawaii, Manoa.