Geometry and topology of symplectic 4-manifolds
April 24-26, 2015, University of Massachusetts, Amherst, MA
Organizing Committee
R. Inanc Baykur (UMass Amherst), Weimin Chen (UMass Amherst) and Daniel Ruberman (Brandeis University)
Invited speakers, participants
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Goals
The aim of this 3-day conference is to bring together active researchers in geometry and topology of symplectic 4-manifolds, ranging from leading experts to recent PhDs and graduate students, and provide a panorama of the field through a variety of talks and discussion sessions.
Registration and support
[Funding applications are now closed.]
We ask all the participants to register at
this link.
If you have troubles with viewing or editing the linked form, contact [ruberman at brandeis dot edu].
Funding is available for conference participation. Support requests should be made at the registration link by March 20, 2015. We particularly encourage graduate
students and recent PhDs to apply. For full consideration, junior participants should also have a reference letter
e-mailed to [baykur at math dot umass dot edu] by the funding request deadline.
We have reserved rooms for all the invited speakers and participants coming from outside of Western Massachusetts at the
UMass Hotel and Conference Center. Please provide your planned
dates of stay and accommodation preferences at the registration link above. Further inquiries regarding
accommodation should be made to [wchen at math dot umass dot edu].
This event is sponsored by the National Science Foundation Grant DMS-1522633.
Travel
The closest airport to Amherst is the Bradley (Hartford/Springfield) Airport, and there are shuttle services from/to UMass Amherst. The second major airport in the area is the Logan (Boston) Airport (1,5-2hrs distance), which however requires an additional bus commute between Boston and Amherst. Greyhound buses serve between Amherst and many other locations such as Boston and New York.
Schedule
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Friday, April 24 9:00 - 9:30am Registration 9:30 - 10:30am Vidussi 10:30 - 11:00am TEA 11:00 - 12:00pm Li 2:00 - 3:00pm Yeung 3:00 - 3:45pm REFRESHMENTS 3:45 - 4:45pm Sivek 5:15 - 6:45pm Discussion |
Saturday, April 25 9:30 - 10:30am Gay 10:30 - 11:00am TEA 11:00 - 12:00pm Wendl 2:00 - 3:00pm Gompf 3:00 - 3:45pm REFRESHMENTS 3:45 - 4:45pm Starkston 5:15 - 6:45pm Discussion |
Sunday, April 26 9:30 - 10:30am Matic 10:30 - 10:45am TEA 10:45 - 11:45am Dimitroglou Rizell |
All talks and the discussion sessions will be held at Lederle Graduate Research Tower, Room 1634. Tea and refreshments will be served in the lounge area next to LGRT 1634.
Titles and abstracts
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David Gay
Symplectic trisections
Trisections are to 4-manifolds as Heegaard splittings are to 3-manifolds. They are also strongly related to PALF's on 4-manifolds with boundary, and there is an appropriate relative notion of a trisection restricting to an open book decomposition on the boundary. All this suggests that the next step forward is a proper notion of a symplectic trisection, and this talk will discuss progress in that direction. -
Georgios Dimitroglou Rizell
The classification of monotone Lagrangian tori in a four-dimensional symplectic vectorspace
We prove that there are exactly two monotone Lagrangian tori in a four-dimensional symplectic vectorspace up to Hamiltonian isotopy and rescaling: the Clifford torus and the Chekanov torus. This is shown by, first, finding a singular symplectic conic linking the torus appropriately and, second, applying a classification result for homotopically non-trivial Lagrangian tori inside the cotangent bundle of a two-torus. The latter result follows using methods due to Ivrii. -
Robert Gompf
Degree raising for Lefschetz pencils
Donaldson's construction of Lefschetz pencils on symplectic 4-manifolds hints that there should be a natural way to raise degree. That is, for every positive integer k, there should be a canonical algorithm that takes a Lefschetz pencil and returns another Lefschetz pencil on the same symplectic manifold, such that the fiber homology class of the new pencil is k times that of the old one. Auroux and Katzarkov produced such an algorithm for k=2. We will discuss preliminary research aimed at understanding what this algorithm is doing, and how it might generalize to arbitrary k. -
Tian-Jun Li
Curve configurations in symplectic 4-manifolds
We investigate the notion of symplectic divisorial compactification for symplectic 4-manifolds with concave or convex boundary, which is motivated by the notion of compactifying divisors in algebraic geometry. For this purpose we systematically analyze the local contact geometry of curve configurations. This is joint work with Cheuk Yu Mak. -
