C5

Abstract: Four manifolds are some of the most intriguing objects in topology. So far, they have eluded any attempt of classification and their behaviour is very different from what one encounters in other dimensions. On the other hand, Einstein metrics are among the canonical types of metrics one can find on a manifold. In this talk I will discuss many of the peculiarities that make dimension four so special and see how Einstein metrics could potentially help us understand more about four manifolds.

I will report on some joint work with Jacob Tsimerman
concerning multiplicative relations among singular moduli.
Our results rely on the ``Ax-Schanuel'' theorem for the j-function
recently proved by us, which I will describe.

 This is joint work with Angus Macintyre. We prove a general model-completeness theorem for Henselian valued fields
stating that a Henselian valued field of characteristic zero with value group a Z-group and with finite ramification is model-complete in the language of rings provided that its residue field is model-complete. We apply this to extensions of p-adic fields showing that any finite or infinite extension of p-adics with finite ramification is model-complete in the language of rings.