Spring 2025: Upcoming Seminars
March 14: Jun Hu (Brooklyn College and the Graduate Center)
Characterizations of circle homeomorphisms of different regularities
In this talk, we give a summary of results on characterizations of circle homeomorphisms of different regularities (quasisymmetric, symmetric, or $C^{1+\alpha }$) in terms of Beurling-Ahlfors extension, Douady-Earle extension, and Thurston's earthquake representation of an orientation-preserving circle homeomorphism, and an integral operator induced by the circle homeomorphism. We also mention corresponding characterizations of the elements of the tangent spaces of these sub Teichmuller spaces at the base point in the universal Teichmuller space. Some open problems will be raised at the end of the talk.
March 21: Biao Wang (Queensborough Community College)
TBA
March 28: Zeno Huang (Graduate Center and College of Staten Island)
The asymptotic Plateau problem
The asymptotic Plateau problem is a set of problems asking the existence and multiplicity for a minimal surface (or disk) in $H^3$ asymptotic to a given Jordan curve at the sphere at infinity. I will describe the problems and current solutions to some of them as well as some open problems. Much of the talk is based on recent work with Lowe and Seppi.
April 11: Jun Luo (Sun Yat-sen University)
TBA
April 25: Saeed Zakeri (Graduate Center and Queens College)
TBA
May 2: Sudeb Mitra (Graduate Center and Queens College)
TBA
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Spring 2025: Previous Seminars
February 14: Jayadev Athreya (University of Washington)
Linear Flows on Translation Prisms
Motivated by the study of billiards in polyhedra, we study linear flows in a family of singular flat 3-manifolds which we call translation prisms. Using ideas of Furstenberg, and Veech, we connect results about weak mixing properties of flows on translation surfaces to ergodic properties of linear flows on translation prisms, and use this to obtain several results about unique ergodicity of these prism flows and related billiard flows. Furthermore, we construct explicit eigenfunctions for translation flows in pseudo-Anosov directions with Pisot expansion factors, and use this construction to build explicit examples of non-ergodic prism flows, and non-ergodic billiard flows in a right prism over a regular n-gon for n = 7, 9, 14, 16, 18, 20, 24, 30. This is joint work with Nicolas Bedaride, Pat Hooper, and Pascal Hubert.
February 28: Karl Winsor (Stony Brook University)
Hecke triangle groups and special hyperbolic elements
We report on some computer-assisted investigations into the (2,q,infinity) Hecke triangle groups, motivated by the dynamics of billiards in regular polygons. In the case q = 18, we will show that special hyperbolic elements are abundant, and we will relate this phenomenon to the long-standing open question of characterizing the cusps of Hecke triangle groups. These results also provide examples of special affine pseudo-Anosov homeomorphisms on the unfoldings of regular polygon billiard tables.
March 7: Zhe Sun (University of Science and Technology of China)
Exponential volumes of moduli spaces of hyperbolic surfaces
Mirzakhani found a remarkable recursive formula for the volumes of the moduli spaces of the hyperbolic surfaces with geodesic boundary, and the recursive formula plays a very important role in several areas of mathematics: topological recursion, random hyperbolic surfaces etc. We consider some more general moduli spaces $M_S(K,L)$ where the hyperbolic surfaces would have crown ends and horocycle decorations at each ideal point. But the volume of the space $M_S(K,L)$ is infinite when $S$ has the crown ends. To fix this problem, we introduce the exponential volume form given by the volume form multiplied by the exponent of a canonical function on $M_S(K,L)$. We show that the exponential volume is finite. And we prove the recursion formulas for the exponential volumes, generalising Mirzakhani's recursions for the volumes of moduli spaces of hyperbolic surfaces. We expect the exponential volumes are relevant to the open string theory. This is joint work with Alexander Goncharov.
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