3. Postdoctoral Program - MSRI

WORKSHOP REPORT 3
isotopic to each other, and which are all non-displaceable. Ono’s talk used
the theory developed by Fukaya, Oh, Ohta and Ono; Cheol-Huyn Cho provided
in his lecture a series of constructions of homological invariants which
one can extract from their theory.
The remaining talks on Lagrangian Floer homology focused on connections
with mirror symmetry. Kenji Fukaya explained how to prove homological
mirror symmetry for K3 surfaces by constructing the mirror as a
moduli space of objects of the Fukaya category coming from the fibres of a
Lagrangian torus fibration. Melissa Liu gave a proof of mirror symmetry for
toric varieties and orbifolds using the correspondence between Lagrangians
in the cotangent bundle of a torus and constructible sheaves on the base. Finally,
Ivan Smith presented a new approach for using ideas from homological
mirror symmetry to prove the faithfulness of the representation of the mapping
class group into the symplectomorphism group of the representation
variety.
Besides talks, the workshop presented an opportunity for numerous group
discussions which were very productive and led to new collaborations. In
particular, one such group discussed different approaches for understanding
the precise relation between Weinstein handlebodies and symplectic Lefschetz
fibrations. Conjecturally these two structures are equivalent, but at
the moment the details remain to be worked out. The discussion of this
problem at MSRI yielded new ideas which should be useful for addressing
this question. A solution would be relevant for many applications.
The MSRI workshop was preceded a week earlier by a meeting at the
American Institute of Mathematics (AIM), which was devoted specifically
to the work of Bourgeois-Ekholm-Eliashberg and its ramifications. The two
meetings were not conceived as continuations of each other; rather, the AIM
workshop was more technical and specialized, while the MSRI workshop had
a wider scope, with more diverse perspectives represented, and addressed itself
to a broader audience. In our opinion, this was a successful combination
which strengthened both workshops.