Integral points and curves on moduli of local systems

Junho Peter Whang
Princeton University
December 8, 2017
The classical affine cubic surface of Markoff has a well-known interpretation as a moduli space for local systems on the once-punctured torus. We show that the analogous moduli spaces for general topological surfaces form a rich family of log Calabi-Yau varieties, where a structure theorem for their integral points can be established using mapping class group descent. Related analysis also yields new results on the arithmetic of algebraic curves in these moduli spaces, including finiteness of imaginary quadratic integral points for non-special curves.

Diophantine analysis in thin orbits

Alex Kontorovich
Rutgers University; von Neumann Fellow, School of Mathematics
December 8, 2017
We will explain how the circle method can be used in the setting of thin orbits, by sketching the proof (joint with Bourgain) of the asymptotic local-global principle for Apollonian circle packings. We will mention extensions of this method due to Zhang and Fuchs-Stange-Zhang to certain crystallographic circle packings, as well as the method's limitations.

Spectral gaps without frustration

Marius Lemm
California Institute of Technology; Member, School of Mathematics
December 6, 2017
In spin systems, the existence of a spectral gap has far-reaching consequences. So-called "frustration-free" spin systems form a subclass that is special enough to make the spectral gap problem amenable and, at the same time, broad enough to include physically relevant examples. We discuss "finite-size criteria", which allow to bound the spectral gap of the infinite system by the spectral gap of finite subsystems. We focus on the connection between spectral gaps and boundary conditions. Joint work with E. Mozgunov.