Weak Prediction Principles
Speaker: Yair Hayut
Abstract: Jensen's diamond is a well studied prediction principle. It holds in L (and other core models), and in many cases it follows from local instances of GCH.
In the talk I will address a weakening of diamond (due to Shaleh and Abraham) and present Abraham's theorem about the equivalence between weak diamond and a weak consequence of GCH. Abraham's argument works for successor cardinals. I will discuss what is known and what is open for inaccessible cardinals.

Dependent theories have now a very solid and well-established collection of results and applications. Beyond first order, the development of "dependency" has been rather scarce so far. In addition to the results due to Kaplan, Lavi and Shelah (dependent diagrams and the generic pair conjecture), I will speak on a few lines of current research around the extraction of indiscernibles for dependent diagrams and on various forms on dependence for abstract elementary classes. This is joint work with Saharon Shelah.

Arbault sets (briefly, A-sets) were first introduced by Jean Arbault in the context of Fourier analysis. One of his major results concerning these sets,asserts that the union of an A-set with a countable set is again an A-set. The next obvious step is to ask what happens if we replace the word "countable" by א_1. Apparently, an א_1 version of Arbault's theorem is independent of ZFC. The aim of this talk would be to give a proof (as detailed as possible) of this independence result. The main ingredients of the proof are infinite combinatorics and some very basic Fourier analysis.

Zilber's trichotomy conjecture, in modern formulation, distinguishes three flavours of geometries of strongly minimal sets --- disintegrated/trivial, modular, and the geometry of an ACF. Each of these three flavours has a classic ``template'' --- a set with no structure, a projective space over a prime field, and an algebraically closed field, respectively. The class of ab initio constructions with which Hrushovski refuted the conjecture features a new flavour of geometries --- non-modular, yet prohibiting any algebraic structure.

The notion of reflection plays a central role in modern Set Theory since the descovering of the well-known Lévy and Montague \textit{Reflection principle}. For any $n\in\omega$, let $C^{(n)}$ denote the class of all ordinals $\kappa$ which correctly interprets the $\Sigma_n$-statements of the universe, with parametes in $V_\kappa$.

Non-equational stable groups.
Speaker: Rizos Sklinos
Abstract: The notion of equationality has been introduced by Srour and further
developed by Pillay-Srour. It is best understood intuitively as a notion
of Noetherianity on instances of first-order formulas. A first-order
theory is equational when every first-order formula is equivalent to a
boolean combination of equations.
Equationality implies stability and for many years these two notions were
identified, as only an "artificial" example of Hrushovski (a tweaked

We will present briefly the "multiverse view" of set theory, advocated by Hamkins, that there are a multitude of set-theoretic universes, and not one background universe, and his proposed "Multiverse Axioms". We will then move on to present the main result of Gitman and Hamkins in their paper "A natural model of the multiverse axioms" - that the countable computably saturated models of ZFC form a "toy model" of the multiverse axioms.

Zilber introduced quasi-minimal classes to generalize the model theory of pseudo exponential
fields. They are equipped with a pregeometry operator and satisfy interesting properties such
as having only countable or co-countable definable sets. Differentially closed fields of
characteristic 0, rich examples of a \omega-stable structures, are good candidates to be
quasiminimal. The difficulty is that a differential equation may have uncountably many
solutions, and thus violate the countable closure requirement of quasiminimal structures.

In an attempt to classify the geometries arising in strongly minimal sets, Zil'ber conjectured them to split into three different types: Trivial geometries, vector space-like geometries and field-like geometries. Soon after, Hrushovski refuted this conjecture while introducing a new construction method, which has been modified and used a lot ever since.

Abstract: In this talk I will discuss some suitable definable Bohr compactification of an ultraproduct of finite groups, and relate it to ultra quasirandom groups.

Abstract: Paul Larson proved that under Martin's axiom and large continuum there are no Laver ideals over aleph_1. He asked about weakly Laver ideals under some forcing axiom.
We shall address two issues:
1. Under Martin's axiom and the continuum is above aleph_2, there are no weakly Laver ideals over aleph_1..
2. Under Baumgartner's axiom, the parallel of Larson's theorem holds for ideals over aleph_2.

Reflections on the coloring and chromatic numbers
Speaker: Chris Lambie-Hanson
Abstract: Compactness phenomena play a central role in modern set theory, and
the investigation of compactness and incompactness for the coloring and chromatic
numbers of graphs has been a thriving area of research since the mid-20th century,
when De Bruijn and Erdős published their compactness theorem for finite chromatic

I will review some recent results in the Borel reducibility on uncountable cardinals of the Helsinki logic group.
Borel reducibility on the generalised Baire space \kappa^\kappa for uncountable \kappa is defined analogously to that for \kappa=\omega. One of the corollaries of this work is that under some mild cardinality assumptions on kappa, if T1 is classifiable and T2 is unstable or superstable with OTOP, then the ISOM(T1) is continuously reducible ISOM(T2) and ISOM(T2) is not Borel reducible to ISOM(T1).

We shall prove that there is a sequence of Boolean algebras for which the ultraproduct of the lengths divided by an ultrafilter is strictly less than the length of the product algebra.
This is a joint work with Saharon Shelah.