
Ahlfors Lecture series
October 31  November 1, 2019, Organizers: YumTong Siu and HorngTzer Yau
Harvard University, Science Center Hall D
Speaker and Program
Peter Sarnak (Princeton University and Institute for Advanced Study)
gives two talks about "Diophantine Analysis and Groups".

October 31, 2019:
Lecture I 4:155:15 PM in SC Hall D

Lecture I:
"Applications of Points on Subvarieties of Tori"
Abstract:
The intersection of the division group of a finitely generated subgroup of a torus with an
algebraic subvariety has been understood for some time (Lang, Laurent). After a brief
review of some of the tools in the analysis and their recent extensions
(AndréOort conjectures), we give some old and new applications;
periodicity of Betti numbers, algebraicity of Painlevé equations, and
the additive structure of the spectra of quantum graphs.

November 1, 2019:
Lecture II 4:145:15 PM in SC Hall D
 Lecture II:
"Integer Points on Affine Cubic Surfaces"
Abstract:
The level sets of a cubic polynomial in four or more variables tends to have many
integer solutions, while ones in two variables a limited number of solutions.
Very little is known in the case of three variables. For cubics which are
character varieties (thus carrying a nonlinear group of morphisms) a
Diophantine analysis has been developed and we will describe it.
Passing from solutions in integers to integers in, say, a real quadratic
field there is a fundamental change which is closely connected to challenging
questions about onecommutators in sl_{2} over such rings.

A reception follows the Thursday lecture at 5:30 pm in
the Math Department common room.
