Abstract 

String can be viewed as higher version of spin, while the latter plays a fundamental role in Atiyah-Singer index theory. People try to develop parallel theory for string, the full story of which is still mystery. Geometry and topology of string manifolds then attract increasing attentions and interests, while the ones of dimensional 24 are quite special among string manifolds. In this talk, we will discuss the algebraic-topological aspect of 24-dimensional string manifolds, following the work of Hirzebruch, Ochanine, Landweber-Stong, Mahowald-Hopkins, Chen-Han, and very recent work of mine joint with Fei Han. We will talk about string cobordism, various index-theoretic genera and some applications.