Seminars & Colloquia

Eli Grigsby (Boston College) will speak at 2:00 p.m. in Carney 309

"On Khovanov and Heegaard Floer homoology"

Abstract: Khovanov and Heegaard Floer homology, two theories inspired by ideas in physics, have transformed the landscape of low-dimensional topology in the past decade. The philosophies underlying the theories' constructions are quite different, yet there are intriguing connections between the two.

In this talk, I will focus on one such connection: a relationship between a reduced version of Khovanov homology and a relative version of Heegaard Floer homology recently developed by Andras Juhasz. This relationship can be used to prove that Khovanov's categorification of the reduced n-colored Jones polynomial detects the unknot when n>1; furthermore, the relationship, in its most general form, satisfies nice naturality properties with respect to standard TQFT-type operations like cutting and stacking.

This is joint work with Stephan Wehrli.

Ken Baker (University of Miami) will speak at 2:00 p.m. in Carney 309

"Rational open books, cabling, and contact structures"

Abstract: The Giroux Correspondence is a one-to-one correspondence between contact structures up to isotopy and open book decompositions up to positive stabilization. An open book decomposition of a 3-manifold is a link with a fibration of its exterior such that each fiber is a Seifert surface for the link. Cabling a link component produces a new open book decomposition (with few exceptions). We will describe how the contact structure supported by an open book changes under cabling, generalizing Hedden's result for open books in S^3. We'll also define rational open books and discuss their cablings.

This is joint work with John Etnyre and Jeremy Van Horn-Morris.

Scott Taylor (Colby College) will speak at 2:00 p.m. in Carney 309

"Levelling edges of Heegaard spines"

Abstract: I will describe recent work (joint with Maggy Tomova) which develops a new kind of thin position for graphs in 3-manifolds. I will outline the theory and describe how it can be used to level edges of certain graphs in 3-maniforld. The main theorem is a generalization of an old theorem by Casson and Gordon to bridge surfaces for graphs in 3-manifolds.

Thursday, October 29

Hank Bai (Boston College) will speak at 2:00 p.m. in Carney 309

"Quantum Teichmuller space and cluster algebra"

Abstract: Cluster algebras were developed by Fomin and Zelevinsky in 2002. Many cluster algebras arise as the coordinate rings of varieties, with the key feature - known as the Laurent Phenomenon - that the transition functions for any pair of charts (clusters) are Laurent polynomials in the coordinates (cluster algebras). The work of Gekhtman, Shapiro and Vainshtein related Teichmuller theory to cluster algebras. There is a non-commutative deformation of the rational functions on the Teichmuller space, called quantum Teichmuller space. In this talk we study the relation between the quantum Teichmuller space and quantum cluster algebra, in accordance with the technique introduced by Berenstein and Zelevinsky. This is joint work with Francis Bonahon.

Adam Levine (Columbia University) will speak at 2:00 p.m. in Carney 309

"Sliceness of Whitehead and Bing doubles"

Abstract: Links obtained using the operations of Whitehead and Bing doubling (and combinations thereof) are of great interest in the study of concordance, since they play a fundamental role in the work of Freedman on topological 4-manifolds. I will discuss recent work on this topic and prove some new results on the smooth sliceness of such links. For example, we can prove that the positive Whitehead double of the Borromean rings is not smoothly slice; whether or not it is topologically slice remains a major unsolved question.

Jeremy Kahn (SUNY Stonybrook) will speak at 4:00 p.m.  in Carney 309

"Essential immersed surfaces in closed hyperbolic 3-manifolds"

Abstract: We prove that fundamental group of a closed hyperbolic 3-manifold contains a surface subgroup. The subgroups are quasifuchsian groups 1 + eplilon close to a fuchsian group. We prove this result by showing via mixing of the geodesic flow that randomly determined pairs of pants are sufficiently uniformly distributed to fit together into a closed almost flat surface. This is joint work with Vladimir Markovic.

