LOG2 - SET THEORY AND MODEL THEORY
Organizers
Andrés Eduardo Caicedo, Boise State University, USA
Alf Onshuus, Universidad de los Andes, Colombia
LOG2-1 Alexander Berenstein, Universidad de los Andes, Colombia
Supersimple theories expanded with a predicate for a forking independent subset
We study expansions of models of a supersimple theory with a new predicate of forking-independent elements
that are dense inside a type G(x), we call such expansions H-structures associated to G. We show that any
two such expansions have the same theory and that under some technical conditions, the saturated models of
this common theory are again H-structures associated to G. We extend this work to the continuous setting
and use it to build examples of continuous simple structures that are not stable.
Joint work with Itai Ben Yaacov, Evgueni Vassiliev, Juan Felipe Carmona.
LOG2-2 Christina Brech, Universidad de Sao Paulo, Brasil
Generalized Schreier families and large Banach spaces with no indiscernible sequences
Subsymmetric sequences in Banach spaces can be seen as indiscernible structures. Tsirelson constructed the
first reflexive (separable) Banach space with no subsymmetric subsequences. We will discuss how to generalize
this result to large densities. The main idea is to obtain generalized Schreier families of finite subsets on large
index sets and we manage to construct those families on any set of cardinality smaller than the first Mahlo
cardinal. This is a joint work with J. Lopez-Abad and S. Todorcevic.
LOG2-3 Xavier Caicedo, Universidad de los Andes, Colombia
On the Model Theory of Sheaves
Sheaves of structures yield a natural model theory of variable, dynamic, or ex- tended structures which provides a geometric foundation for intuitionistic logic, and illuminates classical and continuous model theory
through several natural categorical constructions. We describe some of those connections and their po- tential
applications, as generic models, geometric functors, classifying toposes and Morita equivalence of theories.
LOG2-4 Samaria Montenegro, Universidad de los Andes - Universidad de Costa Rica, Colombia - Costa Rica
Shelah’s classification theory and pseudo real closed fields
One of the objects of study of model theory are complete first-order theories and their classification. Shelah
classified complete first-order theories by their ability to encode certain combinatorial configurations. In this
talk we will explain some of these classes and we will focus in the class of NTP2 theories. In particular we
explain the example of pseudo real closed fields (PRC-fields). PRC-fields are a generalization of real closed
fields and pseudo algebraically closed fields. The main theorem of this talk is a positive answer to the conjecture
by Chernikov, Kaplan and Simon: If M is a PRC-field, then the complete theory of M is NTP2 if and only if
M is bounded.
LOG2-5 Claribet Piña, Universidad de los Andes, Colombia
Admissible trees and homogeneous sets
Ever since 1930, when for the first time F. Ramsey publishes his famous theorem, it has been a concern to
know the smallest value for an integer m satisfying m → (n)rl for given r, n and l positive integers. That is, we
wonder what is the smallest integer m satisfying the following.
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Theorem 0.1. Given r, n and l positive integers, there is an integer m such that if A is a set with |A| = m
and c : [A]r −→ l is any coloring of the r−subsets of A into l colors, there is B ⊆ A with |B| = n such that c
is constant on [B]r .
Many efforts have been put into calculating only few of these numbers. However, what it is pretty clear is that
the sizes m and n depend one on each other, and thus the complexity of A in the theorem depends on the
complexity of B. In this sense, given positive integers r and l, we wondered about the existence of a family
F of finite subsets of N such that given a family (ct )t∈F of colorings ct : [t]r −→ l there is another family G,
depending on F, such that if t belongs to the family G, then t is ct0 −homogeneous for some t0 ∈ F. In other
words, G is a family of homogeneous for the elements of F.
In this talk we will introduce certain type of families F for which the statement above holds when r = 1 and
r = 2. Moreover, we will establish the relation between the complexities of the families F and G in both cases.
One of the main tools to prove the results mentioned above is the concept of admissible trees, for sets in any
of the families that we will work with, which we will introduce in the talk. We will also provide an application
of all of this to a result in partition calculus.
