Workshop: Current Trends in Descriptive Set Theory


Monday, December 12, 2016

09:15 Opening & Registration
09:30 - 10:30 B. Miller
Introductory lecture
10:30 - 11:00 Coffee break
11:00 - 12:00 C. Conley
Følner tilings via matchings
12:00 - 14:00 Lunch break
14:00 - 15:00 S. Jackson
The subshift of finite type and graph homomorphism questions
Given a countable group $G$ acting continuously and freely on $F(2^G)$, and given a subshift $S$ of finite type, does there exist a continuous equivariant map from $F(2^G)$ to $S$? We show that the set of subshifts, for the case $G=\mathbb{Z}^n$, for which the answer is yes is semi-recursive, but not recursive. Thus, the sub-shift question is undecidable. We also consider the corresponding question for graph homomorphisms.
15:00 - 15:30 Break
15:30 - 16:00 V. Kanovey
Definable countable sets of reals with and without definable elements
Does any non-empty OD set, especially a set of reals, necessarily have an OD element? The answer is in the positive in some well-known models, and in the negative in some other generic models.

Tuesday, December 13, 2016

09:30 - 10:30 D. Kerr
Actions of amenable groups on the Cantor set and their crossed products
10:30 - 11:00 Coffee break
11:00 - 12:00 A. Kwiatkowska
Cyclically dense conjugacy classes and topological similarity for groups of measurable functions
12:00 - 14:00 Lunch break
14:00 - 15:00 F. Le Maître
$L^1$ full groups
In this talk I will present $L^1$ full groups, which are the measurable analogue of topological full groups. They are cli Polish groups which provide a complete flip-conjugacy invariant for measure-preserving invertible ergodic transformations. After establishing some of their basic properties, I will explain why the $L^1$ full group of a measure-preserving ergodic invertible transformation is topologically finitely generated if and only if the transformation has finite entropy.
15:00 - 15:30 Break
15:30 - 16:30 D. Lecomte
Applications of the representation theorem for Borel sets
We recall and apply the Debs-Saint Raymond representation theorem for Borel sets to obtain progress concerning problems motivated by the studies of definable countable colourings and Borel equivalence relations.

Wednesday, December 14, 2016

10:00 - 10:30 P. Schlicht
Applications of long games in generalized descriptive set theory
10:30 - 11:00 Coffee break
11:00 - 11:30 M. Malicki
There is no universal Polish metric group with a bi-invariant metric. A short proof
In a recently published paper, Michal Doucha proved that there is no universal Polish metric group with a bi-invariant metric, that is, there does not exist a complete, separable metric group G with a bi-invariant metric such that every Polish metric group with a bi-invariant metric can be isometrically embedded in G. The same holds for locally compact Polish metric groups with a bi-invariant metric. We would like to present a short proof of this result.

Thursday, December 15, 2016

09:30 - 10:30 A. Marks
Descriptive set theory and geometrical paradoxes
10:30 - 11:00 Coffee break
11:00 - 12:00 R. Tucker-Drob
Bounded generation and cost
We show that any group which is boundedly generated by finitely generated subgroups of infinite index must have cost 1. This leads to a strong positive answer to problem 33.2 in Kechris-Miller. This is joint work with Mark Shusterman.
12:00 - 14:00 Lunch break
14:00 - 15:00 M. Zelený
On Borel chromatic numbers
15:00 - 15:30 Break
15:30 - 16:30 J. Zapletal
The Florida diagram and its repercussions
We show how comparison of various generic extensions of a ground model can yield results in Borel reducibility, independence results in choiceless set theory, and restatements of coding principles on $\omega_1$.

Friday, December 16, 2016

09:30 - 10:30 T. Tsankov
Metric Scott analysis
Back-and-forth equivalence relations have been an important tool in model theory and descriptive set theory since their advent in the fifties. They provide Borel approximations to the isomorphism relation that are useful in a variety of settings and are particularly important in infinitary model theory. In this work, we extend the theory to metric structures and describe the connections with infinitary continuous logic. Many classical results have a continuous counterpart (for example, the existence of Scott rank, the Lopez-Escobar theorem, etc.) but also new features appear that have no analogue in the classical setting and allow the study of equivalence relations other than isomorphism. This is joint work with I. Ben Yaacov, M. Doucha, and A. Nies.
10:30 - 11:00 Coffee break
11:00 - 12:00 A. Tserunyan
Edge slides and ergodic actions
We use edge-slidings and saturated disjoint Borel families to give a conceptually simple proof of Hjorth's theorem on cost attained. The same technique yields a strengthening of Hjorth's theorem: the action of the free group can be arranged so that each of the standard generators acts ergodically. The existence of an ergodic action for the first generator immediately follows from a powerful theorem of Tucker-Drob, whose proof uses a recent substantial result in probability theory as a black box. Using asymptotic means on graphs and a new notion of packed disjoint Borel families, we give a constructive proof of a weaker version of Tucker-Drob's theorem, which is enough to yield most of its known corollaries, including what was needed for our strengthening of Hjorth's theorem. This is joint work with B. Miller.
12:00 - 14:00 Lunch break
14:00 - 15:00 A. Törnquist
Definability and disjointness modulo various ideals on $\omega$