3. Postdoctoral Program - MSRI

was also very helpful as a preparation for two workshops of the program devoted to algebraic
structures in the theory of holomorphic curves.
2. Integrable Structures. The Integrable Structures workgroup was organized by Oliver Fabert,
Paolo Rossi, and Dimitri Zvonkine who also gave almost all talks in the workgroup. There
were covered four topics:
- Integrable systems via infinite-dimensional Grassmannians;
- Integrable systems via pseudo-differential operators;
- From Frobenius manifolds to integrable systems: the Dubrovin-Zhang construction;
- Open problems.
The goals were to systematize our knowledge of integrable systems and try to attack the open
problems: explaining the appearence of integrable systems in SFT and in the intersection
theory of the space of r-spin structures; understanding the integrable system that describes
the Gromov-Witten invariants of a P 1 -bundle.
3. Polyfolds. The group was organized by Joel Fish, Oliver Fabert and Roman Golovko to
study the new foundational theory of polyfolds which is currently being developed by Hofer,
Wysocki and Zehnder. Given the size, scope, and quite technical nature of polyfolds and their
applications, the main goal of the working group was not to provide a complete treatment of
the subject, but rather to provide an overview which was accessible to a diverse audience of
symplectic and contact geometers. In particular, a distinct effort was made to present material
in an approachable manner to both specialists and non-specialists alike. The organizers took
turns presenting the material after frequent meetings amongst themselves to resolve difficulties
and enhance clarification. The term began with a discussion of the strengths and weaknesses
of alternate approaches to transversality problems. Next, the main definitions of the polyfold
theory were presented (e.g. sc-Banach spaces, the sc-calculus, sc-retractions, M-polyfolds,
strong bundles etc); illustrative examples were provided in parallel. With the basics established,
the working group moved on to the statements of the main theorems, namely the abstract
perturbation result which resolves transversality issues. Finally, the term culminated with a
discussion of polyfolds in a broader context, with an emphasis on how the abstract analytic
framework of polyfolds drastically reduces the amount of future work needed to build smooth
compact moduli spaces in a wide variety of settings. One of the goal was to create a “User Guide
to Polyfolds” which wold provide an entrance point to this large subject for mathematicians
interested in applications of the theory. Jointly with Katrin Wehrheim Fabert, Fish and
Golovko are working on producing the text.
4. Giroux Correspondence in higher dimensions. In this seminar organized by Vera Vertesi
participants tried to construct a complete proof of the Grioux correspondence between open
book decompositions and contact structures in high dimensions. While this was not ultimately
achieved a thorough outline of the program was understood along with many details and issues
that come up along the way. Speakers at this group included Lev Buhovski, Yang Huang,
Selman Akbulut, and Sonja Hohloch – people with very varied backgrounds.
SPRING: There were two working groups but there was an informal working group and much
participation in a working group for the other program on Boarded Heegaard Floer Homology.
1. Quantitative Symplectic Topology. This was run by Leonid Polterovich and Dusa McDuff.
The idea was to present some open problems and present recent relevant work. Some of the
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