Mahler\u27s Measure and Lehmer\u27s Conjecture

Mahler\u27s measure is a function defined on polynomials, which measures the extent to which their roots are distributed away from the unit circle. It is known to be a continuous function on polynomials with complex coefficients, however when restricted to polynomials with integer coefficients it is expected to have gaps in its values. This has been conjectured by D. H. Lehmer in 1933, and is to this day one of the famous open problems in Number Theory. Mahler\u27s measure and Lehmer\u27s conjecture have fundamental connections and applications within Number Theory as well as in other areas of mathematics, for instance ergodic theory. In this talk I will introduce Mahler\u27s measure and discuss some of its properties and applications. I will also talk about Lehmer\u27s conjecture. If time allows, I may also briefly talk about a higher-dimensional generalization of Mahler\u27s measure