January 31st: Robert Suzzi Valli (CUNY Graduate Center)
Title: Lengths of non-simple closed geodesics on hyperbolic 2-orbifolds
Abstract: Let O =
2/G, where G is a Fuchsian group (possibly with torsion). In general, O is called a hyperbolic 2-orbifolds. Toshihiro Nakanishi, among others, showed that closed geodesics with self-intersection on O cannot be too short. Following Nakanishi, we will give the best possible lower bound on the lengths of such geodesics.
February 7th: Robert Suzzi Valli (CUNY Graduate Center)
Title: Lengths of non-simple closed geodesics on hyperbolic 2-orbifolds (continued)
February 14th: Ozgur Evren (CUNY Graduate Center)
Title: Quasiconformal mappings between triangles, hexagons, and pairs of pants
Abstract: As a prelude for Erina Kinjo's 2011 paper titled ``On Teichmuller Metric and the Length Spectrums of Topologically Infinite Riemann Surfaces'', we will talk about a 2002 paper of C. J. Bishop titled ``Quasiconformal Mapping of Y-pieces''. Namely, we will show that one can find a quasiconformal map with small maximal dilitation between two hyperbolic triangles which maps vertices of one triangle to another and is affine on the edges; provided that the lengths and angles of the triangles are comparable. We will use this map to construct quasiconformal maps with similar properties between hyperbolic hexagons and finally pairs of pants.
February 21th: No meeting
February 28th: Jane Gilman (Rutgers University)
Title: Rewriting systems, hyperbolic geometry, and mapping class groups
Abstract: Let S be a compact Riemann surface of genus g ≥ 2 and G a group of conformal automorphisms of S. G is, of course, isomorphic to a finite subgroup of the mapping class group. The action of G on the first homology group of S can be studied in a number of equivalent ways. The ways include lifting curves from the quotient surface S/G or equivalently using the Schreier-Reidemeister rewriting system to obtain a presentation for the Fuchsian group that uniformizes the quotient surface and cutting nd pasting fundamental polygons to obtain simple actions of elements of G on curves on the surface. In particular, one can find a homology basis adapted to the action of G and intersection numbers for curves in the basis thus yielding the symplectic representation of the group as a subgroup of the embedding of the mapping class group into the groups of symplectic matrices. In this mostly expository talk we connect all of these concepts relating them to Thurston's classification of the elements of the mapping class group.
Recently there has been renewed interest in the action of a conformal automorphism group on a compact Riemann surface as new group theoretic tools have been applied to the conformal, geometric, and topological problem. We summarize older results due to Gilman and Gilman-Patterson about adapted homology bases forprime order automorphisms and time permitting extend these results to new results for arbitrary finite groups using the more recent concept of a geerating vector. We survey more recent results of Wootton, Broughton, Weaver, Rodriguez, and others.
March 6th: Jane Gilman (Rutgers University)
Title: Rewriting systems, hyperbolic geometry, and mapping class groups (continued)
March 13th: Reza Chamanara (CUNY Brooklyn College)
Title: Canonical metrics for projective structures on surfaces
March 20th: Christopher Arettines (CUNY Graduate Center)
Title: Combinatorics of curves on surfaces
Abstract: Surfaces and curves on them can be described combinatorially via a fundamental polygon and an edge-gluing pattern. From this combinatorial data, I will present a simple algorithm that produces a visual representative of a curve (or collection of curves) which has minimal intersections. In addition, this procedure can be used to determine if a collection of curves fills the surface, which is useful when studying actions of the mapping class group on a surface.
March 27th: Patrick Hooper (CUNY City College)
Title: Polynomials whose roots are affinely equidistributed around an ellipse
Abstract: We say that a monic polynomial p(z) has roots which are affinely equidistributed in the complex plane if the roots of p(z) are the images of the roots of zn-1 under an affine map of the plane. In this case, the roots of p(z) are all located on some ellipse in the complex plane. Igor Rivin asked recently if the roots of p'(z) always lie on the major axis of this ellipse.
We will describe how to understand polynomials of this form and answer Igor's question. It should be noted that it turns out that Igor's question had already been answered in the paper ``On the derivatives of a Vertex Polynomial'' by J. Parish. The solution I describe will be very close to the approach of Parish.
April 3rd: No meeting
April 10th: No meeting
April 17th: Gerardo Jimenez (CUNY Graduate Center)
Title: Comparison between the Teichmüller metric and the length spectrum metric under partial twists
Abstract: Let dT and dS denote the Teichmüller metric and the length spectrum metric, respectively, in the Teichmüller space Tg. In 2003, Zhong Li proved that dT and dS are not metrically equivalent. More precisely, he showed that there exist two sequences τn and τn* such that dS(τn, τn*) → 0 as n → ∞ while dT(τn, τn*) ≥ d > 0 for n large. By using earthquakes we will construct two paths α(t) and α*(t), 0 ≤ t ≤ 1, in Tg such that limt→1 dS(α(t), α*(t)) = 0 while limt→1 dS(α(t), α*(t)) = ∞.
April 24th: Ozgur Evren (CUNY Graduate Center)
Title: Kinjo's results on the length spectrum metric
Abstract: In his 2003 paper, Hiroshige Shiga showed that on the Teichmüller space of a Riemann surface with an infinitely generated fundamental group which admits a pants decomposition where all the boundary components are either punctures or geodesics with lengths bounded from above and below; the topologies defined by the Length Spectrum Metric and the Teichmüller metric are the same. The converse to this statement, after remaining open for some time, was proven to be false by Erina Kinjo in 2011. We will talk about this result, namely, we will construct a Riemann surface with an infinitely generated fundamental group which admits no pants decomposition where the lengths of the boundary components are bounded from above, but the Length Spectrum Metric and the Teichmüller Metric defines the same topology on the Teichmüller space of the surface.
May 1st: Howard Masur (University of Chicago)
Title: Introduction to Teichmüller space and Thurston's theory of measured foliations
Abstract: This will be an introduction to the seminar that follows. I will discuss the notion of the Teichmüller space of a closed surface of genus g beginning with the example of the torus. In that case the compactified real line which is the boundary at infinity can also be thought of as the set of straight line flows or foliations on a fixed torus. The reals encode the asymptotic behavior of geodesics in Teichmüller space.
I will then discuss Thurston's famous generalization to higher genus where the notion of straight line flow is replaced by what is called a measured foliation. The space of (projectivized) measured foliations acts as the boundary at infinity for the Teichmüller space and encodes the asymptotic behavior of geodesics.
May 8th: Andrew Silverio (Rutgers University)
Title: Pencils in hyperbolic 4-space
Abstract: A pair of isometries is said to be linked if both of them can be factored simultaneously by three involutions. A core geodesic of a pair is a line symmetric about the axes of the pair's palindromes. In hyperbolic 3-space, any two pairs are linked and have a unique core geodesic. Higher dimensions do not have such nice properties. To study the linked pairs of isometries in hyperbolic 4-space with a core (totally) geodesic (plane), one can resurrect the classic idea of pencils. I will discuss some basic hyperbolic geometry, revisit factoring, and attempt to show some linked pairs in dimension 4 that do not restrict to lower dimensions and those that can possibly have discreteness condition.
May 15th: No meeting
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