Columbia Geometric Topology Seminar

Spring 2017

The GT seminar meets on Fridays in Math 520, at 2 PM (overflow seminars Thursday 2:40pm Room 507).
Organizer: Walter Neumann.
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Date Speaker Title
January 20 Organizational Meeting Room 520, 2pm
February 3 Victoria Akin, Chicago Point-pushing in the mapping class group
February 9, 2:40pm
Room 507		 Dani Wise A graph coloring problem and its algebraic and topological consequences
February 10 cancelled SNOW Rescheduled April 21
February 17 Sarah Mousley, U. Illinois Boundary maps for some hierarchically hyperbolic spaces
February 24 Carolyn Abbott, CUNY and Wisconsin Universal acylindrical actions
March 3 Feng Luo, Rutgers Discrete uniformization for polyhedral surfaces
March 10 Anh Tran, UT Dallas The topology of the Jones polynomial
March 17 No seminar Spring break
March 24 Xinghua Gao, Illinois Orders from $\widetilde{PSL_2(\mathbb{R})}$ Representations and Non-examples
March 31 Ilya Kofman, CUNY Asymptotic knot theory
April 7 Matthew Stover, Temple U Azumaya algebras and hyperbolic knots
April 14 TBA TBA
April 21 Jonah Gaster, Boston College Combinatorial properties of curve graphs
April 28 Kevin Kordek, TAMU Picard groups of moduli spaces of curves with symmetry

 

Abstracts

Victoria Akin, Chicago
February 3
Title: Point-pushing in the mapping class group
The point-pushing subgroup of the mapping class group of a surface with a marked point can be considered topologically as the subgroup that pushes the marked point about loops in the surface. Birman, who discovered the point-pushing map, showed that this subgroup is abstractly isomorphic to the fundamental group of the surface, \pi_1(S). We can characterize this point-pushing subgroup algebraically as the only normal subgroup inside of the mapping class group isomorphic to \pi_1(S). This uniqueness allows us to recover a description of the outer automorphism group of the mapping class group.

Dani Wise, Room 507
February 9, 2:40pm
Title: A graph coloring problem and its algebraic and topological consequences.
I will first describe a simple graph coloring problem and survey some examples of graphs for which the coloring problem has or has no solution. I will then give a quick introduction to Bestvina-Brady Morse theory. Finally, I will describe the relationship between the coloring problem and some amusing virtual algebraic fibering consequences for geometric group theory and hyperbolic 4-manifolds. This is joint work with Kasia Jankiewicz and Sergey Norin.

Sarah Mousley, U. Illinois
February 17
Title: Boundary maps for some hierarchically hyperbolic spaces
There are natural embeddings of right-angled Artin groups $G$ into the mapping class group $Mod(S)$ of a surface $S$. The groups $G$ and $Mod(S)$ can each be equipped with a geometric structure called a hierarchically hyperbolic space (HHS) structure, and there is a notion of a boundary for such spaces. In this talk, we will answer the following question: does every embedding $\phi: G \rightarrow Mod(S)$ extend continuously to a boundary map $\partial G \rightarrow \partial Mod(S)$? That is, given two sequences $(g_n)$ and $(h_n)$ in $G$ that limit to the same point in $\partial G$, do $(\phi(g_n))$ and $(\phi(h_n))$ limit to the same point in $\partial Mod(S)$? No background in HHS structures is needed.

Carolyn Abbott, CUNY and Wisconsin
February 24
Title:Universal acylindrical actions
The class of acylindrically hyperbolic groups, which are groups that admit a certain type of non-elementary action on a hyperbolic space, contains many interesting groups such as non-exceptional mapping class groups and Out(F_n) for n>1. In such a group, a generalized loxodromic element is one that is loxodromic for some acylindrical action of the group on a hyperbolic space. Given a finitely generated group, one can look for an acylindrical action on a hyperbolic space in which all generalized loxodromic elements act loxodromically; such an action is called a universal acylindrical action. I will discuss recent results in the search for universal acylindrical actions, describing a class of groups for which it is always possible to construct such an action as well as an example of a group for which no such action exists.

