Hebrew University topology and geometry seminar

October 28, 2015
Ross building, Seminar Room 70A

Shmuel Weinberger

University of Chicago

Exotic Spheres, Lens spaces, Anharmonic manifolds, and Hilbert's 17th problem

Abstract: I would like to discuss the following 4 different subjects that surprisingly are interrelated.
  1. Milnor's discovery that the seven sphere has exotic differential structures.
  2. 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.
  3. The study of manifolds whose universal covers have no $L^2$ harmonic forms (in joint work with S.Cappell and J.Davis)
  4. 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.