# Columbia Geometric Topology Seminar

## Fall 2015

The GT seminar meets on Fridays in Math 520, at 2 PM.
Organizer: Abhijit Champanerkar.
Other area seminars. Our e-mail list. Archive of previous semesters

Date Speaker Title
September 11 Organizational meeting 2pm Room 520
September 18 Daryl Cooper (UCSB & IAS) Deforming Convex Projective Manifolds
September 25 Balazs Strenner (IAS) Algebraic degrees and Galois conjugates of pseudo-Anosov stretch factors
October 2 Anastasiia Tsvietkova (UC Davis) The number of surfaces of fixed genus in an alternating link complement
October 9 No Seminar Workshop on Geometric Structures on 3-Manifolds, IAS Princeton
October 16 Joseph Maher (CSI & GC, CUNY) Random walks on weakly hyperbolic groups
October 23 No seminar
October 30 Tye Lidman (UT Austin & IAS) Quasi-alternating links with small determinant
November 6 Joel Hass (UC Davis & IAS) Invariants of Random Knots
November 13 Cameron Gordon (UT Austin) Taut foliations, left-orderability and cyclic branched coverings
November 20 Christian Zickert (Maryland) Triangulation independent Ptolemy varieties
November 27 No Seminar Thanksgiving
December 4 Andrei Vesnin (Chelyabinsk, Russia) Around right-angled hyperbolic polyhedra
December 11 No seminar

## Abstracts

Daryl Cooper, UCSB & IAS
September 18th, 2015
Title: Deforming Convex Projective Manifolds
Abstract: This talk concerns a properly convex real projective manifold with (possibly empty) compact, strictly convex boundary, and which consists of a compact part plus finitely many generalized cusps. We extend a theorem of Koszul which asserts that for a compact manifold without boundary the holonomies of properly convex structures is an open subset of the representation variety.

Balazs Strenner, IAS
September 25th, 2015
Title: Algebraic degrees and Galois conjugates of pseudo-Anosov stretch factors
Abstract: Thurston showed that the algebraic degrees of stretch factors of pseudo-Anosov maps on the orientable surface of genus g are bounded by 6g-6. He claimed that his Dehn twist construction produces examples of maximal 6g-6 degree stretch factors, but he did not give a proof. I will discuss a method based on a construction of Penner that produces maximal degree examples, and also examples of all even degrees less than 6g-6. I will also mention related work with Hyunshik Shin, where we study Galois conjugates of stretch factors, and use this to resolve a conjecture of Penner.

Anastasiia Tsvietkova, UC Davis
October 2nd, 2015
Title: The number of surfaces of fixed genus in an alternating link complement
Abstract: Let $L$ be a prime alternating link with $n$ crossings. We show that for each fixed $g$, the number of genus $g$ incompressible surfaces in the complement of $L$ is bounded by a polynomial in $n$. Previous bounds were exponential in $n$. This is joint work with Joel Hass and Abigail Thompson.

Joseph Maher, CSI & GC, CUNY
October 16th, 2015
Title: Random walks on weakly hyperbolic groups
Abstract: Let G be a group acting by isometries on a Gromov hyperbolic space, which need not be proper. If G contains two hyperbolic elements with disjoint fixed points, then we show that a random walk on G converges to the boundary almost surely. This gives a unified approach to convergence for the mapping class groups of surfaces, Out(F_n) and acylindrical groups. This is joint work with Giulio Tiozzo. We will also discuss some work in progress on applications to 3-manifolds.

Tye Lidman, UT Austin & IAS
October 30th, 2015
Title: Quasi-alternating links with small determinant
Abstract: Quasi-alternating links are a generalization of alternating links as a natural class of links with simple Floer and Khovanov homologies. Greene classified the quasi-alternating links with determinant at most 3, and Teragaito classified the determinant 5 links. We will extend this classification further and study some Dehn surgery problems along the way. This is joint work with Steven Sivek.

Joel Hass, UC Davis & IAS
Nov 6th, 2015
Title: Invariants of Random Knots
Abstract: I will discuss a model for generating random knots called the Petaluma model. This model is based on the petal diagram of a knot, introduced by Adams. In this model we obtain exact formulas for the distribution of the linking number of a random 2-component link, as well as for the moments of the distribution of the Casson invariant and other finite type invariants. This is joint work with C. Evan Zohar, N. Linial and T. Nowik.

Cameron Gordon, UT Austin
Nov 13th, 2015
Title: Taut foliations, left-orderability and cyclic branched coverings
Abstract: It is conceivable that for a prime, closed, orientable 3-manifold the following are equivalent: (1) $M$ admits a co-orientable taut foliation, (2) $\pi_1(M)$ is left-orderable, and (3) $M$ is not a Heegaard Floer L-space. We will discuss this in the case where $M$ is the $n$-fold cyclic branched covering of a knot. This is joint work with Tye Lidman.

Christian Zickert, University of Maryland
Nov 20th, 2015
Title: Triangulation independent Ptolemy varieties
Abstract: The Ptolemy variety is an invariant of a triangulated 3-manifold M. It detects SL(2,C)-representations of pi_1(M) in the sense that every point in the Ptolemy variety canonically determines a representation (up to conjugation). It is closely related to Thurston's gluing equation variety for PSL(2,C)-representations. Unfortunately, both the gluing equation variety and the Ptolemy variety depend on the triangulation and may miss several components of representations. We discuss the basic properties of these varieties, how to compute invariants such as trace fields and complex volume, and how to obtain a variety, which is independent of the triangulation.

Andrei Vesnin, Chelyabinsk, Russia
Dec 4th, 2015
Title: Around right-angled hyperbolic polyhedra
Abstract: Right-angled polyhedra in hyperbolic spaces are serving as useful building blocks for constructing hyperbolic manifolds and orbifolds with interesting properties. We will look at dimension three, where we describe existence conditions and the volume set structure. Then we will present a way to construct closed hyperbolic 3-manifolds related to color-epimorphisms of right-angled Coxeter groups and find 2-fold branched coverings of the 3-sphere. As a particular case, we will discuss the first example of a closed orientable hyperbolic 3-manifold constructed by F. Loebell in 1931.

# Other relevant information.

Spring 2016

## Previous semesters:

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.