Time and place: the seminar meets 4:30-6pm, usually on Mondays in 4-265, though some weeks it may switch to Wednesdays in 2-131.
| Monday February 12, 2007 | Organizational meeting at 4:15pm in 4-265. This will be a brief meeting to talk about ideas for the upcoming semester, and settle the question of whether the seminar will regularly meet on Monday or Wednesday. All are welcome! |
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| Wednesday February 21,
2007 4:30-6pm in 2-131 |
Katrin Wehrheim:
Lagrangian correspondences, holomorphic quilts, and a 2-categorification
functor
Abstract: This will be a continuation of the geometry seminar talk on my joint work with Chris Woodward. I will recall the definition of a symplectic category (whose morphisms are generalized Lagrangian correspondences) and extend it to a 2-category whose 2-morphism spaces are Floer homology groups. Next, I will recall the refined Donaldson category associated to a symplectic manifold (whose objects are generalized Lagrangians and morphisms are Floer homology classes). Then, to a Lagrangian correspondence, I will associate a functor between Donaldson categories. Finally, I will extend these constructions to a 2-functor from the symplectic 2-category to the 2-category (categories, functors, natural transformations). These algebraic structures arise naturally from holomorphic quilts and all proofs can be given by pictures. |
| Wednesday February 28,
2007 4:30-6pm in 2-131 |
Chris Wendl:
Gromov compactness and intersection theory in dimension four
Abstract: Gromov's compactness theorem is wonderful, but it does allow some bad things to happen: somewhere injective curves can converge to multiple covers, or arbitrarily large nodal curves with multiply covered components, for which transversality fails and a big analytical headache ensues. In dimension 4, however, there are situations where none of these bad things can happen, due to intersection theory: the general rule seems to be that if the curves in a sequence are somehow "as nice as possible", then so is their limit. I will explain what I mean by that, and prove some partial results to illustrate it, both in closed symplectic 4-manifolds and in symplectizations or symplectic cobordisms. This is work in progress toward a general compactness theory for "good" holomorphic curves in low dimensions. |
| Wednesday March 7,
2007 4:30-6pm in 2-131 |
Tim Perutz (Cambridge, UK):
Broken fibrations and Lagrangian correspondences
Abstract: I'll explain how to use the geometry of a broken (Lefschetz) fibration on a 4-manifold to obtain a moduli space of pseudo-holomorphic curves from which numbers resembling Seiberg-Witten invariants can be extracted. The crucial point is the construction of Lagrangian correspondences between (symplectic) symmetric products of surfaces. |
| Monday March 19,
2007 4:30-6pm in 4-265 |
Yaron Ostrover:
On the uniqueness of Hofer's metric
Abstract: One of the most remarkable facts regarding the group of Hamiltonian diffeomorphisms is that it carries an intrinsic geometry given by a Finsler bi-invariant metric. This metric was first discovered by Hofer in 1990. Hofer's metric yields a geometric intuition for Hamiltonian systems and it can be used in many ways as a tool in symplectic geometry and dynamics. In this talk we will concentrate on the following question: are there other Finsler-type bi-invariant metrics on the group of Hamiltonian diffeomorphisms which are not equivalent to Hofer's metric? Or in other words: whether Hofer's metric is unique. |
| September 19, 2006 | Chris Wendl:
Intersection theory for punctured holomorphic curves Abstract: I'll briefly review the adjunction formula for closed holomorphic curves in symplectic 4-manifolds, and then introduce its extension (due to R. Siefring) to punctured curves in 4-dimensional symplectic cobordisms. The key is to understand what "intersections at infinity" are, how to count them and why they're positive. If time permits, I'll mention applications to the theory of J-holomorphic foliations. |
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| September 26, 2006 | Katrin Wehrheim:
Symplectic Vortex Equations in Gauge Theory and the
Atiyah-Floer conjecture Abstract: Symplectic vortex equations were introduced (by Cieliebak-Gaio-Salamon and recently Mundet-Tian) to define invariants for Hamiltonian group actions on symplectic manifolds. In certain cases, Gaio-Salamon also established an isomorphism between these invariants and the Gromov-Witten invariants of the quotient via an adiabatic limit from symplectic vortices to holomorphic curves in the quotient. The Atiyah-Floer conjecture, relating anti-selfdual instantons to holomorphic curves in a moduli space of flat connections, can be seen as infinite dimensional analogue of this result. The aim of this talk is to provide
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| October 3, 2006 | Yaron Ostrover:
Quasi-morphisms on the group of Hamiltonian Diffeomorphisms Abstract: A distinguished result by A. Banyaga states that the group of Hamiltonian Diffeomorphisms of a closed symplectic manifold is simple. In particular, it does not admit any non-trivial homomorphism to the real line. In this talk, following a work by M. Entov and L. Polterovich, I will discuss the existence of Calabi quasi-morphisms defined on the group of Hamiltonian Diffeomorphisms. I will focus on the spectral invariants used to define them and on the required algebraic properties of the quantum homology algebra. |
| October 10, 2006 | Matt Hedden:
Floer homology for knots and applications to symplectic and
contact geometry in dimensions three and four Abstract: This talk will address some questions related to knot theory in the context of symplectic and contact geometry in dimensions three and four.
