Mathematics

It is a remarkable fact that the category of manifolds maps to a nearly abelian category, the stable homotopy category. It is perhaps even more remarkable that the structure of the stable homotopy category reflects the structure of the category of formal groups. This raises the question: how is the structure of the category of formal groups reflected in the geometry of compact manifolds? The partial answers which are available involve K-theory and Atiyah-Singer index theorem, elliptic cohomology and string theory. I will give a conceptual survey of the setting of work in progress by a number of people.