Date:

Wed, 22/01/202011:00-13:00

Location:

Ross building - Room 63

**Yuval Dor**will speal about

**Transformal Valued Fields**.

__Abstract__

**:**Abraham Robinson characterized the existentially closed valued fields as those which are algebraically closed and nontrivially valued. This theorem is somewhat surprising: it makes no assumption on the topology of the field other than the fact that it is not discrete, and immediately implies a strong from of the Nullstellensatz, asserting that the only obstruction to the solvability of a system of polynomial equations in a neighborhood of a point is the obvious one.

In this talk we give an analogue of Robinson's theorem in the presence of an automorphism; more precisely, a special kind of automorphism, which is topologically infinitely contracting. The motivating example is an ultraproduct of valued fields in which the distinguished automorphism is a power of the Frobenius automorphism, and indeed in a precise sense this example is the universal one: various constructions such as Henselization, strict Henselization, and the higher ramification filtration can be shown to admit meaningful analogues, compatibly with the so called Frobenius reduction functors and obeying formal properties similar to their ordinary algebraic counterparts.

The model companion of the theory of transformal valued fields is called \tilde wVFA. We will discuss the basic geography of types and definable sets in this theory and compare the situation with the two reducts, ACVF and ACFA.

Joint with E. Hrushovski