3. Postdoctoral Program - MSRI

Symplectic Geometry, Noncommutative Geometry and Physics
May 10 to May 14, 2010, MSRI, Berkeley, CA, USA
One of the principal aims of the Hayashibara forum is to bring together
researchers from geometry and physics, both to make a serious attempt to
overcome conceptual barriers between experts and to expose these areas to
younger researchers. A synthesis of ideas from geometry and physics should
prove to be extremely powerful. This program is aimed at enhancing the
understanding of the interaction between these subjects among researchers
from both fields.
The conference included discussions on recent work with the explicit
goal of furthering interactions between mathematicians and physicists. We
anticipate an expanded interest in these interactions and the realization
that experts and students in each field can indeed work in the other. To
this end, this conference contained mini-course lectures aimed at increasing
communication in mathematics and physics.
2 Highlights of the presentations
One striking feature of the workshop was the inclusion of three mini-course
lectures aimed at increasing the interaction between mathematicians and
physicists, and also to encourage young researchers to explore both fields.
The first lecturer, Yan Soibelman (Kansas), presented three lectures on
an overview of his joint work with Maxim Kontsevich on motivic Donaldson-
Thomas invariants for 3D Calabi-Yau categories. Soibelman presented two
approaches, one based on the ideas of motivic integration, and the second
based on the notion of cohomological Hall algebras.
The second mini-course lecturer was Dennis Auroux (MIT), who gave
lectures on special Lagrangian torus fibrations and mirror symmetry. These
lectures focused on the construction of mirror manifolds using special Lagrangian
fibrations, with the Strominger-Yau-Zaslow conjecture as a starting
point. The main goal was the construction of a mirror manifold to a Kähler
manifold with effective anticanonical class, using a special Lagrangian torus
fibration and enumerative geometry data (weighted counts of holomorphic
discs). Auroux’s first talk provided motivation for the SYZ conjecture and
basic examples. In particular, he explained how Landau-Ginzburg models
naturally arise in the non-Calabi-Yau setting, viewing the superpotential as
a mirror counterpart to a Floer-theoretic obstruction. The main example is
toric Fano varieties. In the second talk, Auroux presented a simple example
of the wall-crossing phenomena arising in the non-toric case, to motivate
“instanton corrections”. Finally, he discussed joint work in progress with
Mohammed Abouzaid and Ludmil Katzarkov on the extension of mirror
symmetry to arbitrary hypersurfaces in toric varieties, by considering Lagrangian
fibrations on blow-ups. Here the main examples are pairs of pants
(and their higher-dimensional analogues) and higher-genus curves.
The third mini-course lecturer was Katrin Wehrheim (MIT), who gave
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