# Past Logic Seminar

We say that a real closed field is an IPA-real closed field if it admits an integer part (IP) which is a model of Peano Arithmetic (PA). In [2] we prove that the value group of an IPA-real closed field must satisfy very restrictive conditions (i.e. must be an exponential group in the residue field, in the sense of [4]). Combined with the main result of [1] on recursively saturated real closed fields, we obtain a valuation theoretic characterization of countable IPA-real closed fields. Expanding on [3], we conclude the talk by considering recursively saturated o-minimal expansions of real closed fields and their IPs.

References:

[1] D'Aquino, P. - Kuhlmann, S. - Lange, K. : A valuation theoretic characterization ofrecursively saturated real closed fields ,

Journal of Symbolic Logic, Volume 80, Issue 01, 194-206 (2015)

[2] Carl, M. - D'Aquino, P. - Kuhlmann, S. : Value groups of real closed fields and

fragments of Peano Arithmetic, arXiv: 1205.2254, submitted

[3] D'Aquino, P. - Kuhlmann, S : Saturated o-minimal expansions of real closed fields, to appear in Algebra and Logic (2016)

[4] Kuhlmann, S. :Ordered Exponential Fields, The Fields Institute Monograph Series, vol 12. Amer. Math. Soc. (2000)

In the year 2003 Yuri Ershov gave a talk at a conference in Teheran on

his notion of ``extremal valued fields''. He proved that algebraically

complete discretely valued fields are extremal. However, the proof

contained a mistake, and it turned out in 2009 through an observation by

Sergej Starchenko that Ershov's original definition leads to all

extremal fields being algebraically closed. In joint work with Salih

Durhan (formerly Azgin) and Florian Pop, we chose a more appropriate

definition and then characterized extremal valued fields in several

important cases.

We call a valued field (K,v) extremal if for all natural numbers n and

all polynomials f in K[X_1,...,X_n], the set of values {vf(a_1,...,a_n)

| a_1,...,a_n in the valuation ring} has a maximum (which is allowed to

be infinity, attained if f has a zero in the valuation ring). This is

such a natural property of valued fields that it is in fact surprising

that it has apparently not been studied much earlier. It is also an

important property because Ershov's original statement is true under the

revised definition, which implies that in particular all Laurent Series

Fields over finite fields are extremal. As it is a deep open problem

whether these fields have a decidable elementary theory and as we are

therefore looking for complete recursive axiomatizations, it is

important to know the elementary properties of them well. That these

fields are extremal could be an important ingredient in the

determination of their structure theory, which in turn is an essential

tool in the proof of model theoretic properties.

The notion of "tame valued field" and their model theoretic properties

play a crucial role in the characterization of extremal fields. A valued

field K with separable-algebraic closure K^sep is tame if it is

henselian and the ramification field of the extension K^sep|K coincides

with the algebraic closure. Open problems in the classification of

extremal fields have recently led to new insights about elementary

equivalence of tame fields in the unequal characteristic case. This led

to a follow-up paper. Major suggestions from the referee were worked out

jointly with Sylvy Anscombe and led to stunning insights about the role

of extremal fields as ``atoms'' from which all aleph_1-saturated valued

fields are pieced together.

We shall describe a new proof of the Mordell-Lang conjecture in positive characteristic, in the situation where the variety under scrutiny is a smooth subvariety of an abelian variety. Our proof is based on the theory of semistable sheaves in positive characteristic, in particular on Langer's theorem that the Harder-Narasimhan filtration of sheaves becomes strongly semistable after a finite number of iterations of Frobenius pull-backs. Our proof produces a numerical upper-bound for the degree of the finite morphism from an isotrivial variety appearing in the statement of the Mordell-Lang conjecture. This upper-bound is given in terms of the Frobenius-stabilised slopes of the cotangent bundle of the variety.

I will report on joint work with Arno Fehm in which we apply

our previous `existential transfer' results to the problem of

determining which fields admit diophantine nontrivial henselian

valuation rings and ideals. Using our characterization we are able to

re-derive all the results in the literature. Also, I will explain a

connection with Pop's large fields.

Abstract: (Joint work with Sylvy Anscombe) We consider four properties

of a field K related to the existence of (definable) henselian

valuations on K and on elementarily equivalent fields and study the

implications between them. Surprisingly, the full pictures look very

different in equicharacteristic and mixed characteristic.

If you fix a class of models and a construction method that allows you to construct a new model in that class from an old model in that class, you can consider the Kripke frame generated from any given model by iterating that construction method and define the modal logic of that Kripke frame. We shall give a general definition of these modal logics in the fully abstract setting and then apply these ideas in a number of cases. Of particular interest is the case where we consider the class of models of ZFC with the construction method of forcing: in this case, we are looking at the so-called "generic multiverse".

In the real, p-adic and motivic settings, we will present recent results on oscillatory integrals. In the reals, they are related to subanalytic functions and their Fourier transforms. In the p-adic and motivic case, there are furthermore transfer principles and applications in the Langlands program. This is joint work with Comte, Gordon, Halupczok, Loeser, Miller, Rolin, and Servi, in various combinations.