UWashington-PIMS Mathematics Colloquium: Joseph H. Silverman

An abstract discrete dynamical system consists of a set X and a self-map f from X to itself. Dynamics studies the orbits of points of X under repeated application of f. When X and f are defined by polynomials, algebraic geometry comes into play, and when the polynomials have integer coefficients, number theory joins the game. In the first part of this talk, I will discuss how these three areas of mathematics, dynamical systems, algebraic geometry, and number theory, have come together in the past 30+ years to form the active new field of arithmetic dynamics. In the second part I will give a taste of arithmetic dynamics through a recent conjecture which says that in the number theory setting, there are lots of orbits of rational points and that they are widely spaced throughout the space X.