Hyperbolic Geometry Seminar

Location and Time:

CUNY Graduate Center

365 Fifth Avenue, New York, NY 10016

Room: 4214-03 (Thesis room)

Tuesdays 3:00pm - 4:00pm

Organizers:

Ara Basmajian (CUNY, Gradute Center and Hunter College)
Email: abasmajian@gc.cuny.edu

Daniel White (CUNY Graduate Center)
Email: dwhite2@gradcenter.cuny.edu

Fall 2018 Schedule: 

September 4th: Daniel White (CUNY Graduate Center)

Title: Understanding surface automorphisms by using geodesic laminations

Abstract: This is a sequence of lectures that will be ongoing throughout the semester. I will be following material from Casson and Bleiler’s book Automorphisms of Surfaces After Nielsen and Thurston including geodesic laminations, geodesic foliations, and transverse measures associated to nonperiodic, irreducible surface automorphisms.


September 11th: No meeting (No classes)


September 18th: No meeting (No classes)


September 25th: Daniel White (CUNY Graduate Center)

Title: Understanding surface automorphisms by using geodesic laminations (part 2)


October 2nd: Daniel White (CUNY Graduate Center)

Title: Understanding surface automorphisms by using geodesic laminations (part 3)


October 9th: Daniel White (CUNY Graduate Center)

Title: Understanding surface automorphisms by using geodesic laminations (part 4)


October 16th: Daniel White (CUNY Graduate Center)

Title: Understanding surface automorphisms by using geodesic laminations (part 5)


October 23rd: Ken Bromberg (Utah)

Title: Univalent maps, surfaces in hyperbolic 3-space and Schwarzian derivatives

Abstract: Using the fact that the Riemann sphere is the natural boundary of hyperbolic 3-space, C. Epstein described the construction of a surface in hyperbolic space associated to a (locally) univalent map on a domain in the sphere. We will describe this construction and how one can use facts about one object to understand the other. This talk will be largely expository.


October 30th: Andrew Yarmola (Princeton)

Title: Circle packings and Delaunay circle patterns for complex projective structures

Abstract: At the interface of discrete conformal geometry and the study of Riemann surfaces lies the Koebe-Andreev-Thurston theorem. Given a triangulation of a surface S, this theorem  produces a unique hyperbolic structure  on S and a geometric circle packing whose nerve is the given triangulation. In this talk, we explore an extension of this theorem to the space of complex projective structures -- the family of maximal CP^1-atlases on S up to Möbius equivalence. Our goal is to understand the space of all complex projective structures carrying a circle packing with a fixed nerve. As it turns out, this space is no longer a unique point and evidence suggests that it is homeomorphic to Teichmüller space via the bundle projection --  a conjecture by Kojima, Mizushima, and Tan. In joint work with Jean-Marc Schlenker, we show that this projection is proper, giving partial support for the conjectured result. Our proof relies on geometric arguments in hyperbolic ends and allows us to work with the more general notion of Delaunay circle patterns, which may be of separate interest. I will give an introductory overview of the definitions and results and demonstrate some software used to motivate the conjecture.


November 6th: Dragomir Saric (Queens College and CUNY Graduate Center)

Title: Asymptotics of Modulus of Curve Families and Applications

Abstract: We discuss asymptotic behavior of certain families of curves under affine deformations of Euclidean structures. As an application we find limits of Teichmuller geodesics and Teichmuller disks in the Thurston boundary of the universal Teichmuller space. This is joint work with Hakobyan and Miyachi.

By applying the above methods, we find asymptotic behavior of curve families connecting a compact set with infinity on an infinite Riemann surface. We describe the asymptotic behavior in terms of Fenchel-Nielsen coordinates of the hyperbolic metric. This gives us a sufficient condition for the geodesic flow to be ergodic. A joint work with Basmajian and Hakobyan.


November 13th: No meeting.


November 20th: Daniel White (CUNY Graduate Center)

Title: Understanding surface automorphisms by using geodesic laminations (part 6)


November 27th: Dragomir Saric (Queens College and CUNY Graduate Center)

Title: Asymptotics of Modulus of Curve Families and Applications (part 2)


December 4th: Dragomir Saric (Queens College and CUNY Graduate Center)

Title: Asymptotics of Modulus of Curve Families and Applications (part 3)


December 11th: TBA

 

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