*Note: On Tuesdays 9/18 and 9/25 and Monday 10/8 this seminar will meet at the Simons Center at Stony Brook University at 1:30pm*
September 11th: Christopher Arettines (CUNY Graduate Center)
Title: An introduction to Teichmüller space
Abstract: A discussion of the pair of pants theorem and its relation to surfaces of infinite type. In particular, we will discuss flute surfaces and some of their geometric properties.
September 18th (Doubleheader): (At the Simons Center, Stony Brook)
1:30-2:30pm: Moira Chas (Stony Brook University)
Title: Combinatorial length, geometric length, and self-intersection of curves on surfaces
Abstract: Consider an orientable surface S with boundary and a free homotopy class C of closed oriented curves in that surface. The combinatorial length of C is the minimum number of letters required for a description of C in terms of a set of standard generators of the fundamental group of S. The self-intersection of C is the minimum number of times in which a representative of C crosses itself. If the surface is endowed with a hyperbolic metric, then one can also define the geometric length of C, as the length of the unique geodesic representative in C.
In this talk we will discuss several relations between combinatorial length, geometric length and self-intersection number. If time permits then we will discuss the definition the Goldman-Turaev Lie bialgebra and how this algebraic structure relates to the intersection and self-intersection number of curves on an surface.
Parts of this work are joint work with Fabiana Krongold, Steve Lalley, and Anthony Phillips.
2:45-3:45pm: Robert Suzzi Valli (CUNY Graduate Center)
Title: Figure eights on hyperbolic orbifold surfaces
Abstract: I will discuss my work-in-progress with partial results in finding the shortest closed geodesic with 1 self-intersection on a hyperbolic orbifold surface.
September 25th (Doubleheader): (At the Simons Center, Stony Brook)
1:30-2:30pm: Moira Chas (Stony Brook University)
Title: Minimal intersection of curves on surfaces
Abstract: Consider the set of directed free homotopy classes of curves on a orientable surface and consider the Z-module generated by this set. Goldman proved that there exists a Lie algebra structure on this module, obtained by combining the geometric intersection of curves with the usual loop product.
In this talk, we will first give the definition and properties of the Goldman Lie bracket. Secondly, we will show how to characterize simple closed classes curves in terms of the Lie bracket when the surface has non-empty boundary. Finally we will show that if a and b are two free homotopy classes and either a or b has a simple representative then the bracket of a and b encodes the minimal intersection number of a and b.
If time permits then we will discuss the Lie cobracket discovered by Turaev and its relation with the self-intersection of curves on surfaces and/or presentation for this Lie bialgebra.
2:45-3:45pm: Youngju Kim (Korea Institute for Advanced Study)
Title: Parabolic quasiconformal conjugacy classes in the Heisenberg group
Abstract: We study noncompact 3-dimensional manifolds obtained by quotienting the Heisenberg group by cyclic groups of parabolic automorphisms. In particular, we consider the quasiconformal equivalence classes of such manifolds. This is related to the quasiconformal conjugacy classes of parabolic isometries acting on the complex hyperbolic plane.
October 2nd (Doubleheader): Moira Chas (Stony Brook University)
Joint meeting with the CUNY Geometry and Topology Seminar
3:00pm
Title: The Goldman bracket determines intersection numbers for surfaces and orbifolds
Abstract: In the mid eighties, Bill Goldman discovered a Lie algebra structure on the free abelian group with basis the free homotopy classes of closed oriented curves on an oriented surface S. Goldman also proved that an embedded curve could be isotoped to not intersect a closed geodesic if and only if their Lie bracket (as defined in that work) vanished. He asked for a topological proof and about extensions of the conclusion to curves with self-intersection. Turaev, in the late eighties, asked about characterizing simple closed curves algebraically.
In this talk we will show how the Goldman bracket answers these questions for all finite type surfaces. In fact we will discuss how to count self-intersection numbers and mutual intersection numbers for all finite type orientable orbifolds in terms of a new Lie bracket operation, extending Goldman's. The arguments are purely topological or based on elementary ideas from hyperbolic geometry.
4:15pm
Title: String topology and the geometric decomposition of 3-manifolds
Abstract: In the late nineties, in joint work with Dennis Sullivan, we generalized Goldman's Lie algebra structure to a graded Lie algebra on families of loops (defining the equivariant homology of the free loop space of a manifold of any dimension). This graded Lie algebra, together with other operations in spaces of loops is known now as String Topology.
We will describe the String topology bracket on the free loop space of three manifolds and how this structure can be used to recognize hyperbolic and Seifert vertices and the gluing graph in the geometrization of three manifolds.
If time permits then we will discuss other applications of String Topology to three manifolds (These are different works of Abbaspour, Basu, Gadgil, McGibbon, D. Sullivan, M. Sullivan, and myself).
Monday October 8th (Doubleheader): (At the Simons Center, Stony Brook)
1:30-2:30pm: Viveka Erlandsson (CUNY Graduate Center)
Title: Margulis region in hyperbolic 4-space
Abstract: We discuss the shape of the Margulis region in hyperbolic 4-space corresponding to the fixed point of a screw parabolic element. In particular, we consider the case when the rotational angle of the parabolic element is Diophantine. This is joint work (in progress) with Saeed Zakeri.
2:45-3:45pm: Hiroshige Shiga (Tokyo Insitiute of Technology/CUNY Graduate Center)
Title: On conformal mappings of regions of discontinuity of Kleinian groups
Abstract: Let G be a Kleinian group with a simply connected invariant domain Δ. Then there exists a Riemann mapping f from the unit disc onto Δ. We discuss properties of the mapping f.
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October 9th: No meeting
October 16th: No meeting
November 20 - December 4: Informal meetings
Friday December 14th (Doubleheader):
10:30-11:30pm: Gregory Fein (Rutgers University)
Title: Recognizing polynomially growing outer automorphisms
Abstract: The recognition theorem for Out(Fn) of Feighn and Handel provides qualitative and quantitative invariants which uniquely determine any outer automorphism that has no periodic behavior. I have new invariants which do the same for a large collection of maps where periodic behavior is allowed. I will explain the construction, but first I will give the necessary background with many examples and analogies to the mapping class group of a surface.
11:30-12:30pm: Ozgur Evren (CUNY Graduate Center)
Title: The length spectrum metric on the Teichmüller space of a flute surface
Abstract: Following a construction by Shiga and using additional hyperbolic geometric estimates, we will obtain sufficient conditions in terms of length parameters of a flute surface for the topological inequivalence of the length spectrum metric and the Teichmüller metric on the Teichmüller space of the flute surface. Next, we will construct infinite parameter families of quasiconformally distinct hyperbolic structures for a flute surface, both with fixed and varying boundary data, with the property that the length spectrum metric is not topologically equivalent to the Teichmüller metric.
Diversity and Inclusion at the GC