ABSTRACT. Over the last few decades, a definable refinement of the usual notion of cardinality has been employed to great effect in shedding new light on many classification problems throughout mathematics. In order to best understand such applications, one must investigate the abstract nature of the definable cardinal hierarchy.

It is well known that the initial segment of the hierarchy below R / Q looks quite similar to the usual cardinal hierarchy. On the other hand, if one moves sufficiently far beyond R / Q, the two notions diverge wildly.

After reviewing these results, we will discuss recent joint work with Clinton Conley, seeking to explain the difficulty in understanding definable cardinality beyond R / Q by showing that the aforementioned wild behavior occurs immediately thereafter.

ABSTRACT. The cardinal invariants of the continuum arise from combinatorial, topological and measure theoretic properties of the reals, and are often defined to be the minimum size of a family of reals satisfying a certain property.

An example of such an invariant is the minimum size of a subgroup of $S_\infty$, all of whose non-identity elements have only finitely many fixed points and which is maximal (with respect to this property) under inclusion. This cardinal invariant is denoted $\mathfrak{a}_g$. Another well-known invariant, denoted $\hbox{non}(\mathcal{M})$, is the minimum size of a set of reals which is not meager. It is a ZFC theorem that $\hbox{non}(\mathcal{M})\leq\mathfrak{a}_g$. A third invariant, denoted $\mathfrak{d}$, is the minimum size of a family $\mathcal{F}$ of functions in $^\omega\omega$ which has the property that every function in $^\omega\omega$ is eventually dominated by an element of $\mathcal{F}$. In contrast to the situation between $\mathfrak{a}_g$ and $\hbox{non}(\mathcal{M})$, ZFC cannot prove either of the inequalities $\mathfrak{a}_g\leq \mathfrak{d}$ or $\mathfrak{d}\leq\mathfrak{a}_g$. The classical forcing techniques seem, however, to be inadequate in addressing the consistency of $\mathfrak{d}<\mathfrak{a}_g$ which was obtained only after a ground-breaking work by Shelah and the appearance of his ``template iteration" forcing techniques.

We further develop these techniques to show that $\mathfrak{a}_g$, as well as some of its relatives, can be of countable cofinality. In addition we will discuss other recent developments of the technique and conclude with open questions and directions for further research.

ABSTRACT. Hilbert's Tenth Problem in its original form was to find an algorithm to decide, given a multivariate polynomial equation with integer coefficients, whether it has a solution over the integers. In 1970 Matiyasevich, building on work by Davis, Putnam and Robinson, proved that no such algorithm exists, i.e. Hilbert's Tenth Problem is undecidable. Since then, analogues of this problem have been studied by asking the same question for polynomial equations with coefficients and solutions in other commutative rings. The biggest open problem in the area is Hilbert's Tenth Problem over the rational numbers. In this talk we will construct some subrings R of the rationals that have the property that Hilbert's Tenth Problem for R is Turing equivalent to Hilbert's Tenth Problem over the rationals. We will also discuss some recent undecidability results for function fields of positive characteristic.

ABSTRACT. The third round of the Kurt Gödel Research Prize Fellowships Program, under the title: Connecting Foundations and Technology, aims at supporting young scholars in early stages of their academic careers by offering highest fellowships in history of logic, kindly supported by the John Templeton Foundation. Young scholars being less or exactly 40 years old at the time of the commencement of the Vienna Summer of Logic (July 9, 2014) will be awarded one fellowship award in the amount of EUR 100,000, in each of the following categories:

• Logical Foundations of Mathematics,
• Logical Foundations of Computer Science, and
• Logical Foundations of Artificial Intelligence

The following three Boards of Jurors were in charge of choosing the winners:

• Logical Foundations of Mathematics: Jan Krajíček, Angus Macintyre, and Dana Scott (Chair).
• Logical Foundations of Computer Science: Franz Baader, Johann Makowsky, and Wolfgang Thomas (Chair).
• Logical Foundations of Artificial Intelligence: Luigia Carlucci Aiello, Georg Gottlob (Chair), and Bernhard Nebel.

http://fellowship.logic.at/

ABSTRACT. The aim of the FLoC Olympic Games is to start a tradition in the spirit of the ancient Olympic Games, a Panhellenic sport festival held every four years in the sanctuary of Olympia in Greece, this time in the scientific community of computational logic. Every four years, as part of the Federated Logic Conference, the Games will gather together all the challenging disciplines from a variety of computational logic in the form of the solver competitions.

At the Award Ceremonies, the competition organizers will have the opportunity to present their competitions to the public and give away special prizes, the prestigious Kurt Gödel medals, to their successful competitors. This reinforces the main goal of the FLoC Olympic Games, that is, to facilitate the visibility of the competitions associated with the conferences and workshops of the Federated Logic Conference during the Vienna Summer of Logic.

This award ceremony will host the

• 3rd Confluence Competition (CoCo 2014);
• Configurable SAT Solver Challenge (CSSC 2014);
• Ninth Max-SAT Evaluation (Max-SAT 2014);
• QBF Gallery 2014; and
• SAT Competition 2014 (SAT-COMP 2014).