Midwest Model Theory Meeting ’10–’11
Talks will be in CH240 (sometimes known as MA240), on the west end of the second
floor of the Math Building, the low structure attached to the west side of the Math
Tower (231 W. 18th Ave.) Refreshments, as announced, will be served in the Math
Lounge (MW 724) on the south end of the top floor of the Math Tower.
Saturday May 21
1400–1500.
1510–1530.
1530–1600.
1600–1700.
1710–1730.
Hans Schoutens. O-minimalism.
Margaret Thomas. Rational points and integer-valued definable functions.
Refreshments.
Philipp Rothmaler. Mittag-Leffler modules.
James Freitag. Indecomposability in partial differential fields.
Sunday May 22
0845–0930. Refreshments.
0930–1030. Dan Miller. Diagrams of Lebesgue classes of real constructible functions.
1030–1100. Refreshments.
1100–1200. Chris Laskowski. ‘Automatic’ quantifier elimination.
1200–1400. Lunch break.
1400–1500. Uri Andrews. Decidable models of ω-stable theories.
1510–1530. Maryanthe Malliaris. Comparing the complexity of unstable theories.
1530–1600. Refreshments.
1600–1700. Liviu Nicolaescu. Tame flows.
1710-1730. Sarah Cotter. Forking in VC-minimal theories.
Monday May 23
0845-0930. Refreshments.
0930-1030. Serge Randriambololona. The complex heritage of a real set. (PROGRAM
CHANGE)
1040–1100. Philipp Hieronymi. A dichotomy for expansions of the real field.
1100–1130. Refreshments.
1130-1230. Alf Dolich. O-minimal structures expanded by dense independent sets.
Abstracts on following pages.
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Abstracts
Andrews, U. Decidable models of ω-stable theories.
A natural question from recursive model theory asks when it is the case that every
countable model of T has a decidable presentation. These are the theories which should be
considered completely tame from a recursive model theoretic standpoint. In full generality
the question remains open, but using methods from pure stability theory, I will give a
natural characterization which holds for ω-stable theories. This result can be seen as an
effectivization of Vaught’s conjecture for ω-stable theories.
Cotter, S. Forking in VC-minimal theories.
VC-minimal theories were first defined by Adler in 2008; types of VC-minimal theories
include strongly minimal, o-minimal, and C-minimal theories. After covering some basic
results, we will consider a specific class of VC-minimal theories, those which do not admit
finite covers of their generating sets. We will look at a result characterizing forking in
such theories in terms of definable types, generalizing a result of Dolich for o-minimal
theories.
Dolich, A. O-minimal structures expanded by dense independent sets.
Let T be an o-minimal theory, in a language L, expanding that of ordered groups. We
consider LP the language obtained from L by adding a new unary predicate P and the
LP -theory T i whose axioms include those for T together with an axiom scheme stating
that P is a dense subset which is algebraically independent (in the sense of T ). This
theory behaves similarly to the theory of dense pairs of o-minimal structures, as initially
studied by L. van den Dries [Dense pairs of o-minimal structures, Fund. Math. 157 (1998)
61–78], but with several intriguing differences.
(Joint with C. Miller and C. Steinhorn.)
Freitag, J. Indecomposability in partial differential fields
We will discuss the indecomposability theorems in model theory (especially in the
case of groups definable in partial differentially closed fields). In the case of differential
fields, several model theoretic indecomposability theorems apply, but we will discuss
some challenges regarding their application. Then we will give a new indecomposability
theorem in this context and use it to answer a question of Cassidy and Singer.
Hieronymi, P. A dichotomy for expansions of the real field.
If an expansion of the real field does not define Z, then every nonempty, bounded,
nowhere dense unary definable set has Minkowski dimension zero.
(Joint with A. Fornasiero and C. Miller. Preprint, arXiv 1105.2946.)
Laskowski, M.C. ‘Automatic’ quantifier elimination.
We will survey and discuss a series of results, all of which indicate that if the family of
definable subsets of a model is particularly simple, then one can bound their quantifier
complexity independently of the ambient language.
One application is that a theory T is weakly minimal and trivial if and only if for every
model M of T and for every subset A ⊆ M , the expanded structure (M, A) has nfcp.
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Malliaris, M. Comparing the complexity of unstable theories.
