December 7-11, 2015

Video Lectures can be found here: https://www.ias.edu/video/workshop/2015/1207-11

** All talks will take place in Simonyi Hall Seminar Room

Monday, 12/7

10:00 am - 11:00 am Nathan Dunfield, IAS, "Floer homology, group orders, and taut foliations of hyperbolic 3-manifolds"

Abstract: A bold conjecture of Boyer-Gorden-Watson and others posit that for any irreducible rational homology 3-sphere M the following three conditions are equivalent: (1) the fundamental group of M is left-orderable, (2) M has non-minimal Heegaard Floer homology, and (3) M admits a co-orientable taut foliation. Very recently, this conjecture was established for all graph manifolds by the combined work of Boyer-Clay and Hanselman-Rasmussen-Rasmussen-Watson. I will discuss a computational survey of these properties involving several hundred thousand hyperbolic 3-manifolds. Parts of this are joint with Marc Culler and Mark Bell.

11:00 am - 11:30 am COFFEE BREAK, Simonyi Hall Common Room (2nd floor)

11:30 am - 12:30 pm Jonathan Bowden, Univ Munich, "Contact Structures, foliations and group actions"

Abstract: We use contact deformations to study the topology of the space of taut foliations and group actions on the circle.

12:30 pm - 2:30 pm LUNCH, Dining Hall

2:30 pm - 3:30 pm Jennifer Hom, IAS "Positive-definite symplectic four-manifolds"

Abstract: We give new constraints on the topology of symplectic four-manifolds using invariants from Heegaard Floer homology. In particular, we will prove that certain simply-connected four-manifolds with positive-definite intersection forms cannot admit symplectic structures. This is joint work with Tye Lidman.

3:30 pm - 4:00 pm TEA, Fuld Hall Common Room

4:00 pm - 5:00 pm Andy Wand, Univ Glasgow, "Tight, non-fillable contact structures on 3-manifolds"

Abstract: The modern development of contact geometry in 3 dimensions has seen several (due to Giroux, Wendl, Latschev and Wendl, Hutchings, and others) invariants of contact structures meant in some sense to measure non-(Stein /symplectic)-fillability of the structure. Time permitting, we will discuss two new approaches, which rely on Giroux's theory of open book decompositions: the first a more topological construction generalizing a characterization of tightness in terms of open book decompositions, the second (in joint work with Kutluhan, Matic, and Van Horn-Morris) a refinement of the Heegaard-Floer contact class, inspired by the `algebraic torsion' of Latschev and Wendl, and Hutchings.

 

Tuesday, 12/8

10:00 am - 11:00 am Sergio Fenley, IAS, "Quasigeodesic pseudo-Anosov flows in hyperbolic 3-manifolds

Abstract: We obtain a simple topological and dynamical systems condition which is necessary and sufficient for an arbitrary pseudo-Anosov flow in a closed, hyperbolic three manifold to be quasigeodesic. Quasigeodesic means that orbits are efficient in measuring length up to a bounded multiplicative distortion when lifted to the universal cover. We prove that such flows are quasigeodesic if and only if there is an upper bound, depending only on the flow, to the number of orbits which are freely homotopic to an arbitrary closed orbit of the flow.

11:00 am - 11:30 am COFFEE BREAK, Simonyi Hall Common Room (2nd floor)

11:30 am - 12:30 pm Ciprian Manolescu, UCLA, "Floer homology and covering spaces"

Abstract: I will discuss a Smith-type inequality for regular covering spaces in monopole Floer homology. Using the monopole Floer / Heegaard Floer correspondence, it follows that if a 3-manifold Y admits a p^n-sheeted regular cover that is a Z/p-L-space (for p prime), then Y is a Z/p-L-space. Further, we obtain constraints on surgeries on a knot being regular covers over other surgeries on the same knot, and over surgeries on other knots. This is joint work with Tye Lidman.

12:30 pm - 2:30 pm LUNCH, Dining Hall

2:30 pm - 3:30 pm Thomas Vogel, Ludwig-Maximilians-Universität München, "Uniqueness of the contact structure approximating a foliation"

Abstract: A well known result of Eliashberg and Thurston states that smooth foliations can be approximated by contact structures. We discuss the uniqueness of this contact structure and applications.

