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\title{Math648: Topics in Differential Geometry \\
	Spring 2002 \\
	}
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\noindent
{\bf TIME:} \hspace{5pt}
MWF: 12:50 - 1:40.

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\noindent
{\bf INSTRUCTOR:} \hspace{5pt}
Xiaobo Liu.

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\noindent
{\bf  COURSE DESCRIPTION:} 


{\it Calibrated
geometry} studies distinguished submanifolds of a Riemannian
manifold whose tangent spaces realizes maximum values of some
closed differential forms. Such submanifolds 
necessarily have the smallest volume
among all submanifolds in the same homology classes (In particular,
they are always minimal submanifolds in the sense that mean curvature
is constantly equal to zero).
Well known examples include all complex submanifolds of K\"{a}hler
manifolds, special Lagrangian submanifolds of Calabi-Yau manifolds,
associative cycles and Cayley cycles in manifolds with $G_{2}$ or
$Spin(7)$ holonomy. Recently, calibrated submanifolds
have attracted lots of attention of physicists 
because they provide best candidates for supersymmetric cycles
in string theory. Special Lagrangian submanifolds also get
lots of popularity in mirror symmetry due to a well known conjecture
by Strominger-Yau-Zaslow which predicts that the mirror of a
Calabi-Yau manifold should be obtained by dualizing some
special Lagrangian submanifolds which are diffeomophic to tori. 
In this course we will first study
basic theories of calibrated geometry and related materials. 
If time permits, we will also study some recent papers in this field,
especially about special Lagrangian geometry.


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{\bf REFERENCE:}

1. Reese Harvey,
    {\it Spinors and Calibrations}, Academic Press, Inc., 1990.

2. Reese Harvey, and B. Lawson, 
    {\it Calibrated Geometries}, Acta Math. 148 (1982) 47-157.


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