## Titles and abstracts

## Tutorials

Moti Gitik"Some speculations about Shelah's Weak Hypothesis and the PCF-conjecture"- Let λ be a singular cardinal. S. Shelah defined the following replacement for the power of λ:

The Shelah Weak Hypothesis states that for every cardinal δ the setpp(λ)=sup{cof(∏a/D) |ais a set of cof(λ)

regular cardinals, unbounded in λ,Dis an ultrafilter onaincluding all co--bounded subsets ofa}.

{ λ < δ | λ is singularis at most countable.pp(λ) ≥ δ }

We present new constructions of models such that:Attempts of generalizing this to ω

- there are singular κ
_{0}< κ_{1}withpp(κ_{0}) =pp(κ_{1}) = κ_{1}^{++},- there are singular κ
_{0}<κ_{1}< ... <κ_{n}< ... (n<ω) withpp(κ_{0}) =pp(κ_{1}) = ... =pp(κ_{n}) = ... = (∪_{n<ω}κ_{n})^{+}._{1}many singular cardinals with relations to the PCF-conjecture will be discussed.

Steve Jackson"Some large cardinal properties under AD"- We survey the status of some large cardinal properties assuming the axiom of determinacy. We focus in particular on supercompact, Jonsson, and Rowbottom. We discuss what is known ``globally'' about these notions, and what is known locally (for the smaller cardinals). Building on some joint work with Khafizov and Lowe, we sketch a cardinal representation result which allows us to get an analysis of the (fairly small) Jonsson and Rowbottom cardinals.

Grigor Sargsyan"Core model induction beyond L(R)"- We will sketch how the core model induction works beyond L(R) and we will outline the proof of a recent theorem due to the author that CH+ω
_{1}dense ideal on ω_{1}+ ε (which will be explained) implies that there is a model of AD_{R}+Θ is regular. It should be possible to combine the ideas of this talk with the ideas from Schindler's talk to prove that AD_{R}+Θ is regular is a lower bound for all cardinals are singular.

Ralf Schindler"Mining strength from hypotheses incompatible with choice"- Over the years, the theory of core models and the core model induction has been used successfully to mine strength from set theoretical hypotheses which have their large cardinal strength deep under the surface. We shall focus on hypotheses which are incompatible with the axiom of choice, for instance ones according to which cardinal successors are singular or weakly compact. The essentials of the core model theory will be explained rather than presupposed. This is joint work with Daniel Busche.
## Talks

Arthur Apter"Some remarks on the tree property in a choiceless context"- We discuss the construction of two choiceless models of ZF in which significant instances of the tree property hold. In the first of these models, DC is true, every successor cardinal is regular, every limit cardinal is singular, and every successor cardinal satisfies the tree property. In the second of these models, in which AC fails completely, there is an injective failure of SCH at ℵ
_{ω}, and ℵ_{ω+1}satisfies the tree property.

Ioanna Dimitriou"A modern approach to the first Gitik model"- We'll look at a set sized version of the first Gitik model, i.e., the one constructed in "All uncountable cardinals can be singular". We are going to look at a simplification of this construction. This construction is not only simpler on that it's a set sized forcing, but also on the actual gears of the forcing itself. It uses tree-Prikry forcings and variations of them that interact with each other as little as possible. We'll see as many properties of it as times allows and in the end we'll see some conjectures that will hopefully carry us into the discussion session.

Jip Veldman"On mutual stationarity properties below ℵ_{ω}without AC"- In this talk I will present some equiconsistency results for mutual stationarity below ℵ
without the axiom of choice. This is joint work with Arthur Apter and Peter Koepke._{ω}

Philip Welch"On free subsets of ℵ_{ω}and a question of Pereira"- We consider the following question of Pereira: given F:[ℵ
_{ω}]^{<ω}→ ℵ_{ω}is there an internally approachable N, which is an elementary substructure of H_{ℵω+1}of size some ℵ_{n}, which contains F, but whose characteristic function χ_{N}has range containing a free subset for F?

The question relates to pcf theory, and may be simply be asking for something inconsistent. We observe that if it were to hold then, then the Mitchell core model for measures contains measurables of small Mitchell order.

To state a Conjecture:

A positive answer is consistent with ZF starting from a model with a single measurable..