Gordana Matic
Heegaard Floer contact invariant and algebraic torsion
Ozsvath and Szabo used the Giroux's correspondence between open book decompositions and contact structures to define a contact invariant in Heegaard Floer homology. Latschev and Wendl defined "algebraic torsion" for contact structures in the context of symplectic field theory. We apply Hutchings's prescription for defining algebraic torsion in embedded contact homology to the Heegaard Floer setting by exploiting a parallel between it and a variant of ECH developed by Kutluhan, Lee and Taubes. Given an open book decomposition and a basis of arcs, we define a relative filtration on the corresponding Heegaard Floer chain complex and a non-negative integer quantity associated to this data. This is joint work in progress with Cagatay Kutluhan, Jeremy Van Horn-Morris and Andy Wand. -
Steven Sivek
Augmentations of Legendrian knots and constructible sheaves
Given a Legendrian knot in R^3, Shende-Treumann-Zaslow defined a category of constructible sheaves on the plane with singular support controlled by the front projection of the knot. They conjectured that this is equivalent to a category determined by the Legendrian contact homology of the knot, namely Bourgeois-Chantraine's augmentation category. Although this conjecture is false, it does hold if one replaces the augmentation category with a closely related variant. In this talk, I will describe this category and some of its properties and outline the proof of equivalence. This is joint work with Lenny Ng, Dan Rutherford, Vivek Shende, and Eric Zaslow. -
Laura Starkston
Symplectic embeddings and mapping class monoid relations
For certain contact 3-manifolds supported by a planar open book decomposition, there are two ways of constructing and classifying symplectic fillings whose boundary is that contact 3-manifold. One way involves understanding factorizations of the monodromy of the open book into positive Dehn twists. The other way is to look at embeddings of concave neighborhoods of a collection of symplectic surfaces into a well understood (rational) symplectic 4-manifold. Depending on the bounding contact 3-manifold, each of these methods has different strengths and data from one method can provide information about the other. Therefore it is very useful to understand how to translate the information coming from symplectic surface embeddings to mapping class group relations or vice versa. The goal of this talk will be to discuss some correspondences between these two methods in large classes of examples, where the boundary 3-manifold is Seifert fibered over the 2-sphere. -
Stefano Vidussi
Some remarks on (virtual) properties of Kaehler groups
It is known that there is no infinite, finitely presented group that is simultaneously the fundamental group of a Kaehler manifold and of a closed 3-manifold. I will ignore this fact, and I'll try to convince you (and myself) that it is anyhow interesting to look at what properties could Kaehler groups share with 3-manifold groups. -
Chris Wendl
Spine removal surgery and its applications
Spine removal surgery is a topological operation that can be performed on contact manifolds presented as spinal open books, a generalized notion of open books which arise naturally on boundaries of Lefschetz fibrations over Liouville domains. I will first explain how this operation is defined and why it gives rise to a non-exact symplectic cobordism. Then I will illustrate it with applications, some new, and some that are old but not normally viewed from this perspective, including: (1) Eliashberg's construction of 4-dimensional symplectic caps and its extension to Stein fillable contact manifolds in all dimensions (originally due to Lisca and Matic using very different methods); (2) Gay's cobordism proving that contact 3-manifolds with Giroux torsion are not fillable; (3) Some newer nonfillability results in dimension three; (4) Existence of non-exact symplectic cobordisms in cases where exact cobordisms are obstructed; (5) Constructions of tight but nonfillable contact manifolds in all dimensions. This is mostly joint work with Sam Lisi and Jeremy Van Horn-Morris (except for the part that is instead joint with Patrick Massot and Klaus Niederkrueger). -
Sai-Kee Yeung
Geometric and topological aspects of fake projective planes and finite groups
The purpose of the talk is to explain some geometric and topological results related to fake projective planes, which were classified recently in the work Prasad-Yeung and Cartwright-Steer. Aspects to be discussed include examples of exotic fourfolds arising from fake projective planes, a conjecture related to vanishing theorems on fake projective planes, and an open problem related to surfaces of maximal canonical degree. We would explain the role of finite group actions in such problems.