Number theory deals with properties of the integers, rings of algebraic integers, and a variety of arithmetic objects such as elliptic curves. Many crowning achievements of the human intellect can be found in this beautiful branch of mathematics, with the tradition dating back to the ancient Greeks. The number theory community in the Pure Mathematics department at the University of Waterloo includes seven regular faculty members:
- Kevin Hare
- Wentang Kuo
- Yu-Ru Liu
- David McKinnon
- Michael Rubinstein
- Cameron Stewart
- Jerry Xiaoheng Wang
- as well as postdocs and graduate students
Events
Approximation of rational points and a characterization of projective space
Akash Sengupta, Department of Pure Mathematics, University of Waterloo
Given a real number x, how well can we approximate it using rational numbers? This question has been classically studied by Dirichlet, Liouville, Roth et al, and the approximation exponent of a real number x measures how well we can approximate x. Similarly, given an algebraic variety X over a number field k and a point x in X, we can ask how well can we approximate x using k-rational points? McKinnon and Roth generalized the approximation exponent to this setting and showed that several classical results also generalize to rational points algebraic varieties.
In this talk, we will define a new variant of the approximation constant which also captures the geometric properties of the variety X. We will see that this geometric approximation constant is closely related to the behavior of rational curves on X. In particular, I’ll talk about a result showing that if the approximation constant is larger than the dimension of X, then X must be isomorphic to projective space. This talk is based on joint work with David McKinnon.
MC 5417