# Past Logic Seminar

Ever since the compilers of Euclid's Elements gave the "definitions" that "a point is that which has no part" and "a line is breadthless length", philosophers and mathematicians have worried that the basic concepts of geometry are too abstract and too idealized. In the 20th century writers such as Husserl, Lesniewski, Whitehead, Tarski, Blumenthal, and von Neumann have proposed "pointless" approaches. A problem more recent authors have emphasized it that there are difficulties in having a rich theory of a part-whole relationship without atoms and providing both size and geometric dimension as part of the theory. A possible solution is proposed using the Boolean algebra of measurable sets modulo null sets along with relations derived from the group of rigid motions in Euclidean n-space.

Given a collection F of holomorphic functions, we consider how to describe all the holomorphic functions locally definable from F. The notion of local definability of holomorphic functions was introduced by Wilkie, who gave a complete description of all functions locally definable from F in the neighbourhood of a generic point. We prove that this description is not complete anymore in the neighbourhood of non-generic points. More precisely, we produce three examples of holomorphic functions which each suggest that at least three new definable operations need to be added to Wilkie's description in order to capture local definability in its entirety. The construction illustrates the interaction between resolution of singularities and definability in the o-minimal setting. Joint work with O. Le Gal, G. Jones, J. Kirby.

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.