This schedule is tentative. All talks will be held in 328 Boyd.
Wednesday, May 9
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9:00-9:30
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Arrival/registration
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9:30-10:30
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Sema Salur
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Manifolds with G_2 Holonomy and Contact Structures
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11:00-12:00
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İnanç Baykur
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Surface bundles, Lefschetz fibrations, and their (multi)sections
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2:00-3:00
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Jesse Johnson (slides)
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Uniqueness of high distance open book decompositions |
3:30-4:30
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Ken Baker
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On Dehn surgeries between S^1 x S^2 and lens spaces |
| 7:30
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Reception at the hotel
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Thursday, May 10
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9:30-10:30
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Brad Henry (slides)
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A combinatorial DGA for Legendrian knots from generating families
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11:00-12:0
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Jeremy Van Horn-Morris
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Infinitely many Stein fillings from spinal open books |
2:00-3:00
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Stephen Sivek
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A contact invariant in sutured monopole homology
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3:30-4:30
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Ron Fintushel
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Fake projective planes |
8pm
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Party at Will and Sybilla's
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Rides: meet at Marriott lobby at 7:45. |
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Friday, May 11
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9:30-10:30
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Çağatay Kutluhan
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HF=HM
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10:45-11:45
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Michael Sullivan
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Transverse open string topology for knots |
Afternoon
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Canoe
trip
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Saturday, May 12
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9:30-10:30
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John Etnyre |
Transverse surgery and tight contact structures
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| 11:00-12:00 |
Thomas Mark
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New constructions of symplectic 4-manifolds |
2:00-3:00
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Keiko Kawamuro
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Open book foliation & tightness criteria of contact structures |
3:30-4:30
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Joshua Sabloff
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Obstructions and Constructions for Lagrangian Cobordisms
between Legendrian Submanifolds |
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Sunday, May 13
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(note the times)
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9:00-10:00
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Vera Vertesi
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Invariants for transverse knots in Heegaard Floer homology |
10:15-11:15
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Yi Ni
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Khovanov module and the detection of unlinks
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Abstracts
Ken Baker, University of Miami,
On Dehn surgeries between S^1 x S^2 and lens spaces
We consider knots in S^1 x S^2 with a Dehn surgery to a lens space and how this parallels what happens for knots in S^3. The Cyclic Surgery Theorem enables a classification for non-longitudinal surgeries, while Berge's families of doubly primitive knots in S^3 motivate our conjectural classification for longitudinal surgeries. We will compare this conjectural picture with the Berge Conjecture and propose a means of relating the two.
İnanç Baykur, MPIM-Bonn,
Surface bundles, Lefschetz fibrations, and their (multi)sections
Surface bundles and Lefschetz fibrations over surfaces constitute a rich source of examples of smooth, symplectic, and complex manifolds. Their sections and multisections carry interesting information on the smooth structure of the underlying four-manifold. In this talk we will discuss several problems and recent results on surface bundles, Lefschetz fibrations, and their (multi)sections, which we will tackle, for the most part, using various mapping class groups of surfaces. Conversely, we will use geometric arguments to obtain some structural results for mapping class groups.
John Etnyre, GIT,
Transverse surgery and tight contact structures
One of the fundamental problems in 3-dimensional contact geometry is the construction of tight contact structures on closed manifolds. Two obvious ways to try to construct such structures are via Legendrian surgery and admissible transverse surgery. It was long thought that when performed on a closed tight contact manifold these operations would yield a tight contact manifold. We show that this is not true for admissible transverse surgery. Along the way we discuss the relations between these two surgery operations and construct some contact structures with interesting properties.
Ron Fintushel, MSU,
Fake projective planes
A fake projective plane is a complex surface having the same betti numbers as CP^2 but not diffeomorphic to it. Although these surfaces have been completely classified in the past few years (there are exactly 50 up to diffeomorphism), there is no known geometric or topological construction. This is a report on work in progress, joint with Ron Stern, which leads to the construction of infinitely many smooth fake projective planes, both symplectic and nonsymplectic.
Brad Henry, Siena College,
A combinatorial DGA for Legendrian knots from generating families
A contact structure on a 3-manifold refines traditional knot theory by introducing a geometric criterion for knots. A knot whose tangent space sits within the two-plane field of the contact structure is called Legendrian. In recent decades, Legendrian knot theory has proven to be a useful tool in low-dimensional topology and contact geometry and a fruitful area of study in its own right. Chekanov and Eliashberg's independent work in the late 90s associated a differential graded algebra (DGA) to a Legendrian knot in the standard contact structure on R^3. Numerous Legendrian knot invariants have since been derived from this DGA.
We will outline recent work that assigns a new differential graded algebra to a Legendrian knot in R^3. The definition of the DGA is motivated by considering Morse-theoretic data from a generating family. A generating family f_x for a Legendrian knot is a family of functions whose critical values encode a projection of the knot. The DGA is defined combinatorially using an algebraic analogue of a generating family first introduced by Pushkar. We will discuss the motivation and construction of this DGA and relationships between the new DGA and the Chekanov-Eliashberg DGA.
This work is joint with Dan Rutherford (University of Arkansas).
