Columbia Geometric Topology Seminar

Spring 2016

The GT seminar meets on Fridays in Math 520, at 2 PM.
Organizer: Abhijit Champanerkar.
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Date Speaker Title
January 22 Organizational meeting 2pm Room 520
January 29 Alan Reid (UT Austin & IAS) Profinite rigidity for 1-punctured torus bundles
February 5 Andre de Carvalho (Brazil & IAS) Organizing horseshoe braids
February 12 Richard Kent (UWisconsin & IAS) Spacious knots
February 19 Joan Birman (Barnard) Efficient geodesics and an effective algorithm for distance in the complex of curves
February 26 John Etnyre (GeorgiaTech & IAS) Solid tori and the classification of contact structures on small Seifert fibered spaces
March 4 Jeff Danciger (UT Austin) Proper affine actions of right angled Coxeter groups
March 11 No Seminar
March 18 No Seminar Spring break
March 25 Nancy Wrinkle (Northeastern Illinois University) The ribbonlength of knot diagrams
April 1st Sarah Koch (Michigan) Deforming rational maps
April 6th, 2:30 pm Room 528
Note different date, time and room		 Saddat Oyku Yurttas (Turkey) Counting curves of an integral lamination
April 15th Josh Greene (Boston College) Alternating links and definite surfaces
April 22nd Khalid Bou-Rabee (City College, CUNY) Intersection growths of groups
April 29th Keiko Kawamuro (U of Iowa) Quasi-right-veering braids and quasi-positive braids
May 6th, 2:30 pm Math 507
Note different time and room 		Greg Kuperberg (UC Davis) Knots and 3-manifolds from the viewpoint of NP and BQP

 

Abstracts

Alan Reid, UT Austin & IAS
Jan 29th, 2016
Title: Profinite rigidity for 1-punctured torus bundles
Abstract: This talk will discuss the question: To what extent are the fundamental groups of finite volume hyperbolic 3-manifolds determined by their finite quotients amongst the fundamental groups of compact 3-manifolds. We will show that this is the case for hyperbolic 1-punctured torus bundles.

Andre de Carvalho, Brazil & IAS
Feb 5th, 2016
Title: Organizing horseshoe braids
Abstract: We will discuss the braids associated to periodic orbits of Smale's horseshoe map. Roughly speaking, we say a braid forces another if every homeomorphism of the disk which has a periodic orbit with the first braid type must also have one with the second. We will describe a conjecture about how forcing organizes horseshoe braids. Part of this conjecture concerns the conjugacy problem among horseshoe braids, of which we'll see a recent proof (joint with Toby Hall; as is everything else in the talk). The proof sheds light on the structure of horseshoe braids and knots. Central to it are infinite braids associated to certain horseshoe homoclinic orbits. If time permits, we'll also discuss the complementary (hyperbolic) 3-manifolds and geometric limits of some sequences, all of which is related to the above proof.

Richard Kent, UWisconsin & IAS
Feb 12th, 2016
Title: Spacious knots
Abstract: Brock and Dunfield showed that there are integral homology spheres whose thick parts are very thick and take up most of the volume. Precisely, they show that, given R big and r small, there is an integral homology 3-sphere whose R-thick part has volume (1-r) volume(M). Purcell and I find knots in the 3-sphere with this property, answering a question of Brock and Dunfield.

Joan Birman , Barnard
Feb 19th, 2016
Title: Efficient geodesics and an effective algorithm for distance in the complex of curves
Abstract: This is a report on joint work with Margalit and Menasco. We give an algorithm for determining the distance between two vertices of the complex of curves. While there already exist such algorithms, for example by Leasure, Shackleton, and Webb, our approach is new, simple, and more effective for small distances. Our method gives a new preferred finite set of geodesics between any two vertices of the complex, called efficient geodesics, which are different from the tight geodesics introduced by Masur and Minsky.