Comparing complexity uniformly across theories requires developing tools, and frameworks, which are sensitive to some sufficiently abstract, perhaps combinatorial, notion of
complexity which does not depend on the theories having a common vocabulary. One
example is given by Keisler’s order, a framework for comparing the complexity of theories
via saturation of ultrapowers; while its structure on stable theories is understood by work
of Shelah, its structure on unstable theories has remained elusive. A second framework
is the analysis of the characteristic sequence of hypergraphs associated to a formula, recently developed in a series of papers by the speaker. Roughly speaking, these sequences
describe incidence relations on the formulas parameter space, giving an abstract picture
of its complexity. Notably, in this context, graph-theoretic and combinatorial structure
theorems such as Szemerédi’s celebrated regularity lemma can be leveraged to give modeltheoretic information. These two frameworks are, in fact, intertwined and highlight what
might be called the complexity of asymptotic structure. This talk will briefly survey some
of these connections.
Miller, D. Diagrams of Lebesgue classes of real constructible functions.
A (real) constructible function is a function which can be expressed as a sum of products of globally subanalytic functions and logarithms of globally subanalytic functions.
A major reason that the constructible functions are of interest is that they form the
smallest class of functions which contains all the globally subanalytic functions and is
stable under integration. This talk will discuss a more recent development of the authors in the integration theory of constructible functions. For any constructible function
f : E × Rn → R, we consider LC(f, E) = { (x, p) : x ∈ E, f (x, ·) ∈ Lp (Rn ) }, which we
call the diagram of Lebesgue classes of f over E. Our main theorem (which I will not
state here) shows that LC(f, E) has a relatively simple structure. A corollary of this
theorem is that for any fixed value of p, the set { x ∈ E : f (x, ·) ∈ Lp (Rn ) } is the zero
set of a constructible function on E. Extending the proof of our main theorem allows
us to also prove a preparation theorem for constructible functions which relates closely
to the structure of their Lebesgue diagrams. Obtaining such a preparation theorem was
the initial motivation of our work here, because we use it as a tool to study harmonic
analysis of constructible functions, which is still an ongoing investigation of the authors.
(Joint with R. Cluckers.)
Nicolaescu, L. Tame flows.
The tame flows are flows definable in the pfaffian closure of Ran . In the first part of the
talk I will outline several general methods of constructing such flows, and describe some
of their basic properties. The second part of the talk is devoted to Morse-like tame flows,
i.e., flows with finitely many stationary points that admit a tame Lyapunov function. I
will show how the tameness assumption leads to simpler and more intuitive descriptions
of their Conley indices and to a complete classification of such flows on compact tame sets.
As an application, I will give a tame flow interpretation of R. Forman’s combinatorial
Morse theory.
Rothmaler, Ph. Mittag-Leffler modules.
Mittag-Leffler modules arose in algebra as a very broad generalization of (free and)
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projective modules that is in many respects much easier to work with. This is explained,
among other things, by the fact that they are exactly the positively atomic modules,
i.e. modules in which all positive primitive types are ‘isolated’—the technical term is
‘finitely generated’. As an example, this shows at once that they satisfy a certain positive
elementary chain lemma, more precisely, ascending chains of pure submodules of MittagLeffler modules are Mittag-Leffler. (The intermediate subclass of pure-projective modules
can also be characterized in such terms from the theory of prime models: Mittag-Leffler
is to pure-projective as (positively) atomic to (positively) constructible.)
I will give an introduction to the topic and discuss some recent developments. As a
concrete new result I will exhibit a generalization to Mittag-Leffler modules of the fact
that projective modules are precisely the direct summands of free modules (just as pureprojective modules are precisely the direct summands of what I call pure-free modules).
Schoutens, H. O-minimalism.
O-minimalism is the first-order theory of o-minimal structures, an important class
of models of which are the ultraproducts of o-minimal structures. A complete axiomatization of o-minimalism is not known, but many results are already provable in the
weaker theory DCTC given by definable completeness and type completeness (a small
extension of local o-minimality). In DCTC, we can already prove how many results from
o-minimality (dimension theory, monotonicity, hardy structures) carry over to this larger
setting upon replacing ‘finite’ by ‘discrete, closed and bounded’. However, even then
cell decomposition might fail, giving rise to a related notion of tame structures. Some
new invariants also come into play: the Grothendieck ring is no longer trivial and the
definable, discrete subsets form a totally ordered structure induced by an ultraproduct
version of the Euler characteristic. To develop this theory, we also need another firstorder property, the Discrete Pigeonhole Principle, which I cannot yet prove from DCTC.
Using this, we can formulate a criterion for when an ultraproduct of o-minimal structures
is again o-minimal.
Thomas, M. Rational points and integer-valued definable functions.
We consider the behaviour of analytic functions which are definable in the real exponential field and which take integer values at natural number inputs. This can be
connected to a conjecture of Wilkie concerning the density of rational points on sets definable in this structure. We shall review some results in this latter area and demonstrate
how a proven case of Wilkie’s conjecture, for one-dimensional sets, is able to contribute
to our understanding of integer-valued definable functions.
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