3. Postdoctoral Program - MSRI

prepare for the workshop so they can get the most out of it (like various background readings). In
any event, having this data would be useful in planning the activities of the workshop.
Details on the lecture series
Margaret Symington: Introduction to symplectic geometry. (5 lectures)
The five lectures “Introduction to Symplectic Geometry and Topology” began with some motivating
questions and remarkable results to give a bit of context to the material upcoming in this lecture
series and others. The first topic was linear symplectic algebra, which culminated in the relations
between the symplectic and complex linear groups that imply the equivalence of the classifications
of symplectic and complex vector bundles. After a variety of examples of symplectic manifolds
were given, including a detailed description of the canonical one-form on any cotangent bundle
and the ensuing exact symplectic form, the lack of local invariants near a point or a submanifold
was explained. The major topic of the lectures was constructions of symplectic manifolds, focusing
on blowingup and the symplectic sum, both explained in terms of symplectic cutting (and hence
symplectic reduction). The symplectic sum was then applied in Gompfs proof that every finitely
presented group is the fundamental group of some closed symplectic four-manifold. The last lecture
was devoted to toric manifolds with an emphasis on dimension four. The toric geometry gave further
insight into blowing up and down.
John Etnyre: Introduction to contact geometry. (5 lectures)
In these lectures the basic examples of contact manifolds were presented followed by a proof of
various “local theorems” like Darboux’s theorem and Gray’s theorem. We then proved the existence
of contact structures on 5-manifolds by using open book decompositions. The last two lectures in
the course were an introduction to convex surfaces and culminated in the classification of tight
contact structures on solid tori with various boundary conditions.
Katrin Wehrheim: Introduction to holomorphic curves. (5 lectures)
The course on pseudoholomorphic curves was guided by the proof of Gromov’s nonsqueezing theorem.
The first lecture introduced the geometric ideas and reduced the proof to the existence of a
holomorphic sphere in a certain homology class. The second lecture provided the setup for moduli
spaces of pseudoholomorphic curves: The Cauchy-Riemann operator, (non)-integrability of almost
complex structures, comparison theorems with holomorphic functions, reparametrization of pseudoholomorphic
maps, energy identities. The remaining three lectures introduced the students to
the standard tools for analyzing moduli spaces: Fredholm theory for sections of Banach bundles,
elliptic regularity for the Cauchy-Riemann operator, transversality for simpe curves, the bubbling
phenomenon, and Gromov compactness.
Dusa McDuff: Symplectic embedding problems and capacities. (4 lectures)
The first lecture gave an overview of different ways to make measurements in symplectic topology.
We then concentrated on embedding problems for balls and ellipsoids in 4 dimensions. This is
an interesting, explicit application of J-holomorphic curve techniques that involves understanding
some of the basic facts about symplectic 4-manifolds such as the uniqueness of the symplectic
structure on CP 2 . It also used toric models and the blowing up process introduced by Symington
in the first week.
Lenny Ng: Legendrian knots in contact 3-manifolds (4 lectures)
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