John Etnyre , GeorgiaTech & IAS
Feb 27th, 2016
Title: Solid tori and the classification of contact structures on small Seifert fibered spaces
Abstract: Tight contact structures have been classified on most small Seifert fibered spaces, but there are infinite families that have resisted classification despite many attempts. These are the most interesting families in that they contain examples that do not admit tight contact structures and examples of tight but non-fillable contact structures. In this talk I will discuss an approach to the classification on new infinite subsets of these families and illustrates how studying embeddings of solid tori in contact manifolds can be used to construct and distinguish tight contact structures.

Jeff Danciger , UT Austin
March 4th, 2016
Title: Proper affine actions of right angled Coxeter groups
Abstract: We prove that any right-angled Coxeter group on k generators admits a proper action by affine transformations on \mathbb R^{k(k-1)/2}. As a corollary many interesting groups admit proper affine actions, including surface groups, hyperbolic three-manifold groups, and Gromov hyperbolic groups of arbitrarily large virtual cohomological dimension. Joint work with Francois Gueritaud and Fanny Kassel.

Nancy Wrinkle , Northeastern Illinois University
March 25th, 2016
Title: The ribbonlength of knot diagrams
Abstract: The ropelength problem asks to minimize the length of a knotted space curve such that a unit tube around the curve remains embedded. A two-dimensional analog has a much more combinatorial flavor: we require a unit-width ribbon around a knot diagram to be immersed with consistent crossing information. The ribbonlength is the length of the diagram divided by its width. Attempting to characterize critical points for ribbonlength leads us to new results about the medial axis of an immersed disk in the plane, including a certain topological stability for thin disks. This is joint work with Elizabeth Denne and John M. Sullivan.

Sarah Koch, Michigan
April 1st, 2016
Title: Deforming rational maps
Abstract: For a rational map on the Riemann sphere, there is an associated deformation space (originally defined by A. Epstein). Epstein proved that this deformation space is a complex submanifold of a certain Teichmueller space. The arguments in his proof are local; not much is known about the global topology of the deformation space in general. We present a concrete example of this construction in which the topology can be relatively understood. These deformation spaces are related to dynamically defined subvarieties in the moduli space of rational maps. I will mention some fundamental open questions in complex dynamics concerning the topology of these objects. This is joint work with E. Hironaka.

Saddat Oyku Yurttas, Dicle University, Turkey,
April 6th, 2016, 2:30 - 3:30 pm, Room 528
Note different date, time and room
Title: Counting curves of an integral lamination
Abstract: An integral lamination on the n-punctured disk is a non-empty disjoint union of finitely many essential simple closed curves, up to isotopy. A beautiful method of describing such laminations is given by the Dynnikov coordinate system. In the case n = 3, the Dynnikov coordinates of an integral lamination consist of a pair of integers, and the number of connected components of the lamination is the greatest common divisor of these integers. No analogous formula is known when n > 3. In this talk we describe an efficient algorithm for calculating the number of components of an integral lamination from its Dynnikov coordinates. This is joint work with Toby Hall.

Josh Greene , Boston College
April 15th, 2016
Title: Alternating links and definite surfaces
Abstract: I will describe a characterization of alternating links in terms intrinsic to the link complement and use it to deduce some properties of these links, including new proofs of some of Tait's conjectures that previously required the use of the Jones polynomial.

Khalid Bou-Rabee , City College, CUNY
April 22nd, 2016
Title: Intersection growths of groups
Abstract: Intersection growth concerns the asymptotic behavior of the index of the intersection of all subgroups of a group that have index at most $n$. We motivate studying this growth and explore some examples with a focus on nilpotent groups and zeta functions. This covers joint work with Ian Biringer, Martin Kassabov, and Francesco Matucci.

Keiko Kawamuro , U of Iowa
April 29th, 2016
Title: Quasi-right-veering braids and quasi-positive braids
Abstract: In the first half of this talk, I define a quasi-right-veering braid in a general open books then show its connection to tight contact structure. In the second half of the talk, I compare sub-monoids of the braid group B_n and sub-monoids of the mapping class group Mod(S) and answer some questions raised by Etnyre and Van Horn-Morris. This is joint work with Tetsuya Ito.

Greg Kuperberg , UC Davis
May 6th, 2016, 2:30 pm Math 507
Note different time and room
Title: Knots and 3-manifolds from the viewpoint of NP and BQP
Abstract:

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