Feng Luo, Rutgers
March 3
Title:Discrete uniformization for polyhedral surfaces
We discuss some of the recent work on discrete conformal geometry of polyhedral surfaces and a discrete uniformization theorem.  The relationship among discrete conformal geometry,  the work of Thurston and Alexandrov on convex surfaces in hyperbolic 3-space, and the Koebe circle domain conjecture will be addressed.  This is a joint work with D. Gu, J. Sun,  S. Tillmann and T. Wu.

Anh Tran, UT Dallas
March 10
Title: The topology of the Jones polynomial
We will discuss old and new conjectures about the topology of the Jones polynomial.
These include the AJ conjecture, slope conjecture, and strong slope conjecture.
The AJ conjecture of Garoufalidis relates the A-polynomial and the colored Jones polynomial of a knot. The A-polynomial was introduced by Cooper et al. in 1994 and has been fundamental in geometric topology. A similar conjecture to the AJ conjecture was also proposed by Gukov from the viewpoint of the Chern-Simons theory. The slope conjecture of Garoufalidis and a new conjecture of Kalfagianni and the speaker are about the relationship between the degree of the colored Jones polynomial of a knot and the topology of the knot.
These conjectures assert that certain boundary slopes and Euler characteristics of essential surfaces in a knot complement can be read off from the degree of the colored Jones polynomial.

Xinghua Gao, Illinois
March 24
Title: Orders from $\widetilde{PSL_2(\mathbb{R})}$ Representations and Non-examples
Let M be an integer homology 3-sphere. It is still unknown whether the left-orderability of its fundamental group and it not be a Heegaard-Floer L-space are equivalent. One way to study left-orderability of $\pi_1(M)$ is to construct a non-trivial representation from $\pi_1(M)$ to $\widetilde{PSL_2(\mathbb{R})}$. However this method does not always work. In this talk, I will give examples of non L-space irreducible integer homology 3-spheres whose fundamental groups do not have nontrivial $\widetilde{PSL_2(\mathbb{R})}$ representations.

Ilya Kofman, CUNY
March 31
Title: Asymptotic Knot Theory
For a hyperbolic knot or link K, the volume density is the ratio of hyperbolic volume to crossing number, and the determinant density is the ratio of 2\pi\log\det(K) to crossing number. We explicitly realize limit points of both densities for families of links that asymptotically approach several biperiodic alternating links. We relate these limits using geometric topology, dimer models and Mahler measure of two-variable polynomials. This is joint work with Abhijit Champanerkar and Jessica Purcell.

Matthew Stover, Temple
April 7
Title: Azumaya algebras and hyperbolic knots
I will talk about arithmetic geometry of SL(2,C) character varieties of hyperbolic knots. A simple criterion on roots of the Alexander polynomial determines whether or not a natural construction extends to determine a so-called Azumaya algebra on the so-called canonical component of the character variety, and I'll then explain how this forces significant restrictions on arithmetic invariants of Dehn surgeries on the knot. This is joint work with Ted Chinburg and Alan Reid.

Jonah Gaster, Boston College
April 21
Title: Combinatorial properties of curve graphs
The curve graph of a closed oriented surface of genus $g$ has vertices given by simple closed curves, and edges that correspond to curves that can be realized disjointly. Inquiry into the large scale geometry of these graphs has borne considerable fruit, and lead to the resolution of some of Thurston's conjectures. We will take a more naive perspective and explore instead combinatorial properties of this graph. For instance, what is its chromatic number (finite due to work of Bestvina-Bromberg-Fujiwara)? What are its induced subgraphs? Though precise answers to these questions are currently beyond reach, we will present progress that informs them. In particular, in joint work with Josh Greene and Nick Vlamis we show that the separating curve graph has chromatic number coarsely equal to $g \log(g)$, and the subgraph spanned by vertices in a fixed non-zero homology class is uniquely $g-1$-colorable.

Kevin Kordek, TAMU
April 28
Title: Picard groups of moduli spaces of curves with symmetry
In 1967, Mumford showed that the (orbifold) Picard group of the moduli space of genus g Riemann surfaces is isomorphic to the second integral cohomology of the genus g mapping class group. Technology developed since that time now allows one to productively study various generalizations of Mumford's original calculation. In this talk, I will explain how the theory of symmetric mapping class groups, developed by Birman-Hilden, Harvey, and others, can be used to understand - and sometimes exactly compute - the Picard groups of various moduli spaces of curves with symmetry, for example the moduli spaces of hyperelliptic curves.

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