Columbia Geometric Topology Seminar

Spring/Fall 2018

The GT seminar meets on Fridays at 2:00pm Fridays in room 520 (with room 622 from 11 to 1 as an overflow for additional talks).
Organizer: Walter Neumann.
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Spring 2018

Date Speaker Title
Jan 26 Chen Lei, Chicago From point-picking to sections of surface bundles
Feb 2 Joseph Maher, CSI Random mapping classes have generic foliations
11:15am Room 622
Liming Pang, NYU Some configuration and related covering spaces arising in generalizing intersection theoretic approaches to Jones polynomials
Feb 16 Hoang Nguyen, UWM Distortion of surfaces in 3-manifolds
Feb 23 Abhijit Champanerkar, CSI Geometry of biperiodic alternating links
Mar 2 Jing Tao, OU Geometry of the Thurston Metric on Teichmuller Space
Mar 9. Spencer Dowdall, Vanderbilt Abstract commensurations of big mapping class groups
Mar 16 No seminar Spring Break 12-16 March
Mar 23 CANCELLED. See Mar 27 CUNY seminar. Joe Quinn will talk on Tuesday in the CUNY seminar. His plane was cancelled because of snow storm. Shapes of Hilbert-Blumenthal surfaces and some quaternionic generalizations
Mar 30 Adam Sikora, SUNY Buffalo New Approach to Quantum Teichmuller Theory
Apr 6 Lee Mosher, Rutgers Newark Some hyperbolic actions of subgroups of Aut(F_n) (joint with Michael Handel)
Apr 13 Neil Hoffman, OK state Unifying unexpected lens space surgeries
Apr 20. Olga Plamemevskaya, Stony Brook Planar open books and links of surface singularities
April 27 Pallavi Dani, LSU Subgroup distortion in hyperbolic groups.

Fall 2018

Date Speaker Title
TBA Tullia Dymarz, Wisconsin



Chen Lei, Chicago
Jan 26, 11:15pm
Title:From point-picking to sections of surface bundles
Abstract: Given any n points on a manifold, how can we systematically and continuously find a new point? What if we ask them to be distinct? In this talk, I will try to answer this question in surfaces. Then I will connect this question to sections of surface bundles. The slogan is "there is no center of mass on closed hyperbolic surfaces".

Joseph Maher, CSI
Feb 2
Title: Random mapping classes have generic foliations
Abstract: A pseudo-Anosov element of the mapping class group determines a quadratic differential, which lies in the principal stratum if all zeroes are simple, equivalently, if the corresponding foliations have trivalent singularities. We show that this occurs with asymptotic probability one for random walks on the mapping class group, and furthermore, the hitting measure on the boundary gives weight zero to foliations with saddle connections. This is joint work with Vaibhav Gadre.

Liming Pang, NYU
Feb 9 NOTE TIME: 11:15am Room 622
Title: Some configuration and related covering spaces arising in generalizing intersection theoretic approaches to Jones polynomials
Abstract: In this talk we'll present some results on configuration spaces of points in surfaces with a view towards generalizing Bigelow's intersection theoretic approach to the Jones polynomial by considering an intersection number of two submanifolds in a cover of an associated space. We use an interesting homomorphism sending braids on a punctured torus to a group which is a central extension of Z x Z by Z (very similar to the discrete Heisenberg group). This map can be used to construct two layers of covering spaces analogous to Bigelow's covering spaces.

Hoang Nguyen, UWM
Feb 16
Title:Distortion of surfaces in 3-manifolds
Abstract: In the 3-manifold theory, a great deal of interest has focused on the study of immersed surfaces in 3-manifolds in last decades. One reason is that studying immersed surfaces will help us to understand the structures of 3-manifolds. For instance, cubulation is used in the work of Wise and Agol to resolve the Virtuallly Haken conjecture on the hyperbolic manifolds. Wise observed that the following problem is important in the study of of cubulations of 3-manifold groups: Determine the distortion of surface subgroups in 3-manifold groups. The answer to this problem has been answered by Bonahon-Thurston in the hyperbolic case. In this talk, I will give a solution to this problem in the non-geometric 3-manifold case.

Abhijit Champanerkar, CSI
Feb 23
Title:Geometry of biperiodic alternating links
Abstract: In this talk we will study the hyperbolic geometry of alternating link complements in the thickened torus. We give conditions which imply that such link complements are hyperbolic and admit a positively oriented, unimodular geometric triangulation, and determine upper and lower volume bounds. For links which arise from semi-regular Euclidean tilings, called semi-regular links, we determine the complete hyperbolic structure on their complement. This has a number of nice consequences like determining exact volumes, arithmeticity and commensurability for this class of links. We will also discuss the Volume Density Conjecture and examples.

Jing Tao, OU
March 2
Title: Geometry of the Thurston Metric on Teichmuller Space
Abstract: The Thurston metric is an asymmetric metric on Teichmuller Space defined using Lipschitz constants of maps between hyperbolic surfaces. This metric was introduced by Thurston in the late 80's, who showed this metric is geodesic, though geodesics are not necessarily unique, and induced by an asymmetric Finsler norm on tangent space. In this talk, I will survey some recent advances in this field, particular on the coarse geometry of the geodesics in the Thurston metric, and some finer properties in the case of the punctured torus. This talk is based on joint work with David Dumas, Anna Lenzhen, and Kasra Rafi.

