Peter Sarnak (Princeton University and Institute for Advanced Study)
gives two talks about "Diophantine Analysis and Groups". Talks on youtube.

October 31, 2019:

Lecture I 4:15-5:15 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 sub-variety has been understood for some time (Lang, Laurent). After a brief
review of some of the tools in the analysis and their recent extensions
(André-Oort 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 4:14-5:15 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 one-commutators in sl_{2} over such rings.

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

This is a lecture series in honor of
Lars Ahlfors (1907-1996)
who was William Caspar Graustein Professor of Mathematics at
Harvard University from 1946 to 1977. Ahlfors won the fields medal in 1936 and
the Wolf prize in 1981.