# Past Logic Seminar

I will consider automorphism groups of countable structures acting continuously on compact spaces: the viewpoint of topological dynamics. A beautiful paper of Kechris, Pestov and Todorcevic makes a connection between this and the ‘structural Ramsey theory’ of Nesetril, Rodl and others in finite combinatorics. I will describe some results and questions in the area and say how the Hrushovski predimension constructions provide answers to some of these questions (but then raise more questions). This is joint work with Hubicka and Nesetril.

An abelian l-group G is essentially a partially ordered subgroup of functions from a set to a totally ordered abelian group such

that G is closed under taking finite infima and suprema. For example, G could be the continuous semi-linear functions defined on the open

unit square, or, G could be the continuous semi-algebraic functions defined in the plane with values in (0,\infty), where the group

operation is multiplication. I will show how G, under natural geometric assumptions, can be interpreted (in a weak sense) in its lattice of

zero sets. This will then be applied to the model theory of natural divisible abelian l-groups. For example we will see that the

aforementioned examples are elementary equivalent. (Parts of the results have been announced in a preliminary report from 1987 by F. Shen

and V. Weispfenning.)

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.