American Institute of Mathematics, Palo Alto, California
Robert Bryant, Xiaobo Liu, and Pit-Mann Wong
Mirror symmetry provides a correspondence between the symplectic geometry of one Calabi-Yau manifold and the complex geomery of its mirror partner. A typical example is the correspondence between the combinatorial problem of counting the number of holomorphic curves to the theory of variation of Hodge structures.
One of the most important conjectures in Mirror symmetry is the SYZ conjecture of Strominger, Yau and Zaskow. The conjecture asserts that the mirror of a Calabi-Yau manifold X can be obtained by dualizing the fibers of a special Lagrangian toric fibration of X. The conjecture was partially motivated by the work of McLean on the moduli space of special Lagrangian cycles. Similar conjectures were formulated by Leung for manifolds with exceptional holonomy using other types of calibrated cycles.
The well-developed theory of pseudo-holomorphic curves in almost complex manifolds provides a guide to the sort of results one would like to generalize to calibrated cycles in Riemannian manifolds. Such results would be very useful in higher dimensional geometry. For example, applications of calibrations to gauge theory were proposed by Donaldson and Thomas for Calabi-Yau 3-folds and 4-folds. A unified approach to higher dimensional gauge theory for Riemannian manifolds of any dimension was proposed by Tian using a codimension 4 calibration. A connection of calibrations with Seiberg-Witten equations was found by Akbulut and Salur.
The main goals of the workshop are to clarify the connections between the aforementioned fields, and to identify some target results for both the short and long term. Some specific topics to be discussed include:
The workshop schedule.
A report on the workshop activities.