# Hebrew University topology and geometry seminar

October 28, 2015

1100-1235

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.
- 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.