Nickolas Rollick, Department of Pure Mathematics, University of Waterloo
"Approximation Constants for Closed Subschemes of Projective Varieties"
Diophantine approximation is a branch of number theory with a long history, going back at least to the work of Dirichlet and Liouville in the 1840s. The innocent-looking question of how well an arbitrary real algebraic number can be approximated by rational numbers (relative to the size of the denominator of the approximating rational number) took more than 100 years to resolve, culminating in the definitive Fields Medal-winning work of Klaus Roth in 1955.
Much more recently, David McKinnon and Mike Roth have re-phrased and generalized this Diophantine approximation question to apply in the setting of approximating algebraic points on projective varieties defined over number fields. To do this, they defined an ``approximation constant", depending on the point one wishes to approximate and on a given line bundle. This constant measures the tradeoff between the closeness of the approximation and the arithmetic complexity of the point used to make the approximation, as measured by a height function associated to the line bundle.
In this thesis, we generalize the notion of approximation constant to apply to approximations of arbitrary closed subschemes, as well as introducing a related approximation constant to measure the quality of a ``generic" approximation (allowing us to disregard unusually good approximations along certain Zariski-closed subsets). Much of the subsequent discussion revolves around computing lower bounds and exact values for the approximation constants in various cases, and making conjectures around how things should behave in general.
The presentation will focus on motivating the history behind the topic of this thesis, before moving on to the results of various approximation constant computations and giving some idea as to the methods of proof involved. We will conclude with a brief discussion of the simplest case that does not yield to these proof techniques (thus forming a likely starting point for future work on the topic).