Jonathan Bloom (Columbia University) will speak at 2:00 p.m. in Carney 309

"Link surgery, monopole Floer homology, and odd Khovanov homology"

Abstract: I'll describe new invariants of a framed link in a 3-manifold, which arise as the pages of a spectral sequence generalizing the surgery exact triangle in monopole Floer homology. The construction draws on a surprising connection between the topology of link surgeries and the combinatorics of polytopes called graph associahedra.  For a classical link L in S^3, we obtain a sequence of bigraded vector spaces, interpolating between the reduced, Z/2Z Khovanov homology of L and a version of the monopole Floer homology of the branched double cover.  This perspective also yields a simple, topological proof that odd Khovanov homology is mutation invariant.  I'll emphasize low-dimensional topology through lots of pictures, and not the technical details of Floer homology.

Paper reference:  arxiv.org/abs/0903.3746, arxiv.org/abs/0909.0816

Ruifeng Qiu (East China Normal University, visiting UC Santa Barbara) will speak at 3:00 p.m. in Carney 309

"The amalgamation and self-amalgamation of high distance Heegaard splittings are always efficient"

Abstract: Let M be a compact orientable 3-manifold which contains a closed incompressible surface F. We denote by N(F) an open regular neighborhood of F in M. If each component of M-N(F) has a high distance Heegaard splitting, then M has a unique minimal Heegaard splitting, i.e. the amalgamation or self-amalgamation of the minimal Heegaard splittings of M-N(F).

Sungmo Kang (Boston College) will speak at 2:00 p.m. in Carney 309

"Some hyperbolic knots in S^3 with lens space and Seifert-fibered surgeries"

Abstract: We are interested in some group of hyperbolic knots in S^3 which lie on Heegaard surface of genus 2 of S^3. We define primitive/primitive knots and primitive/Seifert-fibered knots from this group. The former admits lens surgeries and the latter admits small Seifert-fibered space surgeries. The goal of this talk is to provide some idea to give a complete list of all doubly primitive/primitive and all primitive/Seifert knots. The idea is based on the R-R diagrams introduced by Osborne and Stevens.

John Berge will speak at 2:00 p.m. in Carney 103

"On locating and identifying minimal complexity genus two Heegaard diagrams of compact, closed, orientable 3-manifolds"

Abstract: Suppose F is a Heegaard surface of a closed, compact, orientable 3-manifold M, such that F bounds handlebodies H and H'. A choice of complete sets of cutting disks v of H and v' of H' yields a Heegaard diagram carried by F. The complexity of such a diagram is the total number of points of essential intersection of disks in v' with disks in v.

We will show that it is usually possible, and surprisingly easy, to locate and identify all minimal complexity Heegaard diagrams carried by F, when F has genus two.

Some of the consequences of the ability to identify the minimal complexity Heegaard diagrams carried by F are:
1) One can locate all essential tori in M.
2) One can find all alternative Heegaard surfaces in M, up to isotopy.
3) If F' is a genus two Heegaard surface of a closed orientable 3-manifold N, one can determine if there is a homeomorphism h:M->N such that H(F)= F'.
4) If all genus two Heegaard surfaces of M and N have identifiable minimal complexity Heegaard diagrams, which is usually the case, one can determine if M and N are homeomorphic.

Ilker Yuce (Boston College) will speak at 4:00 p.m. in Carney 309

"Two-Generator Free Kleinian Groups and Hyperbolic Displacements"

Abstract: The log 3 Theorem, proved by Culler and Shalen, states that every point in the hyperbolic 3-space is moved a distance at least log 3 by one of the non-commuting isometries ξ or η provided that ξ and η generate a torsion-free, discrete group which is not co-compact and contains no parabolic.
In my talk, I'll introduce a technique which determines a lower bound for the maximum of displacements under a given set of isometries. In particular, I'll show that every point in the hyperbolic 3-space is moved a distance at least (1/2)log(5+3 √ 2) by one of the isometries ξ, η or ξη when ξ and η satisfy the conditions given in the log 3 Theorem.

To be announced

Rob Kirby (University of California, Berkeley) will speak at 4:00 p.m. in McGuinn 521 Lounge

"Broken fibrations for 4-manifolds"

Abstract: I will discuss the existence and uniqueness theorems for broken fibrations of arbitrary orientable, smooth 4-manifolds over either S^2, B^2, or S^1 x I. Existence always holds, and there is a nice set of moves relating different broken fibrations for a given 4-manifold.

Sonal Jain (New York University) will speak at 3:00 p.m. in Carney 309

"The minimum canonical height on an elliptic surface"

Abstract: TBA