CUNY Graduate Center
365 Fifth Avenue, New York, NY 10016
Room: 4214-03 (Thesis room)
Tuesdays 3:00pm - 4:00pm
Ara Basmajian (CUNY, Graduate Center and Hunter College)
Email: abasmajian@gc.cuny.edu
Daniel White (CUNY Graduate Center)
Email: dwhite2@gradcenter.cuny.edu
January 29: Organizational meeting
February 5th: Ara Basmajian (CUNY Graduate Center and Hunter College)
Title: Curve families in Collar neighborhoods of geodesics.
Abstract: Given two periodic functions we compare the modulus of the family of curves between the graphs of these functions to the modulus of the family of vertical lines between the graphs. We next compare these modulus values as the functions degenerate to zero. These comparisons lead to applications for hyperbolic surfaces.
February 12th: No meeting (Lincoln's birthday)
February 19th: Robert Suzzi Valli (Manhattan College)
Title: Growth Rates in the Modular Group
Abstract: The Modular Group G is isomorphic to the free product of Z2 and Z3. Let a and b be the generators of Z2 and Z3, respectively. Fix S={a,b,b-1}, a symmetric generating set for G. In this talk, we will investigate the growth rate of primitive conjugacy classes in G, where growth is measured by word length in terms of the generating set S. We first compute the growth rate of all primitive conjugacy classes in G, and then we specialize to primitive conjugacy classes of reciprocal elements in G, that is, hyperbolic elements which are conjugate in G to their inverses. Geometrically, reciprocal words in G correspond to closed geodesics on the modular surface, H/G, which pass through the order two cone point. Such geodesics are therefore called reciprocal. This is joint work with Ara Basmajian.
February 26th: Robert Suzzi Valli (Manhattan College)
Title: Growth Rates in the Modular Group (continued)
March 5th: No meeting (Department meeting and colloquium)
March 12th: Nicholas Vlamis (CUNY Queens College)
Title: Exploring algebraic rigidity in mapping class groups
Note room change: Room 6496
Abstract: A classical theorem of Powell (with root in the work of Mumford and Birman) states that the pure mapping class group of a connected, orientable, finite-type surface of genus at least 3 is perfect, that is, it has trivial abelianization. We will discuss how this fails for infinite-genus surfaces to the integers. This characterization yields a direct connection between algebraic invariants of pure mapping class groups and topological invariants of the underlying surface. This is joint work with Javier Aramayona and Priyam Patel.
March 19th: Francesco Preta (New York University)
Title: Fibers of hyperbolic mapping tori lying on short geodesics of moduli space
Abstract: A closed geodesic in the moduli space of a genus g surface is said to be L-short if its length is bounded by L/(2g-2). Each of these curves corresponds to a pseudo-Anosov element of the mapping class group with short dilatation. Farb, Leininger and Margalit proved that for a fixed L there is only a finite number of mapping tori with such monodromy. In this way, building on results by Thurston and Fried, such 3-manifolds can be used to construct families of examples for short geodesics of different genus. Motivated by understanding their location in moduli spaces, this talk will provide an explicit method to analyze the topology and geometry of families of surfaces obtained from the same mapping torus. In order to do so, I will build on results by McMullen on their covering spaces and I will propose examples to show proper bounds on their geometry.
March 26th: No meeting (Scheduling conflict)
April 2nd: Seminar Canceled
April 9th: Senia Sheydvasser (CUNY Graduate Center)
Title: Hyperbolic isometry groups and quaternion algebras
Abstract: The accidental isomorphisms of PSL(2,R) and PSL(2,C) with the orientation preserving isometry groups of hyperbolic 2-space and 3-space are well-known. Somewhat less well-known is that there are similar isomorphisms for any hyperbolic n-space, first written down by Vahlen in the early 20th century. We will show how quaternion algebras can be used to give a nice representation of the isometry group of hyperbolic 3-space, and by studying involutions on this algebra, we will produce nice discrete subgroups that are in some sense analogs of PSL(2,O), where O is the ring of integers of an algebraic number field.
April 16th: No meeting (Teichmüller theory school at Stony Brook University)
April 23rd: No meeting (Spring recess)
April 30th: Xinlong Dong (CUNY Graduate Center)
Title: Hyperbolic structure with upper bound condition
Abstract: Following Papadopoulos's paper, we are given the upper bound for the Fenchel-Nielsen distance in terms of quasiconformal distance, between two hyperbolic metrics obtained by a Fenchel-Nielsen multi-twist deformation. Using this upper bound, we will discuss the question of finding sufficient conditions under which the quasiconformal Teichmüller space and the Fenchel-Nielsen Teichmüller space coincide setwise and sufficient conditions under which the spaces are homeomorphic.
May 7th: Jane Gilman (Rutgers University-Newark) [Cancelled. Postponed until Fall 2019]
Title: Computability Models: Algebraic, Topological and Geometric Algorithms
Abstract: This talk is about models of computation for computational problems in hyperbolic geometry and topology. The problems include analysis of the Riley slice and deciding discreteness for subgroups of PSL(2,C). We review the known computational models (BSS machines, bit-computability, symbolic computation), and we discuss two new models which seem to well adapted to the problems at hand: G-Fenchel matrix computability, and extended bit-computability. These models are applied to recent work of Misha Kapovich. Portions of this are joint work with A. Tsvietkova.
May 14th: Bjoern Muetzel (Dartmouth College)
Title: Energy distribution of harmonic 1-forms and Jacobians of Riemann surfaces with a short closed geodesic
Abstract: We study the energy distribution of harmonic 1-forms on a compact hyperbolic Riemann surface S where a short closed geodesic is pinched. If the geodesic separates the surface into two parts, then the Jacobian torus of S develops into a torus that splits. If the geodesic is non-separating then the Jacobian torus of S degenerates. The aim of this work is to get insight into this process and give estimates in terms of geometric data of the initial surface S such as its injectivity radius and the lengths of geodesics that form a homology basis. This is joint work with Peter Buser, Eran Makover and Robert Silhol.
Diversity and Inclusion at the GC