LOGIC AND MATHEMATICS 2008

April 19-20, 2008

Department of Mathematics
University of Illinois at Urbana-Champaign

Marton Elekes (Hungarian Academy of Sciences and Fields Institute)

Additive Erdos-Sierpinski duality, covers by a small number of translates, and locally compact groups
Abstract: We present a method that can be used to generalize certain results about the reals to arbitrary locally compact groups. A key feature is that the statement has to have the property that it holds in a group if it holds in some of its factors. The two applications we describe are Bartoszynski's theorem about the non-existence of an additive Erdos-Sierpinski duality, and a theorem of the speaker and Steprans stating that consistently the reals can be covered by less than continuum many translates of a suitable compact Lebesgue nullset.

Ilijas Farah (York University)

Nonseparable UHF algebras
Abstract: This is a joint work with Takeshi Katsura. Uniformly Hereditarily Finite (UHF) algebras are those C* algebras in which every finite subset is `near' a finite-dimensional full matrix subalgebra. This can be formalized in three different ways, all three being equivalent in the separable case. Separable UHF algebras were classified in the 1960s by Glimm and Dixmier. Dixmier asked whether three definitions are equivalent in the nonseparable case. I will give a complete answer to this question as well as some remarks on extending the Glimm-Dixmier theorem to the nonseparable case.

Bradd Hart (McMaster University)

Simple continuous theories
Abstract: Simplicity can be developed in the context of first order continuous logic in a manner similar to that of first order logic. First order simple theories can be thought of naturally as discrete continuous theories. Stable continuous theories are simple as well. One missing element to date is a continuous analog of the first order construction of adding a generic predicate. In fact, Ben-Ya'acov has asked the question if all simple "essentially" continuous theories are stable. I will define essential continuity and give examples of how simple continuous theories can be formed.

Justin Moore (Cornell University)

Structure within the class of Aronszajn lines.
Abstract: An Aronszajn line is an uncountable linear order which does not contain an uncountable separable or scattered suborder. I will give an exposition of a number of results -- some new and some old -- concerning the relation of embeddability within the class of Aronszajn lines.

Lionel Nguyen Van The (University of Calgary)

Ramsey properties of the dense local order
Abstract: This is a joint work with Claude Laflamme and Norbert Sauer. In 84, Lachlan proved that there are only three countable tournaments (complete oriented graphs) that are ultrahomogeneous (where any isomorphism between finite subtournaments can be extended to an automorphism of the whole structure): The random tournament, the rationals and the so-called dense local order. For the first two, finite and infinite Ramsey properties have been well understood for quite some time thanks to the work of Devlin, Nesetril-Rodl and Laflamme-Sauer-Vuksanovic. The purpose of this talk is to present similar results in the latter case.

Simon Thomas (Rutgers University)

The Large Scale Geometry of Finitely Generated Groups
Abstract: Gromov's geometric group theory seeks to classify finitely generated groups in terms of the "large scale geometry" of their Cayley graphs. In this talk, I will discuss some foundational aspects of this program from the perspective of the theory of Borel equivalence relations. In particular, I will show that the quasi-isometry relation is not smooth and also explain why it is almost certainly not hyperfinite.