3. Postdoctoral Program - MSRI

and projective spaces. Abreu announced some new computation concerning classical Lagrangian
intersection problems. It is interesting that these computations have been approached from several
related, but algebraically different, points of view. Seidel outlined a breath-taking route from the
classical singularities theory to a novel count of Lagrangian intersection points.
Relations with low-dimensional topology: Contact topology has been surprisingly useful in
illuminating the nature of new invariants of 3-manifolds, most notably the Heegaard-Floer invariant
of Ozsváth and Szabó. In Matić’s talk she outlined the construction of invariants of contact
structures on a three manifold with boundary that live in sutured Heegaard-Floer homology groups.
These invariants are not only useful invariants of contact structures but also help one define various
gluing maps in Heegaard Floer theory itself. Honda’s talk sketched the proof, done in collaboration
with Colin and Ghiggini, that � HF (M) is isomorphic to �ECH(M). This long sought result, coupled
with Hutchings and Taubes’s identification of �ECH(M) and Seiberg-Witten Floer Homology,
proves that Heegaard-Floer homology is really the same as Seiberg-Witten Floer homology. A key
insight in this proof involves the contact invariants discussed by Matić in her talk.
Relations with other areas: In Abouzaid’s talk he gave a beautiful construction of a paralellizable
bounding manifold of an exotic smooth sphere that embeds as a Lagrangian submanifold in the
cotangent bundle of the standard sphere. This results shows that the symplectic geometry of the
cotangent bundle to manifolds can detect subtle differences in the smooth topology of the manifold.
In addition, this construction involves an understanding of delicate features of the moduli space of
holomorphic curves.
Abreu returned to a classical topic of symplectic and contact manifolds which admit Hamiltonian
group actions of maximal dimension. Surprisingly, some of these lead to exotic contact structures
on products of spheres. Tori actions on contact manifolds are less understood than their symplectic
counterpart. Some foundational issues related to convexity of the image of the contact moment
maps were addressed in the talk by Karshon. Tolman presented new topological restrictions on
symplectic manifolds admitting Hamiltonian circle actions with “small” fixed point sets.
Finally, Tamarkin described an intriguing approach to the classical symplectic intersection problem
based on micro-local analysis of sheaves in the spirit of Kashiwara-Shapira.
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