The GT seminar meets on Fridays
in Math
520, at 2 PM (overflow seminars Thursday 2:40pm Room 507).

Organizer:
Walter Neumann.

Other
area seminars. Our e-mail
list. Archive of previous semesters

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.

# Other relevant information.

## Previous semesters:

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.## Other area seminars.

- Columbia Symplectic Geometry/Gauge Theory Seminar
- All Columbia Math Dept Seminars
- CUNY Geometry and Topology Seminar
- CUNY Complex Analysis & Dynamics Seminar
- CUNY Magnus Seminar
- Princeton Topology Seminar.