Brett Nasserden, Department of Pure Mathematics, University of Waterloo
"The valuative tree in action"
To kick off the learning seminar we will investigate how valuative methods can be used to study the dynamics of polynomial endomorphisms of the affine plane. Specifically, we will introduce the valuative tree and related dynamical notions such as the dynamical degree of a morphism. With the preliminaries in place we will sketch why a dominant polynomial endomorphism of the affine plane defined over an algebraically closed field of characteristic zero has a point with a forward dense orbit if it does not preserve a non-constant rational function. This result is known if the base field is uncountable, therefore the arguments presented here will be most interesting to those willing to work over countable algebraically closed fields such as the algebraic closure of the rational numbers.
MC 5479