LOG2-6 Carlos Uzcátegui, Universidad Industrial de Santander, Colombia
Descriptive set theoretic properties of partial actions of Polish groups
A Polish G-space is a Polish space X (i.e. a completely metrizable and separable space) with a continuous
action of a Polish group G on X. This class of spaces has received considerable attention in the last decades,
in particular, a lot of work has been done for the classification of the orbit equivalence relations induced by
Polish group actions using the tools of descriptive set theory [2][6][7][8]. In this talk, in stead of global actions,
we consider partial actions of Polish groups and discuss some of its descriptive set theoretic properties.
A partial action of a group G on a set X is a collection of partial maps {mg }g∈G on X satisfying m1 = idX
and mg ◦ mh ⊆ mgh , for all g, h ∈ G. Partial actions were introduced by R. Exel [3][4][5] for the study of
C ∗ -algebras. We are interested in partial actions of a topological group over a topological space, in this case
the domain and range of the functions mg are required to be open subsets of X and each mg has to be a
p
homeomorphism. The partial orbit equivalence relation is then naturally defined by letting xEG
y, if there is
g ∈ G such that x is in the domain of mg and mg (x) = y.
An interesting feature of topological partial actions is that they admit globalizations, that is to say, every
topological partial G-action over a space X is the restriction of a global G-action over a larger space Y ⊇ X.
Moreover, there is a minimal globalization, denoted XG , called the enveloping space of X [2][9]. We show that
(under very mild conditions) XG is a standard Borel space, when X and G are Polish spaces. This allows to
transfer some results about Polish G-spaces into the context of partial actions. For instance, the existence of
universal actions, Effros’s theorem, etc.
An important tool for comparing equivalence relations is given by the pre-order of Borel reducibility, which is
also the basis for defining a notion of complexity for equivalence relations [6][7][8]. If E and F are equivalence
relations over two Polish spaces X and Y , respectively, we say that F Borel reduces to E, in symbols F ≤B E,
if there is a Borel function f : Y → X such that xF y iff f (x)Ef (y) for all x, y ∈ X. We will illustrate how
partial actions fit into that framework by showing that some results about orbit equivalence relations induced
by Polish group actions also hold for by partial actions.
This is a joint work with Hector Pinedo Tapia.
References
[1] F. Abadie, Enveloping actions and Takai duality for partial actions. Journal of Func. Anal. 197 (2003),
14-67.
[2] H. Becker, A. Kechris, The Descriptive Set Theory of Polish Group Actions. London Math. Soc. Lect.
Notes. Cambridge University Press, 1996
[3] M. Dokuchaev, Partial actions: a survey, Contemp. Math., 537 (2011), 173–184.
[4] R. Exel, Circle actions on C ∗ -algebras, partial automorphisms and generalized Pimsner-Voiculescu exact
sequences, J. Funct. Anal. 122 (1994), 361–401.
[5] R. Exel, Partial actions of groups and actions of inverse semigroups, Proc. Am. Math. Soc. 126 (1998),
3481–3494.
[6] Su Gao, Invariant Descriptive Set Theory. Chapmann and Hall, 2009.
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[7] G. Hjorth. Classification and orbit equivalence relations, volume 75 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2000.
[8] A. S. Kechris. Actions of Polish groups and classification problems. In Analysis and logic (Mons, 1997),
volume 262 of London Math. Soc. Lecture Note Ser., pages 115–187. Cambridge Univ. Press, Cambridge,
2002.
[9] J. Kellendonk and M. V. Lawson, Partial actions of groups. Internat. J. Algebra Comput. 14 (2004),
87–114.
LOG2-7 Carlos Videla, Mount Royal University, Canadá
Undecidable fields of algebraic numbers
After Julia Robinson’s breakthrough result of 1949 in which she proved the undecidability of the field of rational
numbers the problem of classifying ( in some unspecified sense) the undecidable /decidable fields of algebraic
numbers was raised. In my talk I will review old and new results concerning this Tarski-Robinson question.
LOG2-8 Rafael Zamora, Institut Mathematique de Jussieu, Francia
Injectivity in tests for separability by potentially Lavrentieff sets
In descriptive set theory, we study classes of subsets of a Polish space, such as those in the Borel hierarchy.
When you consider product spaces, generalizations of these classes arise naturally. One such generalization
is the class of potentially Γ sets, where Γ is a class of Borel sets. For example, potentially open sets are
exactly those sets that are the countable union of Borel rectangles. We will provide injective test to check for
separability of analytic sets by potentially Lavrentieff sets (also known as difference classes).
This is joint work with Dominique Lecomte.
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