Spencer Dowdall, Vanderbilt
March 9
Title: Abstract commensurations of big mapping class groups
Abstract: It is a classic result of Ivanov that the mapping class group of a finite-type surface is equal to its own automorphism group. Relatedly, it is well-known that non-homeomorphic surfaces cannot have isomorphic mapping class groups. In the setting of ``big mapping class groups'' of infinite-type surfaces, the situation is more complicated due tot he fact that the sheer enormity and variety of behavior prevents group elements from having canonical descriptions in terms of normal forms. This talk will present work with Juliette Bavard and Kasra Rafi overcoming these difficulties and extending the above results to big mapping class groups. In particular, we show that any isomorphism between big mapping class groups is induced by a homeomorphism of the surfaces and that each big mapping class group is equal to its abstract commensurator.

Joe Quinn, UANM,
March 23
Title: Shapes of Hilbert-Blumenthal surfaces and some quaternionic generalizations (joint work with Alberto Verjovsky)
Abstract: I will explain a generalization of the classical Hilbert-Blumenthal surfaces to quaternionic surfaces over products of hyperbolic 4- and 5-space. I will present some new results improving on the geometric accuracy and visualizability of fundamental domains for classical Hilbert-Blumenthal surfaces, and then I'll discuss our progress on an ongoing project generalizing this to the higher dimensional quaternionic cases. i

Adam Sikora, SUNY Buffalo
March 30
Title: New Approach to Quantum Teichmuller Theory
Abstract: The Jones polynomial invariant of links in R^3 extends to links in thickened surfaces, leading to the notion of the skein algebra of a surface, which can be thought as quantizating the Teichmuller space.  The algebraic structure of skein algebras remain rather mysterious. We will prove some fundamental properties of these objects using the theory of measured foliations and pseudo-Anosov diffeomorphisms of surfaces.

Lee Mosher, Rutgers Newark
April 6
Title:Some hyperbolic actions of subgroups of Aut(F_n) (joint with Michael Handel)
Abstract: n the course of our theorem on the H^2_b-alternative for Out(F_n) - every finitely generated subgroup of Out(F_n) is either virtually abelian or has second bounded cohomology of uncountable dimension - thinking about natural embeddings of Aut(F_k) into Out(F_n) led us to subgroups of Aut(F_k) which have interesting new hyperbolic actions arising from “suspension” constructions, generalizing a thread of hyperbolic suspension constructions which goes back to a theorem of W. Thurston. In this talk we will describe these suspension constructions, emphasizing how they arise in the context of proving the H^2_b-alternative, and we will speculate on what may unify them.

Neil Hoffman, OK state
April 13
Title:Unifying unexpected lens space surgeries
Abstract: The Berge Conjecture attempts to classify knots in $S^3$ which admit a (non-trivial) lens space surgery. After providing some background for this problem, we will discuss why the natural generalization of this conjecture to knots in lens spaces which admit a second lens space surgery fails. Following this discussion, we will see how to make some sense of these counter-examples. This is joint work with Ken Baker, Nathan Dunfield, and Joan Licata.

Olga Plamemevskaya, Stony Brook
April 20
Title: Planar open books and links of surface singularities
Abstract: Due to work of Giroux, contact structures on 3-manifolds can be topologically described by their open books decompositions (which in turn can be encoded via fibered links). A contact structure is called planar if it admits an open book with fibers of genus 0. Symplectic fillings of such contact structures can be understood, by a theorem of Wendl, via Lefschetz fibrations with the same planar fiber. Using this together with topological considerations, we prove new obstructions to planarity (in terms of intersection form of a Stein filling or presence of certain symplectic surfaces in a weak symplectic filling) and obtain a few corollaries. In particular, we show that the canonical contact structure on a link of a normal surface singularity is planar if and only if the singularity is minimal. For hypersurface singularities, planarity is equivalent to having singularity of type A_n. (Joint work with P. Ghiggini and M. Golla.)

Pallavi Dani, LSU
Apr 27
Title: Subgroup distortion in hyperbolic groups.
Abstract: The distortion function of a subgroup measures the extent to which the intrinsic word metric of the subgroup differs from the metric induced by the ambient group. Ol'shanskii showed that there are almost no restrictions on which functions arise as distortion functions of subgroups of finitely presented groups. This prompts one to ask what happens if one forces the ambient group to be particularly nice, say, for example, to be hyperbolic. I will survey which functions are known to be distortion functions of subgroups of hyperbolic groups. I will then describe joint work with Tim Riley which adds to this list: we construct free subgroups of hyperbolic groups with distortion functions $2^{n^{p/q}}$, for all integers $p > q > 0$.

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Previous semesters:

Fall 2017, Spring 2017, Fall 2016, Spring 2016, Fall 2015, Spring 2015, Fall 2014, Spring 2014, Fall 2013, Spring 2013, Fall 2012, Spring 2012, Fall 2011, 2010/11, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006.

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