CARDINAL INVARIANTS OF THE CONTINUUM AND THEIR APPLICATIONS

这个项目的研究致力于连续统的基本不变量和迭代强迫理论之间的相互作用，以及这两个领域在代数中的应用。我们简要地概述了主要的主题和结果。沿着模板迭代。利用Shelah最近的沿着模板迭代技术的公理化方法，我们得到了几个新的结果，该技术最初是为了证明σ < a的一致性而开发的。例如，我们证明了a_g>max{a,non(M)}是一致的，其中a_g表示最小的最大共有限群的大小。我们还证明了几乎不相交数a可以一致地是可数共合性的一个奇异基数。破碎的迭代。利用一种复杂的新迭代技术，证明了cov(M)=non(N)=N_2和cov(N)=non(M)=c=N_3.3的同时一致性。完美集合公理。假设PSP（k，Γ）持有大小至少为k的pointclass Γ中的每个集合都有一个完美子集。设G_N_1是由（最多）n_1个开集的交集组成的集合。我们证明了在CH模型上迭代加入N_2后，P5P（N_2, G_N_1）成立。与逃避和预测有关的基数不变量。我们在ZFC中证明了b【小于或等于】b_2，其中b为非边界数，b_2为常数预测数，从而回答了Kamo问题。在与Shelah的联合工作中，我们获得了几个相关的结果，例如不同k的k常数预测数可以一致地不同。Sym（ω）的共性。对称群Sym（ω）的共性c（Sym(ω)）是最小基数k，使得Sym（ω）可以写成长度为k的严增真子群链的并。在与Losada的联合工作中，我们证明了g【小于或等于】c（Sym(ω)）在ZFC中成立，从而部分回答了Thomas提出的一个问题。

Research in this project was devoted to the interplay between cardinal invariants of the continuum and iterated forcing theory, as well as to applications of both areas to algebra. We briefly sketch the main topics and results.1. Iteration along a template. Using an axiomatic approach to Shelah's recent technique of iteration along a template which was originally developed to show the consistency of σ < a, we obtained several new results. For example, we proved it is consistent that a_g>max{a,non(M)} where a_g denotes the size of the smallest maximal cofinitary group. We also showed that the almost disjointness number a can consistently be a singular cardinal of countable cofinality.2. Shattered iteration. Using a sophisticated new iteration technique built from a system of compete Boolean algebras which add both Cohen and random reals, we proved the simultaneous consistency of cov(M)=non(N)=N_2 and cov(N)=non(M)=c=N_3.3. Perfect set axioms. Say that PSP(k,Γ) holds if every set in the pointclass Γ of size at least k has a perfect subset. Let G_N_1 be the class of sets which are intersections of (at most) N_1many open sets. We showed that after iteratively adding N_2 Sacks reals over a model for CH, P5P(N_2, G_N_1) holds.4. Cardinal invariants related to evasion and prediction. We proved in ZFC that b【less than or equal】b_2 where b is the unbounding number and b_2 is the constant prediction number, thus answering a question of Kamo. In joint work with Shelah, we obtained several related results, for example that the k-constant prediction numbers for different k can consistently be different.5. The cofinality of Sym(ω). The cofinality c(Sym(ω)) of the symmetric group Sym(ω) is the least cardinal k such that Sym(ω) can be written as the union of a strictly increasing chain of proper subgroups of length k. In joint work with Losada, we proved that g【less than or equal】c(Sym(ω)) holds in ZFC, thus partially answering a question addressed by Thomas.