Colloquium
3:30 p.m., Friday
Math 100
Professor David Boyd
Department of Mathematics
UBC
Mahler's measure and the volume of hyperbolic manifolds
The logarithmic Mahler measure m(P) of a polynomial P(x,y) is the
average of \logP(x,y) over the unit 2torus. Given a knot K,
Cooper, Culler, Gilet, Long and Shalen have defined a certain
polynomial invariant A_K(x,y) of K which is defined in terms of
representations of the fundamental group of K. For hyperbolic
knots, the complement of K can be given the structure of a 3manifold
which has a welldefined volume V(K), and it appears that \pi m(A_K)
has an interesting relationship with V(K). For certain simple knots,
\pi m(A_K) = V(K), but the general picture is considerably more
interesting. We will describe how m(A_K) seems to be related to
another invariant of the knot, the Borel regulator, an arithmetic
invariant introduced by Neumann and Yang. We will present a precisely
formulated conjecture supported by a number of highly accurate
computations.
Refreshments will be served in Math Annex Room 1115, 3:15 p.m.
