Mathematics Colloquium
Eko Hironaka
Emeritus Professor at Florida State University
Title: Lehmer's number in Topology, Geometry and Dynamics
Date: Friday, March 28, 2025
Place and Time: Love 101, 3:05-3:55 pm
Abstract. A Salem number is an algebraic integer s with the property that all roots of its minimal polynomial P(x) besides s and 1/s have complex norm equal to one (i.e. lie on the unit circle). In 1933, after extensive computer search, Lehmer asked whether the larger real root of P(x) = x^10 + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1 (roughly 1.17625) is the smallest Salem number. The problem is still open. Lehmer's query has led to an on-going study of deep relations between number theory, and the study of the topology, geometry and dynamical properties of a variety of mathematical objects. In this talk we will discuss some results in these directions that have come to light in recent decades involving knot theory, hyperbolic geometry, Coxeter theory, and the dynamics of rational maps.