#
Calibrations

June 26 to June 30, 2006
at the

American Institute of Mathematics,
Palo Alto, California

organized by

Robert Bryant,
Xiaobo Liu,
and Pit-Mann Wong

## Original Announcement

This workshop will be devoted to
problems in calibrated geometry related to mirror symmetry and
gauge theory.
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:

- Calibrated cycles and mirror symmetry
- Calibrated geometry and gauge Theory
- Moduli spaces of calibrated cycles

## Material from the workshop

A list of participants.
The workshop schedule.

A report on the workshop activities.