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