Dynamics on geometrically finite hyperbolic manifolds with applications to Apollonian circle packings and beyond

We present recent results on counting and distribution of circles in a given circle packing invariant under a geometrically finite Kleinian group and discuss how the dynamics of flows on geometrically finite hyperbolic $3$ manifolds are related. Our results apply to Apollonian circle packings, Sierpinski curves, Schottky dances, etc.Comment: To appear in the Proceedings of ICM, 201

Levels of Distribution and the Affine Sieve

This article is an expanded version of the author's lecture in the Basic Notions Seminar at Harvard, September 2013. Our goal is a brief and introductory exposition of aspects of two topics in sieve theory which have received attention recently: (1) the spectacular work of Yitang Zhang, under the title "Level of Distribution," and (2) the so-called "Affine Sieve," introduced by Bourgain-Gamburd-Sarnak.Comment: 34 pages, 2 figure

On periodic homeomorphisms of spheres

The purpose of this paper is to study how small orbits of periodic homemorphisms of spheres can be.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-22.abs.htm