Geometry MMA Seminar: A 0-1 law for circle packings of coarsely hyperbolic metric spaces and applications to cusp excursion (logarithm law)

Speaker: Harry Bray, George Mason University

Title:  A 0-1 law for circle packings of coarsely hyperbolic metric spaces and applications to cusp excursion (logarithm law)

Abstract:  On the cusp of the 100 year anniversary, Khinchin’s theorem implies a strong 0-1 law for the real line; namely, the set of real numbers within distance q^{-2-\epsilon} of infinitely many rationals p/q is Lebesgue measure 0 for \epsilon>0, and full measure for \epsilon=0. In these lectures, I will present an analogous result for circle packings in Gromov hyperbolic metric spaces. As an application, we prove a logarithm law; that is, we prove asymptotics for the depth in the packing of a typical geodesic.  The first talk will be background, fundamentals, and the statements of the results. In subsequent lecture(s) we will prove the results. We will use some massive black boxes, such as quasi-independence and ergodicity of the natural boundary measures associated to these spaces.  This is joint work with Giulio Tiozzo.

Time: Monday, March 18, 2024 – 10:30am-11:45am

Place: Exploratory Hall, Room 4106