3. Postdoctoral Program - MSRI

REPORT ON THE MSRI WORKSHOP: ALGEBRAIC
STRUCTURES IN THE THEORY OF HOLOMORPHIC
CURVES
Organizers.
• Mohammed Abouzaid* (Clay Mathematics Institute)
• Yakov Eliashberg (Stanford University)
• Kenji Fukaya (Kyoto University)
• Eleny Ionel (Stanford University)
• Lenny Ng (Duke University)
• Paul Seidel (MIT).
Modern symplectic topology arose in the early 1980’s from a combination
of different sources and techniques. This was soon followed by Gromov’s introduction
of holomorphic curves into the subject. These have now become
the mainstream technique, both for questions internal to symplectic topology,
and for interactions with other disciplines (such as enumerative geometry,
through Gromov-Witten theory, and dynamics, through problems such
as the the Weinstein and Arnold conjectures). Progress in these questions
has become increasingly tied to an ever more sophisticated understanding of
the algebraic structures which may be constructed from the moduli spaces
of holomorphic curves. The workshop brought together experts on different
aspects of these algebraic structures, and a fruitful discussion of ongoing
and future developments took place during it.
There was a significant representation of several branches of symplectic
topology, including Gromov-Witten theory, Symplectic Field Theory, and
the study of Lagrangian Floer homology. Most of the lectures spanned
more than one of these areas, providing an interesting forum for interactions
among mathematicians working on similar problems using different
techniques. In addition, connections with tropical geometry (the topic of
the other MSRI program at the time) were explored.
Workshop Activities
The workshop started with a talk by Paolo Rossi, who was one of three
MSRI postdocs giving lectures. He described the integrable systems which
appear in symplectic field theory. These extend the well-studied topological
recursion formulae in Gromov-Witten theory, whose connections with mirror
symmetry formed the subject of Boris Dubrovin’s talk. Gromov-Witten
invariants conjecturally have additional structures beyond those of topological
type: Alexander Givental gave a proof of the Virasoro conjecture for
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