# Logic Colloquium 2014, July 14-19, 2014

Logic Colloquium 2014 will take place during the Vienna Summer of Logic.

The Program Committee members are: Z. Adamowicz, J. Avigad (Chair), M. Bezem, S. Friedman, J. Koenigsman, K. Lodaya, P. Oliva, T. Slaman, and R. Zach. The Local Organizing Committee includes: M. Baaz (Chair), A. Ciabattoni, S. Eberhard, S. Hetzl (Co-chair), and M. Goldstern.

The invited speakers include:

A. Bauer, P. Blanchette, V. Fischer, N. Greenberg, L. Kolodziejczyk, B. Miller, M. Reynolds, M. Soskova, and A. Visser.

Tutorials will be offered by K. Apt and A. Miquel.

### Karp Prize Lectures

• Matt Foreman will speak about Moti Gitik’s work
• Matthias Aschenbrenner will speak about the work of Ya’acov Peterzil, Jonathan Pila, Sergei Starchenko and Alex Wilkie

• Julia Knight

### Special Sessions (with organizers in parentheses) that are planned include:

• #### Set Theory (Organizers: J. Kellner, M. Goldstern)

• Daisuke Ikegami
• Philipp Lücke
• Diego Mejía
• Konstantin Slutsky

SPEAKER: Diego A. Mejía
TITLE:  Matrix iterations and Cichoń’s diagram
ABSTRACT. Using matrix iterations of ccc posets we prove the consistency, with ZFC, of some constellations of Cichoń’s diagram where the cardinals on the right hand side assume three different values. We also discuss the influence of the constructed models on other classical cardinal invariants of the continuum.

SPEAKER: Konstantin Slutsky
TITLE: Regular cross-sections of Borel flows
ABSTRACT. When working with measurable flows, it is sometimes convenient to choose a countable cross-section and to reduce a problem of interest to a similar question for the action induced by the flow on this cross-section. In some cases, one wants to impose additional restrictions on the cross-section, usually by restricting possible distances between points within each orbit.

Historically, cross-sections of flows were studied mainly in the context of ergodic theory. One of the most important results here is a theorem of D. Rudolph, which states that any free measure preserving flow, when restricted to an invariant subset of full measure, admits a cross-section with only two possible distances between adjacent points.

Borel dynamics deals with actions of groups on standard Borel spaces, when the latter is not equipped with any measure. In this more abstract context, one needs to construct cross-sections that are regular on all orbits without exceptions, and methods of ergodic theory, which tend to produce cross-sections only almost everywhere, are therefore frequently insufficient. In this regard, M. G. Nadkarni posed a question whether the analog of Rudolph’s Theorem holds true in the Borel setting: Does every free Borel flow admit a cross-section with only two different distances between adjacent points?

The talk will provide an overview of these and other results concerning the existence of regular cross-sections, and a positive answer to Nadkarni’s question will be given. As an application of our methods, we give a classification of free Borel flows up to Lebesgue Orbit Equivalence, by which we understand orbit equivalence preserving Lebesgue measure on each orbit. This classification is an analog of the classification of hyperfinite equivalence relations obtained by R. Daugherty, S. Jackson, and A. S. Kechris.

SPEAKER: Daisuke Ikegami
TITLE: Large cardinals, forcing axioms, and the theory of subsets of \omega_1
ABSTRACT. The goal of this research is to rule out “natural” independence phenomena in Set Theory by maximizing your theory in terms of large cardinals and forcing axioms. Using large cardinals in ZFC, by the results of Woodin, we have a clear understanding of the theory of the second-order structure (\omega, P(\omega), \in)) and what it should be.

In this talk, we try to extend this understanding to the theory of the structure (\omega_1, P(\omega_1), \in) by using large cardinals, forcing axioms, and some hypothesis from inner model theory in ZFC. This is joint work with Matteo Viale.
SPEAKER: Philipp Lücke
TITLE: Locally definable well-orders
ABSTRACT. A classical theorem of Mansfield shows that there exists a well-ordering of the set \omega^\omega of all functions from \omega to \omega$that is definable over the collection H(\omega_1) of all hereditarily countable sets by a \Pi_1-formula without parameters if and only if every such function is contained in Gödel’s constructible universe L. In particular, the existence of such a well-ordering implies that the continuum hypothesis holds. We consider the question whether this implication generalizes to higher cardinalities: does the existence of a well-ordering of the set \omega_1^{\omega_1} of all functions from \omega_1 to \omega_1 that is definable over H(\omega_2) by a \Pi_1-formula without parameters imply that the GCH holds at \omega_1? This is joint work with Peter Holy (Bristol). ### Attachments ### 2 responses to “Logic Colloquium 2014, July 14-19, 2014” 1. saf 2013 Karp Prize to be Awarded at Logic Colloquium 2014. The eighth Carol Karp Prize is to be jointly awarded to Moti Gitik for his work in set theory, especially applications of large-cardinal forcings to pcf-theory, and to Ya’acov Peterzil, Jonathan Pila, Sergei Starchenko and Alex Wilkie for their work in model theory, especially as applied to questions in number theory. The prizes will be awarded at the ASL European Summer Meeting (Logic Colloquium) in July. The Karp Prize, established in 1973 in memory of Professor Carol Karp, is awarded every five years. The award is made by the Association, on recommendation of the ASL Committee on Prizes and Awards, for a “connected body of research, most of which has been completed in the time since the previous prize was awarded.” The winners will share a$5,000 cash award.
Lectures on the prize-winning work will be delivered at the award ceremony.

2. saf

Marcin Sabok, Matteo Mio and Gianluigi Greco will be awarded Kurt Goedel Fellowships gold medals designed by the Austrian Mint at the special Vienna Summer of Logic Award Ceremony on July 17, 2014.