Monday, February 25 and Monday, April 7
Davidson Lecture
Hall at Claremont McKenna, 3:00 PM
Title: Mahler's measure, Lehmer's conjecture, and some connections - I and II
Abstract: Mahler's measure of polynomials in one variable with complex
coefficients is a function measuring the extent to which the roots
of a polynomial are distributed outside of the unit circle. I
will start by introducing Mahler's measure and stating a famous
conjecture of D. H. Lehmer about the values it assumes when restricted
to monic polynomials with integer coefficients. This conjecture has
great significance in mathematics, especially in Number Theory and
Ergodic Theory.
Lehmer's conjecture is closely connected to the so called
minimization problem for Salem numbers. In fact, any insight into the
size of the smallest possible Salem number would constitute a major
step towards Lehmer's conjecture. Salem numbers appear as largest poles
of the rational growth functions for certain Coxeter groups, and hence
are related to the asymptotic growth rates of these groups. This
fascinating connection, coming from the work of Cannon, E. Hironaka,
and others, provides additional evidence in support of Lehmer's
conjecture. I will review some results in this direction.
This will be an informal expository talk with all the necessary notation and background information provided.