Title:  Expanders, groups and applications

		   Eyal Rozenman
   Center for the Mathematics of Information
       California Institute of Technology

Abstract: Expanders are graphs which behave randomly in the following
sense: A random walk on an expander forgets its starting point very
fast. *Random* graphs have this property, but expanders can be
constructed very efficiently and with no randomness. One way to build
expanders is to use groups with good generating sets. Different groups
yield expanders with different properties, and we will give some
examples. The simple "forgetful random walks" property of expanders is
extremely useful. We will describe the following applications:

- Constructions of good error correcting codes, and the relation to
expanders on the group (Z_2)^n.
- Deterministic simulation of a random walk a graph and its
application for solving connectivity questions on the graph.
- Constructing Permutations of {1,2,...n} that look random as long as
you look only at k of their values, and the relation to expanders on the
group of permutations of {1,2,...n}.
- How to generate a random element in a group given only generators of
the group, and the relation to expanders on the group of symmetries of
infinite trees.