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| October 17, 2006 | Matt Hedden:
Introduction to knot Floer homology and applications Abstract: Last time I introduced two questions I'm interested in and which I have made progress on using a Floer homology theory for knots introduced by Ozsvath and Szabo, and independently by Rasmussen (see abstract from last week). The questions deal with knots that arise in the context of symplectic and contact geometry in dimensions three and four. If you know something about contact geometry in dimension three then you need not have attended last week. This week I will restate the questions and theorems from last time and then, as requested, I'll delve into an introduction to knot Floer homology. I will try to focus on the algebraic structure of knot Floer homology and on the many applications this knot invariant has to geometric questions in low-dimensional topology. In particular, I will define and discuss the Ozsvath-Szabo contact invariant, a powerful invariant of a contact structure living in the Ozsvath-Szabo Floer homology of the three-manifold supporting the contact structure. I will survey some major results about the contact invariant and also concerning knot Floer homology. I hope to conclude by defining the invariant I alluded to last time which I use to provide upper bounds for the Thurston-Bennequin and rotation numbers of Legendrian knots in contact manifolds with non-vanishing Ozsvath-Szabo contact invariant. |
| October 24, 2006 | Matt Hedden:
Introduction to knot Floer homology and applications (continued
from last week) |
| October 31, 2006 | Brett Parker:
Exploded Torus Fibrations Abstract: This is a talk for non-experts. You may like this talk if you are interested in any of the following things:
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| November 7, 2006 | Brett Parker:
Exploded torus fibrations (part 2) Abstract: I will explain why log smooth structures are useful for encoding the smooth structure at infinity of a manifold with products of cylindrical ends. I will define the normal neighbourhood bundle, which in the case of a manifold with a cylindrical end, gives a formal "cylinder at infinity". On the large scale, these log smooth spaces look like stratified affine cones. I will use this to motivate the definition the explosion of a log smooth space, and define the category of exploded fibrations. |
| November 14, 2006 | NO SEMINAR THIS WEEK |
| November 21, 2006 | Brett Parker:
Exploded torus fibrations (part 3) |
| November 28, 2006 | NO SEMINAR THIS WEEK
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| December 5, 2006 |
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| December 12, 2006 | Nuno Romao:
Gauged vortices and symplectic geometry Abstract: Gauged vortices have recently entered the symplectic world as an ingredient to construct invariants of hamiltonian group actions: the so-called symplectic vortex invariants are maps counting certain points on the moduli spaces of solutions to the vortex equations on a compact Riemann surface. In this talk, I will tell a different story about these moduli spaces, starting from the fact that they support natural Kaehler structures. From the viewpoint of physics, they can be regarded as finite-dimensional truncations of configuration or phase spaces for dynamics in gauge theories. My aim is to explain some recent results on the geometry of vortex moduli spaces and their physical implications, focusing on aspects related to symplectic geometry. |
| December 19, 2006 | only if we really feel like it
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The Symplectic Working Group Seminar is an informal venue for discussing topics of current research in symplectic geometry and J-holomorphic curves. The longer than average time slot is designed to accomodate a relaxed pace with plenty of room for questions and discussion, but potentially also more detail than the average weekly 1-hour seminar. The speakers are usually (unless otherwise noted) from MIT, and we will try to keep the schedule flexible so that talks can span multiple weeks when necessary.
Most weeks, seminar participants go out to dinner (somewhere nearby and inexpensive) after the seminar. All are welcome to join!
For information contact Chris Wendl at wendlc@REMOVETHISmath.mit.edu.