Abstract: I would like to discuss the following 4 different subjects that surprisingly are interrelated.
- Milnor's discovery that the seven sphere has exotic differential structures.
- DeRham's theorem that quotients of the sphere by different free linear actions are only diffeomorphic if there is a linear conjugacy — as reproved by Atiyah and Bott.
- The study of manifolds whose universal covers have no $L^2$ harmonic forms (in joint work with S.Cappell and J.Davis)
- Hilbert's 17th problem, that non-negative rational functions are sums of squares of rational functions, solved qualitatively by Artin, and (more importantly for us) the quantitive form, initiated by Pfister.