相关论文: Sheva-Sheva-Sheva: Large Creatures
We show Shelah's original creature forcing from 1984 strongly preserves tight mad families. In particular, answering questions of Fischer and Friedman and Friedman and Zdomskyy, we show the constellation $\aleph_1 = \mathfrak{a} <…
A system of nonlinear ordinary differential equations with forcing function is developed to model evolution processes in complex systems. In this system R, C, and P are the resource, consumption, and production functions correspondingly. F…
We investigate two variants of splitting tree forcing, their ideals and regularity properties. We prove connections with other well-known notions, such as Lebesgue measurablility, Baire- and Doughnut-property and the Marczewski field.…
Remarkable cardinals were introduced by Schindler, who showed that the existence of a remarkable cardinal is equiconsistent with the assertion that the theory of $L(\mathbb R)$ is absolute for proper forcing. Here, we study the…
Reinforcement learning techniques achieved human-level performance in several tasks in the last decade. However, in recent years, the need for interpretability emerged: we want to be able to understand how a system works and the reasons…
We propose developing the theory of consequences of morasses relevant in mathematical applications in the language alternative to the usual one, replacing commonly used structures by families of sets originating with Velleman's neat…
A narrow system is a combinatorial object introduced by Magidor and Shelah in connection with work on the tree property at successors of singular cardinals. In analogy to the tree property, a cardinal $\kappa$ satisfies the \emph{narrow…
The Rabin tree theorem yields an algorithm to solve the satisfiability problem for monadic second-order logic over infinite trees. Here we solve the probabilistic variant of this problem. Namely, we show how to compute the probability that…
This is an exposition of the first two sections of Chapter VI of Shelah's book Proper and Improper Forcing. It covers various preservation theorems for CS iteration of proper forcing (omega-omega bounding, Sacks property, P-point property,…
Ben-David and Shelah proved that if $\lambda$ is a singular strong-limit cardinal and $2^\lambda=\lambda^+$, then $\square^*_\lambda$ entails the existence of a normal $\lambda$-distributive $\lambda^+$-Aronszajn tree. Here, it is proved…
Recent research has recognized interpretability and robustness as essential properties of trustworthy classification. Curiously, a connection between robustness and interpretability was empirically observed, but the theoretical reasoning…
We construct a sheaf-theoretic analogue of the wrapped Fukaya category in Lagrangian Floer theory, by localizing a category of sheaves microsupported away from some given $\Lambda \subset S^*M$ along continuation maps constructed using the…
A module is called absolutely indecomposable if it is directly indecomposable in every generic extension of the universe. We want to show the existence of large abelian groups that are absolutely indecomposable. This will follow from a more…
We consider the task of properly PAC learning decision trees with queries. Recent work of Koch, Strassle, and Tan showed that the strictest version of this task, where the hypothesis tree $T$ is required to be optimally small, is NP-hard.…
Answering a question of Usuba, we show that an extendible cardinal can be preserved by a set forcing that is not a small forcing.
We answer a question of Woodin by showing that assuming an inaccessible cardinal $\kappa$ which is a limit of ${<}\kappa$-supercompact cardinals exists, there is a stationary set preserving forcing $\mathbb{P}$ so that $V^{\mathbb…
Shelah's own proof to his recent polarized partition theorem involving a singular strong limit that violates the GCH is presented. The proof is slightly re-arranges so that no use of the ideal I[\lambda] is made. The proof should be…
We use Shelah's theory of possible cofinalities in order to solve a problem about ultrafilters. THEOREM. Suppose that $ \lambda $ is a singular cardinal, $ \lambda ' < \lambda $, and the ultrafilter $D$ is $ \kappa $-decomposable for all…
We investigate the relation between the theory of the iterations in the sense of Shelah-Stupp and of Muchnik, resp., and the theory of the base structure for several logics. These logics are obtained from the restriction of set…
The goal of these lectures is to survey some of the recent progress on the description of large-scale structure of random trees. We use the framework of Markov-Branching sequences of trees and discuss several applications.