Meeting in Dependent Theories.


Bogota, Colombia - March 28th to March 29th of 2008




8:30. Alex Usvyatsov, Strong dependence, indiscernible sequences and coordinatization.

Abstract: I will talk about existence of indiscernible sequences and various related coordinatization results (that is, analysing arbitrary types in terms of simpler types) in dependent and strongly dependent theories.

10:00. Isaac Goldbring, Nonstandard Methods in Lie Group Theory.


Abstract: Hilbert's Fifth Problem (H5) asked whether every locally euclidean topological group is a Lie group. The affirmative solution to this question was given by Gleason, Montgomery, and Zippin in the 1950s and a simplified proof using nonstandard methods was given by Hirschfeld in the 1990s. In 1957, Jacoby claimed to have proven the local H5, namely that every locally euclidean local group is locally isomorphic to a Lie group. (A local group is a topological space with a group operation defined for elements sufficiently close to the identity.) However, Jacoby's proof is incorrect as it overlooks a subtlety of local groups pointed out by Olver. I was able to adapt Hirschfeld's methods to give the first correct proof of the local H5. I shall try and give the idea of the nonstandard proof of the H5 and explain some of the intricacies involved in extending these methods to the local situation.

11:30 Alice Medvedev, Model Theory and Rational Dynamics

Abstract: Consider the following rational dynamical system: $V$ is an affine space, and $F: V \rightarrow V$ acts polynomially coordinate-wise: $F(x, y, \ldots, z) = f(x), g(y), \ldots, h(z))$ for polynomials $f, g, \ldots, h$. What subvatieties of $V$ are invariant under this action? In a model of ACFA, consider a union $S$ of definable minimal sets $ S_i := \{ x: \sigma(x) = f_i(x) \}$ for some polynomials $f_i$. What is the pregeometry on $S$ given by the model-theoretic algebraic closure operator? This is the same question, and I have an answer.

3:00 Azadeh Neman (joint work with E. Jaligot and A. Muranov) The Independence Property and Hyperbolic Groups

Abstract: A group is CSA (Conjugately Separated Abelian) if all of its maximal abelian subgroups are malnormal. This includes the torsion-free hyperbolic groups as well as many others. E. Jaligot and A. Ould-Houcine have shown that existentially closed CSA-groups without involutions are not omega-stable by showing the existence of aleph_1 types over the empty set. Using similar arguments A. Ould Houcine proved that they are not superstable. We will demonstrate that they have in fact the independence property, using an elementary combinatorial group theory.  The proof shows that most words have the I.P. relative to the class of hyperbolic torsion-free groups. This is a joint work with E. Jaligot and A. Muranov.

4:00 Alex Berenstein (joint work with Alf Onshuus) Lovely pairs of dependent rosy theories of rank one.

Abstract. Lovely pairs of rank one simple theories were studied by Vassiliev. He proved that under these assumptions, the theory of lovely pairs is supersimple. We prove that if the original theory is dependent, rosy of rank one and eliminates the quantifier exists infinity, then the corresponding theory of lovely pairs is also dependent. Some of the results presented were obtained independently by Alf Dolich.

9:00 PM Party at Alex Berenstein's apartment.


10:00 Itaï Ben Yaacov, Dependent metric structures.

Abstract: I will discuss a characterisation of dependence in metric structures. It is very similar to the Vapnik-Chervonenkis/Shelah dichotomy of polynomial vs. exponential growth rate, but instead of counting points it measures the (Gaussian) mean width of a generated convex set. Among other things, this can be used to show that the Keisler randomisation of a dependent theory is dependent, allowing to view Keisler measures as ordinary types in a dependent theory.

11:00 Janak Ramakrishnan, Classifying types in o-minimal theories: two applications.

Abstract: O-minimal theories have a sharp classification of types into definable and undefinable, with undefinable types further separated by the number of realizations in a prime model.  We present two results obtained by exploiting this classification, one extending a result in a paper of van den Dries and Miller on bounding definable functions, and the other answering a question of Speissegger's on extending a
function to a closed set containing a type.

12:30 Anand Pillay (joint work with Ehud Hrushovski), Compact domination.

Abstract: I will state the compact domination conjecture (for definably compact groups in o-minimal structures), and sketch parts of is proof.


Association for Symbolic Logic US National Comittee for Mathematics - Colciencias
Universidad Nacional - Universidad de los Andes