January 28th: Erina Kinjo (Tokyo Institute of Technology)
Title: On the length spectrum of Riemann surfaces and asymptotic Teichmüller space
Abstract: Teichmüller space has a complete metric called Teichmüller metric. We consider another metric on Teichmüller space defined by the length spectrum of Riemann surfaces. We explain a sufficient condition for the length spectrum metric and Teichmüller metric to define the same topology on Teichmüller space. It is generalization of a result given by H. Shiga in 2003. Also we consider the length spectrum metric on asymptotic Teichmüller space.
February 4th: No meeting
February 11th: Blanca Marmolejo (CUNY Graduate Center)
Title: On Poincaré's theorem (for fundamental polygons)
Abstract: A discussion of material from Alan Beardon's The Geometry of Discrete Groups covering locally finite fundamental domains, convex fundamental polygons, Dirichlet polygons, and side pairing transformations, leading up to Poincaré's theorem for fundamental polygons.
February 18th: Blanca Marmolejo (CUNY Graduate Center)
Title: On Poincaré's theorem (part 2)
February 25th: No meeting
March 4th: Blanca Marmolejo (CUNY Graduate Center)
Title: On Poincaré's theorem (part 3)
March 11th: Viveka Erlandsson (Aalto University)
Title: Distortion of spheres under quasi-isometries of hyperbolic space
Abstract: Cooper has shown that quasi-isometries of hyperbolic n-space (n>2) are nearly isometries. More precisely, he showed that most points (all except a set of measure ε) on a sphere of radius r get mapped to a neighborhood of a sphere of the same radius. The size of the neighborhood is independent of r and only depends on ε and the quasi-isometric constant. Cooper's proof is analytic in nature and uses the absolute continuity of the associated quasi-conformal map. In this talk we outline an alternative proof of the same result using hyperbolic geometry instead. This is work in progress, joint with Ara Basmajian.
March 18th: No meeting (GEAR retreat at University of Maryland)
March 25th: Blanca Marmolejo (CUNY Graduate Center)
Title: On Poincaré's theorem (part 4)
April 1st: No meeting
April 8th: Jenya Sapir (Stanford University)
Title: Counting non-simple closed curves on surfaces
Abstract: We show how one might get coarse bounds on the number of (non-simple) closed geodesics on a surface, given upper bounds on both length and self-intersection number. Recent work by Mirzakhani and by Rivin has produced asymptotics for the growth of the number of simple closed curves and curves with one self-intersection (respectively) with respect to length. However, no asymptotics, or even explicit bounds, are known for other bounds on self-intersection number. We show how one might reduce the problem of coarse bounds on the number of closed curves on an arbitrary surface to the problem of counting curves on a pair of pants. We then give explicit bounds in that case.
April 15th: No meeting (Spring recess)
April 22nd: No meeting (Spring recess)
April 29th: No meeting
May 5th: Ara Basmajian (CUNY Graduate Center and Hunter College)
Title: Bers pants decompositions
Abstract: We will go through the main ideas in the proof of the Bers Pants decomposition theorem and other related matters if there is time.
May 13th: Christopher Arettines (CUNY Graduate Center)
Title: Angles of intersection on the punctured torus
Abstract: I will prove that for a certain class of curves on the punctured torus, angles of intersection can be used to determine the hyperbolic metric on the surface.
Diversity and Inclusion at the GC