** **

**Abstract:**

Any
smooth map of compact oriented manifolds

induces
a map in homology, which by PoincarĂ©

duality,
corresponds to a map in cohomology,

called
the push-forward map or the Gysin map. The

calculation
of the push-forward map for various

flag
bundles plays an important role in

enumerative
algebraic geometry.

Suppose
a Lie group acts on a compact oriented

manifold
with isolated fixed points.

The
Atiyah--Bott--Berline--Vergne localization

formula
converts an integral over the manifold to

a
finite sum over the fixed points. Surprisingly,

this
localization formula provides a systematic

method
for calculating the Gysin map of a fiber

bundle,
allowing us to recover some beautiful

classical
formulas. This talk assumes no

knowledge
of the localization formula. I will

explain
the background in equivariant cohomology,

the
localization formula, and how to apply it to

the
problem of the Gysin map.