April 10, 2014



Mathematics Colloquium lecture April 10

By Reta McDermott

Igor Shparlinski, University of New South Wales in Australia, will present "Distribution of Points on Varieties Over Finite Fields" as part of the mathematics department Colloquium lecture series at 2:30 p.m. Thursday, April 10, in 102 Cardwell Hall.

The abstract for the lecture is:

We give a survey of recent results on the distribution of rational points on algebraic varieties that belong to a given box. Generally, these results fall into two categories:

-large boxes, where asymptotic results are possible;
-small boxes, where only upper bounds are possible.

We discuss results of both types and also indicate the underlying methods. Some of these work for very general varieties, some apply only to very specific varieties such as hypersurfaces x_1^{-1} +....+ x_n^{-1} = a that appear in the Erdös-Graham problem or "hyperbolas" x_1x_2...x_n = a.

Besides intrinsic interest, the aforementioned results have a wide scope of applications in number theory and beyond. These include: bounds of Kloosterman sums, shifted power identity-testing, distribution of elliptic curves in isogeny and isomorphism classes, polynomial dynamical systems in finite fields, and several others.

We conclude with several open problems.