Alex Smith, Northwestern University
Given a polynomial in n variables with rational coefficients, a fundamental and very difficult problem in number theory is to try to find n-tuples of rational numbers where the polynomial is zero. In other words, we try to find rational points lying in the vanishing locus of the polynomial.
We can try to generalize this to arbitrary smooth submanifolds M of R^n. Given such a manifold, can we find rational points lying on it? This is generally too much to hope for, so we instead look for rational points as close to M as we can manage, subject to some bound on the joint denominator of the point's coordinates.
The best results in this direction are due to Beresnevich. For hypersurfaces, Beresnevich's results are known to be as strong as possible, as any stronger result would fail for the vanishing locus of certain quadratic polynomials.
In this talk, we will show that, by excepting these known counterexamples, one can strengthen Beresnevich's result. Our methods are dynamical, involving the application of Ratner's theorems for unipotent orbits, and we will show how our work relates to the dynamical resolution of the Oppenheim conjecture by Margulis.