During the winter term 1987/88 I gave a course at the University of Bonn under the title
"Manifolds and Modular Forms". Iwanted to develop the theory of "Elliptic Genera"
and to leam it myself on this occasion. This theory due to Ochanine, Landweber, Stong
and others was relatively new at the time. The word "genus" is meant in the sense of
my book "Neue Topologische Methoden in der Algebraischen Geometrie" published in
1956: A genus is a homomorphism of the Thom cobordism ring of oriented compact
manifolds into the complex numbers. Fundamental examples are the signature and the
A-genus. The A-genus equals the arithmetic genus of an algebraic manifold, provided
the first Chem class of the manifold vanishes. According to Atiyah and Singer it is the
index of the Dirac operator on a compact Riemannian manifold with spin structure.