Conference on Model-Theoretic Algebra in Honor of Carol Wood

**Angus Macintyre, ***Zero sets of exponential polynomials*

As part of ongoing work with Paolo D’Aquino and Giusy Terzo, comparing the complex exponential to Zilber’s, I look at limitations in both cases on the zero sets of exponential polynomials. In the complex case these limiations are often seen easily via topology, not available in Zilber’s case. Proofs in Zilber’s case can often be found using Schanuel’s conjecture and diophantine geometry. These proofs sometimes work for the complex case, assuming Schanuel, and on occasion yield information not known to be obtainable by classical methods.

**Charles Steinhorn,** *Some model theory for classes of finite structures*

This talk surveys work on model theory for classes of finite structures developed primarily by Macpherson and the speaker, and several of Macpherson’s students. The underlying theme is to bring to the model-theoretic study of classes of finite structures aspects of the model theory of infinite structures.

**John Baldwin, ***Completeness and Categoricity (in power): Formalization without Foundationalism*

Formalization has three roles: 1) a foundation for an area (perhaps all) of mathematics; 2) a resource for investigating problems in ‘normal’ mathematics; and 3) a tool to organize various mathematical areas so as to emphasize commonalities and differences. We focus on the use of theories and syntactical properties of theories in roles 2) and 3). Formal methods enter both into the classification of theories and the study of definable sets of a particular model. We regard a property of a theory (in first or second order logic) as virtuous if the property has mathematical consequences for the theory or for models of the theory. We rehearse some results for Marek Magidor, H. Friedman and Solovay to argue that for second order logic, ‘categoricity’ has little virtue. For first order logic, categoricity is trivial. But ‘categoricity in power’ illustrates the sort of mathematical consequences we mean. One can lay out a schema with a few parameters (depending on the theory) which describes the structure of any model of any theory categorical in uncountable power. Similar schema for the decomposition of models apply to other theories according to properties, which essentially involve formalizing mathematics, to obtain results in ‘mainstream’ mathematics. We consider discussions on method by Kazhdan and Bourbak as well as such logicians as Hrushovski and Shelah.

**Daniel Saracino,** *The 2-color Rado Number of x _{1}+x_{2}+* ∙ ∙ ∙ +

*x*

_{m-1}= ax_{m}In 1982, Beutelspacher and Brestovansky proved that for every integer *m *≥3, the 2-color Rado number of the equation

*x _{1}+x_{2}+* ∙ ∙ ∙ +

*x*

_{m-1}= ax_{m}is *m ^{2} *–

*m*– 1. In 2008, Schaal and Vestal proved that, for every

*m*≥ 6, the 2-color Rado number of

*x _{1}+x_{2}+* ∙ ∙ ∙ +

*x*2

_{m-1}=*x*

_{m}is . Here we prove that, for every integer *a* ≥ 3 and every *m *≥ 2*a** ^{2}* –

*a +*2, the 2-color Rado number of

*x _{1}+x_{2}+* ∙ ∙ ∙ +

*x*

_{m-1}= ax_{m}is . For the case *a=*3, we show that our formula gives the Rado number for all *m *≥ 7, and we determine the Rado number for all *m* ≥ 3.

**Zo****é**** Chatzidakis, ***Some remarks on fluent completions*

Let *K* be a valued field, that is, a field with a homomorphism *v *from the multiplicative group of *K* onto an ordered abelian group *G* (called the value group), and which satisfies in addition that *v*(*a+b*) is greater than or equal to the minimum of *v*(*a*), *v*(*b*), for *a, b * in *K*. We extend it by setting *v*(0) = ∞, an element greater than all elements of *G*. The residue field of *K* is the quotient *k* of the subring {*a*|*v *(*a*) ≥ 0} by its maximal ideal

{*a*|*v *(*a*) > 0}. In what follows we will assume that *k* has characteristic 0.

It is known that *K* embeds into *k*((*G*)), the field of generalized power series, defined as follows: every element *a* of *k((G)), *is represented by a formal (infinite) sum of monomials *a _{g}t^{g}*, with

*a*in

_{g}*k*and

*g*in

*G*,

*t*being just a formal symbol, and where the support of

*a*(the set of

*g*with

*a*non-zero) is well-ordered. Then

_{g}*k*((

*G*)) becomes naturally a valued field, with addition and multiplication defined in the natural way, and with the valuation being defined by

*v*(

*a*) = the minimum element of the support of

*a*.

Note that *K* and *k*((*G*)) have the same value groups and residue fields: one then says that *k*((*G*)) is an immediate extension of *K*. The field *k*((*G*)) is maximally complete, i.e., *k*((*G*)) has no proper immediate extension.

Matthias Aschenbrenner, Lou van den Dries and Joris van den Hoeven introduced the concepts of fluent pc-sequences and of fluent completions, and asked whether the fluent completion of the valued field *K* as above is unique to *K*-isomorphism. Carol Wood and I thought about the problem, and made some easy remarks, which I will tell you about. All definitions will be introduced.

**David Marker, ***Uncountable Real Closed Fields with PA Integer Parts*

D’Aquino, Knight and Starchenko showed that a countable nonarchimedian real closed filed has an integer part that’s a model of PA if and only if it is recursively saturated. We examine some questions arising from trying to understand how this work might generalize to the uncountable. This is joint work with Charles Steinhorn.