Columbia Geometric Topology Seminar

Spring 2013

The GT seminar meets on Fridays in Math. 520, at 1:20PM.
Organizer: Walter Neumann.
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Spring 2013 (Fall 2012, Fall 2013).

Date Speaker Title
January 25 Organizational meeting  
February 1 Johanna Kutluhan Some Schottky subgroups of mapping class groups
February 8 BoGwang Jeon Hyperbolic 3-manifolds of bounded volume and degree
February 15. No seminar
February 22 Ilya Kofman Lorenz and horseshoe knots
March 1 Nathan Broaddus Homology of the curve complex and the Steinberg module of the mapping class group
March 8 Dan Margalit Generating the kernel of the integral Burau representation
March 15.  
March 22 No seminar Spring Break and Hot Topics
March 29 Matthias Kreck Codes and low-dimensional manifolds
TUESDAY APRIL 2, 1:15pm Room 528 Saul Schleimer The boundary of the arc complex
April 5 Burglind Jöricke Braids, Conformal Module and Entropy
April 12 Abhijit Champanerkar Towards the classification of quasi-alternating Montesinos links
April 19.  
April 26 Cameron Gordon L-spaces and left-orderability
May 3 Paul Melvin Spherical projections of 4-manifolds

Fall 2013

Date Speaker Title
September 6 Organizational meeting  
September 27 Spencer Dowdall TBA



February 1.

Johanna Kutluhan “Some Schottky subgroups of mapping class groups”

Abstract: Farb and Mosher defined a notion of "convex cocompact" for subgroups of mapping class groups that models the original definition of convex cocompact for Kleinian groups; free groups of either kind are called Schottky. I'll describe a way to construct examples of Schottky mapping class subgroups that is (at least, a priori), different from the original "abundant" examples Farb and Mosher described. These examples grow out of one way, described by Clay, Leininger, and myself, to quasi-isometrically embed free groups (and more generally, right-angled Artin groups) into mapping class groups.

February 8.

BoGwang Jeon “Hyperbolic 3-manifolds of bounded volume and degree”

Abstract: In the study of hyperbolic 3-manifolds, the following question is natural: "For an n-cusped manifold M and a constant D>0, are there only finitely many Dehn fillings of M whose trace fields have degree< D?" Although it is commonly believed that the answer is yes and Hodgson proved it for the 1-cusped case, little was previously known in general. In the talk, I'll discuss some further steps to answer the question.

February 22.

Ilya Kofman “Lorenz and horseshoe knots”

Abstract: The Lorenz flow is the original chaotic dynamical system with a "strange attractor". Lorenz knots are periodic orbits in the Lorenz flow on R^3. Horseshoe knots are periodic orbits in the flow on R^3 given by the suspension of Smale's horseshoe map. In this talk, I will provide some background, and discuss some surprising relationships between these knots and the simplest hyperbolic knots. Much of this is joint work with Joan Birman.

March 1.

Nathan Broaddus “Homology of the curve complex and the Steinberg module of the mapping class group”

Abstract: The homology of the curve complex is of fundamental importance for the cohomology of the mapping class group. It was previously known to be an infinitely generated free abelian group, but to date, its structure as a mapping class group module has gone unexplored. I will give a resolution for the homology of the curve complex as a mapping class group module. From the presentation coming from the last two terms of this resolution I will show that this module is cyclic and give an explicit single generator. As a corollary, this generator is a homologically nontrivial sphere in the curve complex.

March 8.

Dan Margalit “Generating the kernel of the integral Burau representation”

Abstract: In joint work with Tara Brendle and Andrew Putman, we prove that the kernel of the Burau representation evaluated at t=-1 is equal to the group generated by squares of Dehn twists about odd numbers of marked points. This theorem, which was conjectured by Richard Hain, can either be recast as giving a generating set for the hyperelliptic Torelli group or as describing the topology of the branch locus of the period mapping from Torelli space to the Siegel upper half-plane. I will focus on the main new ingredient, which is a method for transforming infinite presentations for groups into finite ones.

March 29.

Matthias Kreck “Codes and low-dimensional manifolds”


April 2.

Saul Schleimer “The boundary of the arc complex”

Abstract: This is partly joint work with Alexander Webb. We give another proof of Klareich's theorem: the Gromov boundary of the curve complex is homeomorphic to the space of ending laminations. Our proof, based on the train-track machine, generalizes to also cover the Gromov boundary of the arc complex. I'll explain the ideas involved and sketch as much of the proofs as time permits.

April 5.

Burglind Jöricke “Braids, Conformal Module and Entropy”

Abstract: We will discuss two invariants of conjugacy classes of braids and some of their applications. The first invariant is the conformal module which already occurred in papers of Gorin and Lin in connection with their interest in the 13. Hilbert problem. The second is a popular dynamical invariant, the entropy. It occurred in connection with Thurston's theory of surface homeomorphisms.? It turns out that these invariants are related: They are inverse proportional. If time permits we will discuss the basic idea of proof of the latter fact.

April 12.

Abhijit Champanerkar “Towards the classification of quasi-alternating Montesinos links”

Abstract: Quasi-alternating links are a generalization of alternating links from a knot homological perspective. They are homologically thin for both Khovanov homology and knot Floer homology. Recent work of Greene and my joint work with Kofman resulted in the classification of quasi-alternating pretzel links in terms of their integer tassel parameters. Replacing tassels by rational tangles generalizes pretzel links to Montesinos links. In this talk I will establish conditions on the rational parameters of a Montesinos link to be quasi-alternating. Using recent results on left-orderable groups and Heegaard Floer L-spaces, I will also establish conditions on the rational parameters of a Montesinos link to be non quasi-alternating. Finally I will discuss some open problems and conjectures about quasi-alternating links. This is joint work with Philip Ording.

April 26.

Cameron Gordon “L-spaces and left-orderability”

Abstract: We will discuss evidence for the conjecture that a prime rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable. This is joint work with Steve Boyer and Liam Watson.

May 3.

Paul Melvin “Spherical projections of 4-manifolds”

Abstract: Subtle information about the differential topology of a smooth 4-manifold can be gained from a study of its maps to the 2-sphere. As a framework for this study, I will describe the homotopy classification of such maps - recent joint work with Rob Kirby and Peter Teichner, and independently Larry Taylor - and illustrate this classification through examples.

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Previous semesters:

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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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 Walter Neumann.