Lie Groups, Representations and Discrete Mathematics
(2005-2006 Academic Year)
In recent years new and important connections have emerged between discrete
subgroups of Lie groups, automorphic forms and arithmetic on the one hand, and
questions in discrete mathematics, combinatorics, and graph theory on the other.
One of the first examples of this interaction was the explicit construction of
expanders (regular graphs with a high degree of connectedness) via Kazhdan's
property T or via Selberg's theorem (lambda1 is greater than 3/16). Some other
important fruits of this interaction were the construction of Ramanujan graphs,
using the Jacquet-Langlands correspondence and Deligne's theorem on Hecke
eigenvalues (Ramanujan conjecture), construction of new finitely presented
simple groups via ergodic theory of lattices (a la Margulis) in a product of two
trees, and a conceptual approach to the Product Replacement Algorithm of
computational group theory (the convergence properties of the algorithm are
related to the question of whether automorphism groups of free groups have property
τ). In each of these cases, an unexpected application to discrete mathematics
was found by using ideas from number theory, Lie groups, representation theory
and ergodic theory. Applications flow in the reverse direction as well.
A new combinatorial construction of expander graphs was used recently to resolve a
group theoretic question on expansion in Cayley graphs.
One reason these connections have been slow to emerge is that the fields
involved are quite far apart, at least from the traditional viewpoint. People
working in one side of the Lie Group/Discrete Math dichotomy are often not
aware of the relevance of their work to the other side. Furthermore, each field
has its own language and conceptual framework, so there is often a formidable
language barrier to communication.
The program will be led by Alexander Lubotzky of The Hebrew University of Jerusalem
who will be in residence at the Institute for the academic year. The goal of the program is
to bring together mathematicians from several of the above-mentioned areas in order to
strengthen the ties between the fields and generate further collaborations.
Information about a workshop in a related topic, "Automorphic Forms, Group Theory and Graph Expansion" to be held in February 2004 at IPAM can be found at: http://www.ipam.ucla.edu/programs/agg2004/.
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