INTRODUCTION 3The lectures of Robert MacPherson show that real algebraic geometry is an-
other natural source and consumer of symplectic-geometric constructions and ideas.
A stratification of a manifold gives rise to a Lagrangian subvariety of its cotan-
gent bundle, called the characteristic variety of the stratification. Morse theory on
the stratification is a kind of group-valued intersection theory of Lagrangian vari-
eties. If the manifold is complex, intersection homology and more generally perverse
sheaves arise naturally from this point of view. The "centerpiece theorem" proved
in MacPherson's lectures states that when given two transverse, semi-algebraic
Whitney stratifications of an oriented, real algebraic manifold and two functions
which are constant on the respective strata then one can compute the intersection
product of the Lagrangian characteristic cycles of the functions in terms of the
Euler characteristic of the product of the functions.
Lisa Jeffrey and Jerrold Marsden explore in their lectures yet different areas of
symplectic geometry that overlap with physics.
Jeffrey's course describes the rich theory resulting from having a symplectic
action of compact, connected Lie group on a symplectic manifold. Jeffrey covers
the basic results about moment maps and symplectic quotients, defines equivariant
cohomology, and describes the important abelian localization theorem which says
that the equivariant cohomology classes with respect to torus actions can be calcu-
lated in terms of data from the fixed point set of the torus action. She then goes on
to discuss the nonabelian localization principle and the residue formula which relate
intersection pairings on the symplectic quotient to data on the original symplectic
manifold. Nonabelian localization has had two major applications. The first is that
the residue formula has been used to give a proof of formulas for intersection num-
bers in moduli spaces of vector bundles on Riemann surfaces. The second is that
nonabelian localization underlies some proofs of Guillemin and Sternberg's conjec-
ture that "quantization commutes with reduction". In her last lecture, she gives a
brief introduction to how the notions of group actions and symplectic quotients can
be generalized to infinite dimensions, with the motivating example being moduli
spaces of vector bundles over Riemann surfaces.
Marsden's lectures serve as an introduction to mathematical methods in me-
chanics with a broad overview of some recent developments. Reduction of a mechan-
ical system with symmetries is the original birthplace of the theory of symplectic
group actions and the theory of moment(um) maps. In his lectures, Marsden de-
velops an idea of reduction in the context of Hamiltonian, Poisson, and Lagrangian
mechanics and explores its relation with the theory of geometric phases. Next he
develops the theory of relative equilibria. All these theoretical issues are illustrated
and applied to underwater vehicle dynamics. Geometric phases are also discussed
in detail in the context of nonholonomic mechanics, in particular for systems with
rolling constraints. The last two lectures are devoted to the optimal control theory
and numerical methods in computational mechanics, namely the theory of vari-
ational integrators. The theory of mechanical integrators develops a numerical
scheme for systems with symmetries which attempts to preserve the symmetry ex-
actly by the integration process. One would also like to preserve the Hamiltonian or
the symplectic structure. It turns out that one cannot preserve both. The mechan-
ical integrators have been successfully used in a number of important applications.
Many people contributed to the enormous success of the 1997 Graduate Sum-
mer School. We thank the lecturers, who all worked hard to give clear and accessible
presentations, and we thank all the teaching assistants who led problem sessions