Large fields, Galois groups, and NIP fields

A field K is "large" if every smooth curve over K with at least one K-rational point has infinitely many K-rational points. In this talk, I'll discuss what we know about the relations between the arithmetic condition of largeness and the model-theoretic conditions of stability and NIP. Stable large fields are separably closed. For NIP large fields, we know something much weaker: there is a canonical field topology satisfying a weak form of the implicit function theorem for polynomials. Conjecturally, any stable or NIP infinite field should be large. I will discuss these results, as well as the following conjecture: if K is a field and p is a prime and every separable extension of K has degree prime to p, then K is large. This conjecture would imply that NIP fields of positive characteristic are large, and would classify stable fields of positive characteristic. I will present some (very weak) evidence for this conjecture.