This is the page from Fall 2009. The current seminar schedule is at http://math.columbia.edu/~gtseminar/.

The GT seminar meets on Fridays in Math 520, at 1:15PM.

Previous semesters: Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006.

Other area seminars. Our e-mail list.

Date Speaker Title
Sept. 18, 10:45 AM Organizational meeting
Sept. 25 Noah Snyder (Columbia) Knot polynomial identities and coincidences of small quantum groups
Oct. 2 No seminar: NY Joint Symplectic Geometry Seminar at Columbia,
Finitely presented groups conference at City College
Oct. 9 Dylan Thurston (Columbia) Cluster algebras and triangulated surfaces
Oct. 16 Joel Zablow (Long Island Unversity) The Dehn Quandle and its Homology
Oct. 23 Thomas Koberda (Harvard) Representations of mapping class groups and residual properties of 3-manifold groups
Oct. 30 Roland van der Veen (University of Amsterdam) Asymptotics of the colored Jones polynomial of knotted graphs
Nov. 6 No seminar (NY Joint Symplectic Geometry Seminar)
Nov. 13 Daniel Moskovich (RIMS, Kyoto) Surgery and bordism for coloured knots
Nov. 20 Jeremy Kahn (Stony Brook) Essential Immersed Surfaces in Closed Hyperbolic 3-Manifolds
Nov. 27 No seminar (Thanksgiving)
Dec. 4 Cameron Gordon (U Texas Austin) Knots with small rational genus
Dec. 7, 5:30 PM, Math 507 Matthias Kreck (University of Bonn) De Rham cohomology from the empty set
Dec. 11 Peter Brinkmann (CCNY) Characteristic polynomials of pseudo-Anosov maps
Dec. 18 Igor Rivin (Temple) Self-intersections of curves on surfaces

 

Abstracts.

September 25, 2009.

Noah Snyder, Columbia University, “Knot polynomial identities and coincidences of small quantum groups”.

I'll introduce skein theoretic knot invariants coming from the “D2n subfactor planar algebra”. These are a slight modification of the colored Jones polynomial. Using these knot invariants I'll derive some polynomial identities between colored Jones polynomials and other polynomial knot invariants. For example, the 6th colored Jones polynomial at a 28th root of unity of any knot (but not any link!) is always twice the value of a certain specialization of the HOMFLY polynomial. The reason for this identity is that both sides of this equation are equal to a knot invariant coming from the D2n planar algebra. All the proofs are purely skein theoretic and no prior experience with subfactors, planar algebras, or quantum groups will be assumed for the main part of the talk. If time permits, I'll explain briefly at the end what these identities have to do with level-rank duality and coincidences of small quantum groups.

October 9, 2009.

Dylan Thurston, Columbia University, “Cluster algebras and triangulated surfaces”.

Cluster algebras are a relatively new algebraic structure with many intriguing properties. For instance, they are related to total positivity of matrices. We will show how some examples naturally arise from the hyperbolic geometry of punctured Riemann surfaces. These cluster algebras give almost all ”mutationally finite“ cluster algebras, cluster algebras with a certain finiteness property. (Cluster algebras with a stronger finiteness property are classified by Dynkin diagrams.) Finally, we look at what this might tell us about how to quantize and categorify Teichmuller space.

October 16, 2009.

Joel Zablow, Long Island Unversity, “The Dehn Quandle and its Homology, with some initial Connections to Lefschetz Fibrations”.

The Dehn quandle relates Dehn twists to their actions on (isotopy classes of) circles on a surface of genus g. I'll describe and apply a quandle homology theory, and characterize a large family of 2-homology classes corresponding to certain Dehn quandle relations. Using further algebraic properties, I'll show the 2-homology yields an invariant of certain Lefschetz fibrations over the disk D2, with some extra information. Time allowing, I'll look at how the homology recognizes other simple "related" fibrations, and discuss work in progress, extending the Dehn quandle to laminations.

October 23, 2009.

Thomas Koberda, Harvard University, “Representations of mapping class groups and residual properties of 3-manifold groups”.

I will talk about homological representations of mapping class groups, namely ones which arise from actions on covering spaces. I will prove that these are asymptotically faithful and indicate how the Nielsen-Thurston classification can be obtained from these representations. I will then discuss how mapping tori of mapping classes can be used to analyze the image of these representations. As a corollary, I will exhibit a class of compact 3-manifolds whose fundamental groups are, for every prime p, virtually residually finite p.

October 30, 2009.

Roland van der Veen, University of Amsterdam, “Asymptotics of the colored Jones polynomial of knotted graphs”.

After showing how to combinatorially evaluate the colored Jones polynomial of any knotted graph, we discuss two conjectures on the asymptotics at roots of unity. If the order of the root of unity increases with the color, the volume conjecture predicts a relation to the hyperbolic volume of the graph complement. If one fixes the root of unity as the colors go to infinity, we formulate a new conjecture that states roughly that the leading asymptotics do not depend on the embedding of the graph.

November 13, 2009.

Daniel Moskovich, Research Institute for Mathematical Science, Kyoto, “Surgery and bordism for coloured knots”.

For a group G, a G coloured knot is an oriented knot K together with a representation of its knot group π1(S3-K) onto G. We consider the question of how to determine whether or not two G coloured knots are related by moves analogous to crossing changes for knots. For a certain class of finite metabelian groups, we can show that two G-coloured knots are thus related if and only if they are in the same relative bordism class. Motivations include finding G symmetric surgery descriptions of manifolds, using Dehn surgery techniques to investigate twisted Alexander polynomials, finding new invariants of G coloured knots, and surgery presentations of covering links in non-abelian covering spaces.

November 20, 2009.

Jeremy Kahn, Stony Brook, “Essential Immersed Surfaces in Closed Hyperbolic 3-Manifolds”.

Given any closed hyperbolic 3-manifold M and ε > 0, we find a closed hyperbolic surface S and a map f: S → M such that f lifts to a 1+ε-quasi-isometry from the universal cover of S to the universal cover of M. This is joint work with Vladimir Markovic.

December 4, 2009.

Cameron Gordon, Univerity of Texas at Austin, “Knots with small rational genus”.

December 11, 2009.

Peter Brinkmann, City College New York, “Characteristic polynomials of pseudo-Anosov maps”.

Motivated by computer experiments, I will present a new structure theorem for characteristic polynomials of train track maps for pseudo-Anosov homeomorphisms of surfaces. After a brief review of the classification of surface homeomorphisms, I will discuss the algorithmic approach to train tracks due to Bestvina and Handel as well as skew-symmetric forms on train tracks and their surprising degeneracies, and I will sketch the proof of the structure theorem for characteristic polynomials. The approach is largely elementary, with many examples provided by the software package XTrain.

This result is joint work with Joan Birman and Keiko Kawamuro.

December 18, 2009.

Igor Rivin, Temple University, “Self-intersections of curves on surfaces”.

I will discuss questions pertaining to the geometry of simple curves and curves with few self-intersections on surfaces, and related question in geometry and topology of surfaces and geometric group theory (extending coverings, residual finiteness, McShane's identity, and whatever superset (subset?) time permits).

Other relevant information.

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Announcements for this seminar, as well as for related seminars and events, are sent to the GT seminar mailing list. You can subscribe directly or by contacting D. Thurston.