Jesse Johnson, Oklahoma State,
Uniqueness of high distance open book decompositions
An open book decomposition for a 3-dimensional manifold M is a decomposition of M into a link and a surface bundle structure for the link complement. An open book can be described by a self-homeomorphism of a page of the surface bundle and the distance of the open book decomposition is a measure of how "complicated" this self-homeomorphism is. I will describe a recent result showing that if M has an open book decomposition with sufficiently high distance then every other distinct open book decomposition for M has pages of strictly lower Euler characteristic, or in other words high distance open books are the unique minimal open books for their ambient manifold.
Keiko Kawamuro, University of Iowa,
Open book foliation & tightness criteria of contact structures
In this talk, we introduce open book foliation and study the fractional Dehn twist coefficient of open books. As application we discuss how to detect tightness of the contact structure supported by an open book. One of the results is Honda, Kazez and Matic's celebrated tightness criterion. This is a joint work with Tetsuya Ito.
Çağatay Kutluhan, Harvard,
HF=HM
I will talk about joint work with Yi-Jen Lee and Clifford H. Taubes on the equivalence of Heegaard Floer and Seiberg-Witten Floer homologies.
Thomas Mark, University of Virginia,
New constructions of symplectic 4-manifolds
There are a number of unanswered questions about the basic topology of irreducible smooth 4-manifolds, e.g., what constraints must be satisfied by their Euler characteristic and signature, and how many diffeomorphism types represent given topological types. Some progress has been made in this area using a limited number of constructions from smooth and symplectic topology applied in ingenious ways, notably fiber sums and rational blowdowns. We describe a broad variety of new constructions, some of which can be described as fiber sums along singular surfaces, and discuss some examples.
Yi Ni, Caltech,
Khovanov module and the detection of unlinks
It is a long-standing problem whether the Jones polynomial detects the unknot, and it has been known that the Jones polynomial does not detect unlinks. In the knot homology world, Kronheimer and Mrowka proved that Khovanov homology, the categorification of Jones polynomial, detects the unknot. On the other hand, the question whether Khovanov homology detects unlinks remains open. In this talk, we will show that Khovanov homology with an additional natural module structure detects unlinks. This is joint work with Matt Hedden.
Joshua Sabloff, Haverford,
Obstructions and Constructions for Lagrangian Cobordisms between Legendrian Submanifolds
I will discuss obstructions to and constructions of Lagrangian cobordisms between Legendrian submanifolds. The obstructions arise from a long exact sequence relating invariants derived from generating families. This long exact sequence not only provides obstructions, but also illuminates the geometric meaning of generating family invariants. The most interesting of the constructions takes the form of ambient Lagrangian handle attachment to a Legendrian submanifold. Its relation to the long exact sequence of a cobordism allows us to tackle questions involving the geography and botany of Legendrian submanifolds in higher dimensions. All of this is joint with Lisa Traynor, and the constructions material is also joint with Frederic Bourgeois.
Sema Salur, Rochester,
Manifolds with G_2 Holonomy and Contact Structures
A 7-dimensional Riemannian manifold (M,g) is called a G_2 manifold if the holonomy group of its Levi-Civita connection of g lies inside G_2. In this talk, I will first give brief introductions to G_2 manifolds, and then discuss relations between G_2 and contact structures. This is a joint work with Hyunjoo Cho and Firat Arikan.
Steven Sivek, Harvard,
A contact invariant in sutured monopole homology
Kronheimer and Mrowka recently used monopole Floer homology to define an invariant of sutured manifolds, following work of Juhász in Heegaard Floer homology. Contact 3-manifolds with boundary are natural examples of such manifolds. In this talk, I will construct an invariant of a contact structure as an element of the associated sutured monopole homology group. I will discuss several interesting properties of this invariant, including gluing maps which are analogous to the Heegaard Floer sutured gluing maps of Honda, Kazez, and Matic, and a bypass exact triangle relating the homology groups for different choices of sutures. This is joint work with John Baldwin.
Michael Sullivan, University of Massachusetts,
Transverse open string topology for knots
There is an open-ended question about whether string topology is a homeomorphism invariant or homotopy invariant. I will introduce a variant of string topology which is the former and not the latter, in that it produces non-trivial knot invariants, including a piece of knot contact homology (albeit in a rather indirect way). Part of this is joint with D. Sullivan, and part also with S. Basu and J. McGibbon.
Jeremy Van Horn-Morris, Stanford,
Infinitely many Stein fillings from spinal open books
Spinal open book decompositions are a slight modification of the notion of an open book decomposition used by Thurston-Winkelnkemper and Giroux for constructing and studying contact manifolds. I'll give a brief introduction to spinal open books and discuss how they can be used to construct contact manifolds in all dimensions that admit infinitely many distinct Stein symplectic fillings. This is joint with Sam Lisi and Chris Wendl.
Vera Vertesi, MIT,
Invariants for transverse knots in Heegaard Floer homology
Using the grid diagram formulation of knot Floer homology, Ozsvath, Szabo and Thurston defined an invariant of transverse knots in the tight contact 3-sphere. Shortly afterwards, Lisca, Ozsvath, Stipsicz and Szabo defined an invariant of transverse knots in arbitrary contact 3-manifolds using open book decompositions. It has been conjectured that these invariants agree where they are both defined. In this talk I prove this fact by defining yet another invariant of transverse knots, showing that this third invariant agrees with the two mentioned above. This is a joint work with John A. Baldwin and David Shea Vela-Vick.
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