By the Cantor-Bendixson theorem, subtrees of the binary tree on $\omega$ satisfy a dichotomy - either the tree has countably many branches or there is a perfect subtree (and in particular, the tree has continuum manybranches, regardless of the size of the continuum). We generalize this to arbitrary regular cardinals $\kappa$ and ask whether every $\kappa$-tree with more than $\kappa$ branches has a perfect subset. From large cardinals, this statement isconsistent at a weakly compact cardinal $\kappa$. We show using stacking mice that the existence of a non-domestic mouse (which yields a model with a proper class of Woodin cardinals and strong cardinals) is a lower bound. Moreover, we study variants of this statement involving sealed trees, i.e. trees with the property that their set of branches cannot be changed by certain forcings, and obtain lower bounds for these as well. This is joint work with Yair Hayut.[-]

Generalized descriptive set theory has mostly been developed for uncountable cardinals satisfying the condition $\kappa ^{< \kappa }=\kappa$ (thus in particular for $\kappa$ regular). More recently the case of uncountable cardinals of countable cofinality has attracted some attention, partially because of its connections with very large cardinal axioms like I0. In this talk I will survey these recent developments and propose a unified approach which potentially could encompass all possible scenarios (including singular cardinals of arbitrary cofinality).[-]

The combinatorics of successors of singular cardinals presents a number of interesting open problems. We discuss the interactions at successors of singular cardinals of two strong combinatorial properties, the stationary set reflection and the tree property. Assuming the consistency of infinitely many supercompact cardinals, we force a model in which both the stationary set reflection and the tree property hold at $\aleph_{\omega^2+1}$. Moreover, we prove that the two principles are independent at this cardinal, indeed assuming the consistency of infinitely many supercompact cardinals it is possible to force a model in which the stationary set reflection holds, but the tree property fails at $\aleph_{\omega^2+1}$. This is a joint work with Menachem Magidor.
Keywords : forcing - large cardinals - successors of singular cardinals - stationary reflection - tree property[-]

Universally Baire sets play an important role in studying canonical models with large cardinals. But to reach higher large cardinals more complicated objects, for example, canonical subsets of universally Baire sets come into play. Inspired by core model induction, we introduce the definable powerset $A^{\infty }$ of the universally Baire sets $\Gamma ^{\infty }$ and show that, after collapsing a large cardinal, $L(A^{\infty })$ is a model of determinacy and its theory cannot be changed by forcing. Moreover, we show a similar result for adding a club filter to the model constructed over universally Baire sets.[-]

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.[-]