3:30 pm - 4:00 pm TEA, Fuld Hall Common Room

4:00 pm - 5:00 pm Patrick Massot, CMLS, "Curvature and contact topology"

 

Wednesday, 12/9

10:00 am - 11:00 am Laura Starkston, Univ Texas, "Symplectic fillability of contact graph manifolds via line arrangements"

Abstract: An interesting aspect of the classification of symplectic fillings of Seifert fibered spaces is the appearance of complex and symplectic line arrangements in CP2. Line arrangements have been studied classically for decades and have intricate combinatorial and geometric realization properties, but little is known about their topological and symplectic realizations. In joint work with Danny Ruberman, we find obstructions to topological realizations of certain configurations. I will discuss how various types of information about symplectic realizations of line arrangements can be used to produce interesting symplectic fillings of Seifert fibered spaces and contact graph manifolds which are tight but not sympletically fillable.

11:00 am - 11:30 am COFFEE BREAK, Simonyi Hall Common Room (2nd floor)

11:30 am - 12:30 pm Cameron Gordon, Univ Texas, "Taut foliations and cyclic branched covers"

12:30 pm - 2:30 pm LUNCH, Dining Hall

2:30 pm - 3:30 pm Ivan Dynnikov, MSU, "Convex surfaces and grid diagrams"

Abstract: I will speak about a simple way to represent surfaces in the three-space by diagrams similar to rectangular diagrams of links. A set of moves will be given by which one can connect any two diagrams representing isotopic surfaces. By forbidding some of them we get a similar setup for isotopy classes of convex surfaces (with respect to the standard contact structure), which sometimes can be used to distinguish Legendrian knots. This is a joint work with Maxim Prasolov.

3:30 pm - 4:00 pm TEA, Fuld Hall Common Room

4:00 pm - 5:00 pm Tye Lidman, IAS, "Contact structures and reducible surgeries"

 

Thursday, 12/10

10:00 am - 11:00 am John Pardon, Stanford Univ, "Contact homology and virtual fundamental cycles"

Abstract: Contact homology is a powerful invariant of contact manifolds introduced by Eliashberg--Givental--Hofer. The definition involves certain counts of pseudo-holomorphic curves, however these are usually only "virtual" counts since the moduli spaces of such curves are often not cut out transversally. I will discuss one way to construct these counts rigorously.

11:00 am - 11:30 am COFFEE BREAK, Simonyi Hall Common Room (2nd floor)

11:30 am - 12:30 pm Steve Boyer, UQAM, "Foliations on graph manifolds, orderability and L-spaces"

12:30 pm - 2:30 pm LUNCH, Dining Hall

2:30 pm - 3:30 pm Steven Frankel, IAS, "Coarse hyperbolicity and closed orbits for quasigeodesic flows"

3:30 pm - 4:00 pm TEA, Fuld Hall Common Room

4:00 pm - 5:00 pm Josh Greene, Boston College, "Definite surfaces and alternating links"

 

Friday, 12/11

10:00 am - 11:00 am Rachel Roberts, WUSTL, "Taut co-oriented foliations"

Abstract: Eliashberg and Thurston proved that the tangent plane field of any $C^2$ taut oriented foliation $\mathcal F \ne S^1\times S^2$ can be $C^0$ approximated by a pair of particularly nice smooth contact structures. Kazez and Roberts proved that the requirement that $\mathcal F$ be $C^2$ can be replaced by the weaker condition that $\mathcal F$ have continuous tangent plane field. A very similar result was obtained independently by Bowden.

I will discuss some constructions of taut, co-oriented foliations in the context of this theorem.

11:00 am - 11:30 am COFFEE BREAK, Simonyi Hall Common Room (2nd floor)

11:30 am - 12:30 pm Andras Stipsicz, Alfred Reyni Inst of Math, "Tight contact structures on Seifert fibered 3-manifolds"

12:30 pm - 2:30 pm LUNCH, Dining Hall

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