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.

Integral points on Markoff-type cubic surfaces

Amit Ghosh
Oklahoma State University
December 8, 2017
We report on some recent work with Peter Sarnak. For integers $k$, we consider the affine cubic surfaces $V_k$ given by $M(x) = x_1^2 + x_2 + x_3^2 − x_1 x_2 x_3 = k$. Then for almost all $k$, the Hasse Principle holds, namely that $V_k(Z)$ is non-empty if $V_k(Z_p)$ is non-empty for all primes $p$. Moreover there are infinitely many $k$'s for which it fails. There is an action of a non-linear group on the integral points, producing finitely many orbits. For most $k$, we obtain an exact description of these orbits, the number of which we call "class numbers".

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.