**String manifolds, homotopy groups and topological modular forms**

Many mathematical invariants of manifolds can be formulated in terms of
homotopy groups. In practice, they are very hard to calculate and require
the skillful use of cohomology theories. For some years there is a new
family of cohomology theories which uses the theory of elliptic curves and
modular forms. It was initiated by the physicist and mathematician Ed Witten
who investigated the behavior of small loops (strings) in manifolds. It is
the purpose of elliptic cohomology theories to build a mathematical
framework for these ideas and to determine the rich algebraic structure of
homotopy groups. The talk gives a brief insight into this new world.

*Gerd Laures*

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