We report on a joint work in progress with Rahman Mohammadpour in which we study the problem of the possible existence of a universal tree under weak embeddings in the classes of $\aleph_{2}$-Aronszajn and wide $\aleph_{2}$-Aronszajn trees. This problem is more complex than previously thought, in particular it seems not to be resolved under ShFA $+$ CH using the technology of weakly Lipshitz trees. We show that under CH, for a given $\aleph_{2}$-Aronszajn tree $\mathrm{T}$ without a weak ascent path, there is an $\aleph_{2^{-\mathrm{C}\mathrm{C}}}$ countably closed forcing forcing which specialises $\mathrm{T}$ and adds an $\aleph_{2}$-Aronszajn tree which does not embed into T. One cannot however apply the ShFA to this forcing.[-]

Kiesler and Tarski characterized weakly compact cardinals as those inaccessible cardinals such that for every $\kappa$-complete subalgebra $\mathcal{B}\subseteq P(\kappa))$ every $\kappa$-complete filter on $\mathcal{B}$ can be extended to a $\kappa$-complete ultrafilter on $\mathcal{B}.$ Welch proposed a variant of Holy-Schlict games where, for a fixed $\gamma$, player I and II take turns, with I playing an increasing sequence of subalgebras $\mathcal{A}_{\mathrm{i}}$ and II playing an increasing sequence of ultrafilters $\mathcal{U}_{\mathrm{i}}$ for $ i<\gamma$. Player II wins if she can continue playing of length $\gamma.$
By Kiesler-Tarski, player II wins the game with $\gamma=\omega$ if and only if $\kappa$ is weakly compact. It is immediate that if $\kappa$ is measurable, then II wins the game of length $2^{\kappa}$. Are these the only cases?
Nielsen and Welch proved that if II has a winning strategy in the game of length $\omega+1$ then there is an inner model with a measurable cardinal. Welch conjectured that if II has a winning strategy in the game of length $\omega+1$ then there is a precipitous ideal on $\kappa$ .
Our first result confirms Welch's conjecture: if II has a winning strategy in the game of length $\omega+1$ then there is a normal, $\kappa$-complete precipitous ideal on $\kappa$ . In fact if $\gamma\leq\kappa$ is regular and II wins the game of length $\gamma$, then there is a normal, $\kappa$-complete ideal on $\kappa$ with a dense tree that is $<-\gamma$-closed.
But is this result vacuous? Our second result is that if you start with a model with sufficient fine structure and a measurable cardinal then there is a forcing extension where:
1. $\kappa$ is inaccessible and there is no $\kappa^{+}$-saturated ideal on $\kappa$,
2. for each regular $\gamma\leq\kappa$, player II has a winning strategy in the game of length $\gamma,$
3. for all regular $\gamma\leq\kappa$ there is a normal fine ideal $\mathcal{I}_{\gamma}$ such that $P(\kappa)/\mathcal{I}\gamma$ has a dense, $<-\gamma$ closed tree.
The proofs of these results use techniques from the proofs of determinacy, lottery forcing, iterated club shooting and new techniques in inner model theory. They leave many problems open and not guaranteed to be difficult.
This is joint work of M Foreman, M. Magidor and M. Zeman.[-]

I will talk about my result joint with S. Shelah establishing that the Borel space of torsion-free Abelian groups with domain ω is Borel complete, i.e., the isomorphism relation on this Borel space is as complicated as possible, as an isomorphism relation. This solves a long-standing open problem in descriptive set theory, which dates back to the seminal paper on Borel reducibility of Friedman and Stanley from 1989. After this I will survey some recent results (also joint with S. Shelah) on the existence of uncountable Hopfian and co-Hopfian abelian groups, and on the problem of classification of countable co-Hopfian abelian and 2-nilpotent groups.[-]

We characterize hyperfinite bipartite graphings that admit measurable perfect matchings. In particular, we prove that every regular hyperfinite one-ended bipartite graphing admits a measurable perfect matching. We give several applications of this result. We extend the Lyons-Nazarov theorem by showing that a bipartite Cayley graph admits a factor of iid perfect matching if and only if the group is not iso-morphic to the semidirect product of Z and a finite group of odd order, answering a question of Kechris and Marks in the bipartite case. We also answer an open question of Bencs, Hruskova and Toth arising in the study of balanced orientations in graphings. Finally, we show how our results generalize and lead to a simple approach to recent results on measurable circle squaring.[-]