This course is a loose continuation of the course Algebraic Topology offered by me in the fall semester 1998. Thus some knowledge about singular homology theory, homological algebra and the like will be a useful, but not at all a necessary prerequisite. Even more helpful is to have some prior knowledge about differentiable manifolds (at least, in the guise of curves and surfaces in $\Bbb R^3$) and differential forms.
To begin with we will follow the excellent book by Madsen and Tornehave but probably we will also pick material from other sources, among them the notes by Hörmander (see the references below). I will likewise distribute material of my own.
Already in the mid 19th century Riemann possessed basic insights about differential forms and their usefulness in matters of topology of differentiable manifolds. More specifically, he represented the cohomology of a compact Riemann surface with the help of holomorphic differential forms of degree one, that is, in terms of a local coordinate $z$, forms of the type $f(z)\,dz$ where $f$ is holomorphic. One can show that dimension of the vector space of such forms equals the genus $g$ of the surface (= the number of handels).
In the 20th century these ideas have been generalized to higher dimensions. De Rham showed that for any differentiable manifold cohomology can be defined with the help of smooth differential forms. In the case of compact Riemannian and Kählerian manifolds he and Hodge proved a result which is even much closer to Riemann's, that cohomology in any dimension has a unique representation in terms of harmonic differential forms. This has also important applications to algebraic geometry.
To present our second theme, characteristic classes, let us recall that the fundamental theorem of algebra says something about the number of roots of an algebraic equation of a given degree. Viewed as a result about holomorphic line bundles on the Riemann sphere it has a farreaching generalization in the shape of a Riemann-Roch theorem: If $L$ is a holomorphic line bundle over a compact Riemann surface $X$, the one has the formula
where $c$ is the Chern number (= degree) of $L$ and $g$ the genus of $X$. For instance, applied to a the case $g=1$ (a complex torus) it tells us that there are no elliptic functions (of given modulus) with a single simple one but, up to a constant, exactly one with a double pole at a presribed point. There are also Riemann-Roch theorems in higher dimensions.
Generally speaking, a characteristic class arise as an obstruction to a vector bundle over a manifold to be a trivial one. Characteristic classes are usually defined with the aid of differential form using de Rhams theorem. By integration one obtains characteristic numbers which are (numerical) topological invariants.
Characteristic classes appear in connection with index theorems for elliptic partial differential equations. They play a r\^ole also in modern physics (instantons and such). There characteristic numbers naturally arise in connection with quantization. Geometrically speaking, quantization conditions appear often as arithmetic conditions on suitable constants (modulo material constants such as Planck's constant).
ALL THIS IS SOMETHING THAT EVERY YOUNG MATHEMATICIAN SHOULD MASTER!