Past Logic Seminar

3 March 2016
Salma Kuhlmann

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.

[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)

25 February 2016
Franz-Viktor Kuhlmann

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.

4 February 2016
Damian Rössler

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.

28 January 2016

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.

3 December 2015
Franziska Jahnke

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.

26 November 2015
Benedikt Loewe

 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".

19 November 2015
Raf Cluckers

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.