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
Nick Vlamis (CUNY, Queens College)
Email: nicholas.vlamis@qc.cuny.edu
September 10th: Edgar Bering (Temple University)
Title: Special covers of alternating links
Abstract: The “virtual conjectures” in low-dimensional topology, stated by Thurston in 1982, postulated that every hyperbolic 3-manifold has finite covers that are Haken and fibered, with large Betti numbers. These conjectures were resolved in 2012 by Agol and Wise, using the machine of special cube complexes. Since that time, many mathematicians have asked how big a cover one needs to take to ensure one of these desired properties.
We begin to give a quantitative answer to this question, in the setting of alternating links in S^3. If an alternating link L has a diagram with n crossings, we prove that the complement of L has a special cover of degree less than 72((n-1)!)^2. As a corollary, we bound the degree of the cover required to get Betti number at least k. We also quantify residual finiteness, bounding the degree of a cover where a closed curve of length k fails to lift. This is joint work with David Futer.
September 17: Vincent Alberge (Fordham University)
Title: Measured laminations and stratifications
Abstract: I will introduce a natural stratification of the projective measured lamination space of a hyperbolic surface of finite area and I will show that most of the time the group of self-homeomorphisms that preserve such a stratification is induced by the extended mapping class group.
September 24: Jane Gilman (Rutgers University, Newark)
Title: Computability Models: Algebraic, Topological and Geometric Algorithms
Abstract: How does one translate a topological algorithm to one that can be implemented on a computer? In this talk we survey various computational models for problems, procedures and algorithms in low dimensional topology and hyperbolic geometry. The survey includes BSS machines, bit-computability, symbolic computation, G-Fenchel matrix computability, and extended bit-computability (EBC) , a model currently under development by Tsvietkova and Gilman. We apply these to Riley’s procedure, the Riley slice, and discreteness algorithms and tests extending the old computational analysis of Gilman and the more recent work of M. Kapovich.
October 1: No classes today.
Otober 8 : No classes today.
October 15: No meeting today.
October 22 : Dragomir Saric (CUNY Queens College / Graduate Center)
Title: Generating the mapping class group of the punctured solenoid
Abstract: The inverse limit of the directed system of all pointed finite unbranched covers of the punctured torus is called a punctured solenoid S_p. The baseleaf preserving mapping class group MCG(S_p) of the solenoid is isomorphic to the virtual automorphism group Vaut(F_2) of the free group F_2 on 2 generators. We define a decorated Teichmuller space T(S_p) and introduce a natural tessellation of T(S_p). By discarding certain non-regular parts of the tessellation of T(S_p) we obtain a triangulation complex of S_p on which MCG(S_p) acts. Using this action we obtain an explicit set of generators of MCG(S_p) and its representation. Finally we show that MCG(S_p) has no center. This comprises two papers, the first is a joint work with R. Penner and the second is a joint work with S. Bonnot and R. Penner.
October 29: Xinlong Dong (CUNY Graduate Center)
Title: Fenchel–Nielsen coordinates on Teichmüller spaces of surfaces of infinite type
Abstract: We define the Fenchel–Nielsen Teichmüller space relative to a geometric pair of pants decomposition. We study a metric, called the Fenchel–Nielsen metric, on such a Teichmüller space, and we compare it to the (quasiconformal) Teichmüller metric. We study conditions under which there is an equality between the Fenchel–Nielsen Teichmüller space and the familiar Teichmüller space defined using quasiconformal mappings, and we study topological and metric properties of the identity map between these two spaces when this map exists.
November 5: Greg McShane (University of Grenoble)
Title: Isometry classes of surfaces with boundary and the orthospectrum
Abstract: We consider the ortho spectrum of hyperbolic surfaces with totally geodesic boundary. We show that in general the ortho spectrum does not determine the systolic length but that there are only finitely many possibilities for the the systolic length for a given ortho spectrum. In fact we show that, up to isometry, there are only finitely many hyperbolic structures on a surface that share a given ortho spectrum.
This extends results of McKean and Wolpert for closed geodesics.
November 12: Nicholas Vlamis (CUNY, Queens College)
Title: Groups acting on hyperbolic surfaces
Abstract: It is a classical theorem that every finite group can be realized as the isometry group of an orientable, closed hyperbolic surface. This result was extended by Allcock who showed that every countable group can be realized as the isometry group of an orientable hyperbolic surface. Allcock's construction does not concern itself with the underlying topology of the surface, which leads us to ask: given a topological surface S, provide a classification of the groups which are realized as the isometry group of some hyperbolic structure on S. In general, this question is too difficult; however, there is a class of surfaces -- the infinite-genus surfaces with no planar ends -- in which we can provide a nearly complete answer. I will discuss the solution in this setting and provide an application to mapping class groups. This is ongoing joint work with Tarik Aougab and Priyam Patel.
(Note to the CUNY grad students who were at the recent AMS Sectional Meeting in Binghamton: this is an extended version of the talk I gave at the meetings.)
November 19: Yan Mary He (University of Luxembourg)
Title: A Riemannian metric and Hausdorff dimension on the Mandelbrot set
Abstract: In this talk, we introduce a Riemannian metric on the main cardioid of the Mandelbrot set and as an application, we show that the Hausdorff dimension function has no local maximum on the main cardioid. Our work is vastly inspired by the work of Bridgeman-Taylor and Bridgeman which established an extension of the Weil-Peterson metric to the quasi-fuchsian space. Along the way, we introduce multiplier functions for invariant probability measures on Julia sets, which is the counterpart in complex dynamics of length functions for geodesic currents. This is joint work with Hongming Nie.
November 26: Hrant Hakobyan (Kansas State University)
Title: Slits, pillowcases, and Poincare inequalities
Abstract: What do slit carpets, pillowcase surfaces, and Poincare inequalities have in common? In this talk, we will describe and study a class of metric spaces called dyadic slit Sierpinski carpets. Corresponding to such a slit carpet one may construct a topological ``pillowcase" sphere . We show the following.
Theorem: Let X be a dyadic slit carpet and Y be the corresponding pillowcase sphere. Then the following conditions are equivalent:
3. Y is Ahlfors -regular.
4. X supports a p-Poincare inequality for some 1<p<2.
No prior knowledge of these terms is needed, everything will be defined during the talk. The talk is partly based on joint work with Wenbo Li.
December 3: Peter Shalen (UIC)
Title: Quantitative Mostow rigidity I
December 10: Peter Shalen (UIC)
Title: Quantitative Mostow rigidity II
Diversity and Inclusion at the GC