Nick Rollick, Department of Pure Mathematics, University of Waterloo
"Approximating projective subvarieties"
Given an algebraic number, how well can we approximate it by rational numbers, relative to the "complexity" of those rational numbers? Questions of this sort have long preoccupied number theorists. However, it was only recently that this question was formulated and studied geometrically. Identifying numbers with points on the projective line, the analogous geometric question is: given a projective variety X and a closed subvariety Z defined over $\overline{\mathbb{Q}}$, how well can we approximate Z with rational points on X, relative to the "complexity" of the points in question?
In this talk, I will discuss McKinnon and Roth's formulation of this question in terms of "approximation constants" (in the special case where Z is a single point), including lower bounds on these constants. Afterwards, I will outline a few results of my own, involving the computation of approximation constants for certain closed subvarieties of projective space.
MC 5403