September 10th: Youngju Kim (Korea Institute for Advanced Study)
Title: Quasiconformal deformations of Schottky groups in complex hyperbolic space
Abstract: We generalize a Schottky group construction to complex hyperbolic space and study its quasiconformal deformation in a complex hyperbolic quasi-Fuchsian space. In particular, we construct a fundamental domain whose sides consist of disjoint non-asymptotic packs for the action of the Schottky group acting on complex hyperbolic space. Then we prove that a smooth deformation of such a Schottky group is quasiconformally stable.
September 17th: No meeting
September 22nd: Ara Basmajian (CUNY Graduate Center and Hunter College)
Title: Length and intersection number for geodesics on a closed hyperbolic surface
October 1st: Hugo Parlier (University of Fribourg)
Title: Bounds on Bers's Constant
October 8th: No meeting
October 15th: No meeting
October 22nd: Ara Basmajian (CUNY Graduate Center and Hunter College)
Title: Topics in hyperbolic geometry
October 29th: Christopher Arettines (CUNY Graduate Center)
Title: Configuration spaces of polygons
November 5th: Bernard Maskit (Stony Brook University)
Title: New matrix computations for low dimensional hyperbolic geometry
Abstract: One of our goals is to give relatively easy computational answers for questions such as: given hyperbolic motions a and b of the hyperbolic plane, which pair of axes, that of (a,b) or that of (b,a), is positively oriented. We start by defining the shadow of a 2 by 2 matrix A; if A has unit determinant, then its shadow is given by A - A-1. We show that if A and B are any 2 by 2 matrices, then the shadow of the product of their shadows is 4 times the Jørgensen commutator, AB - BA. We explore the meaning of this identity, with reference to both singular and non-singular matrices. We also show, for example, that if A and B are real matrices with unit determinant corresponding to the hyperbolic transformations a and b, where the axes of a and b intersect, and A and B both have positive trace, then the sign of either of the off-diagonal entries in the matrix for the product of their shadows answers the above question concerning the orientation of their axes.
November 12th: Maxime Fortier Bourque (CUNY Graduate Center)
Title: Inserting long cylinders in a surface: the effect on Fenchel-Nielen coordinates
November 19th: No meeting
November 25th: No meeting
December 3rd: No meeting
December 10th (Doubleheader):
Ivan Levcovitz (CUNY Graduate Center)
Title: Moments of the boundary hitting function for the geodesic flow on a hyperbolic manifold
Abstract: I will talk about Bridgeman and Tan's paper which calculates the moments of the distribution of times for the geodesic flow to hit the boundary. This formula is given in terms of the orthospectrum. This formula allows an explicit calculation for the average time for the geodesic flow to hit the boundary for surfaces. I will also discuss how the first two moments correspond to well known identities for the orthospectrum, as is shown in the paper.
Wolfgang Yassiyevich (CUNY Graduate Center)
Title: Geodesics on the modular surface via continued fractions
Abstract: The group SL2(Z) in the upper-half plane is a Fuschian group. The hyperbolic surface associated to this group is known as the ``modular surface''. The problem we examine is what lines in H correspond to the same geodesics on S and which of those correspond to closed geodesics of S. This is answered by looking at continued fractions of the endpoints circle in H determining modular geodesics.
December 17th: Jonah Gaster (University of Illinois at Chicago)
Title: A non-injective skinning map with a critical point
Abstract: Following Thurston, certain classes of 3-manifolds yield holomorphic maps on the Teichmüller spaces of their boundary components. Inspired by numerical evidence of Kent and Dumas, we present a negative result about these maps. Namely, we construct a path of deformations of a hyperbolic structure on a genus-2 handlebody with two rank-1 cusps. We exploit an orientation-reversing isometry to conclude that the skinning map sends a specied path to itself, and use estimates on extremal length functions to show non-monotonicity and the existence of a critical point. Time permitting, we will indicate some surprising unexplained symmetry that comes out of our calculations.
Diversity and Inclusion at the GC