CARDINAL INVARIANTS OF THE CONTINUUM AND THEIR APPLICATIONS

这个项目的研究致力于连续统的基数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。简要概述了本文的主要研究课题和研究成果。沿着模板迭代。利用公理方法对谢拉最近提出的沿着模板迭代的方法，我们得到了几个新的结果。模板最初是为了显示σ&lt；a的一致性而开发的。例如，我们证明了a_g&gt；max{a，non(M)}是一致的，其中a_g表示最小极大余有限群的大小。我们还证明了几乎不交数a可以一致地是可数余项式的奇异基数。破碎的迭代。利用竞争布尔代数系统构造的复杂的新迭代技巧，证明了cov(M)=non(N)=N_2和cov(N)=non(M)=c=N3.3的同时相合性。完美集合公理。假设Psp(k，Γ)成立，如果至少k的点类Γ中的每个集合都有一个完美子集。设G_N_1是(至多)N_1个开集的交集。我们证明了在迭代添加N_2个Sack之后，在CH的一个模型上，P_5P(N_2，G_N_1)保持。与回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是去界数，b_2是常预测数，从而回答了Kamo的一个问题。在与谢拉的联合工作中，我们得到了几个相关的结果，例如，对于不同的k，k-常数预测数可以一致地不同。Sym(ω)的余完备性。对称群Sym(ω)的余定性c(sym(ω))是使得Sym(ω)可以写成长度为k的真子群的严格增链的并的最小基数k。在与Losada的合作中，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.