Topics in Applications of Set Theory

索莱茨基的研究兴趣在于应用数学逻辑（模型论，集合论，主要是描述性集合论）的分析和拓扑。该项目由几个部分组成。例如，Solecki提出研究波兰群上普遍可测同态的自动连续性问题。一个可能的解决这个问题肯定会涉及描述波兰组的集合论。然而，它也很可能严重依赖，一方面，在集理论假设（如连续统假设），另一方面，在代数性质的基础组（顺从性，自由度）。这个问题之间的联系和集合论和代数的迹象可以发现在早期的工作克里斯滕森，Mokobodzki和Solecki。在该项目的另一部分，Solecki将应用方法开发首先在集合论（在博雷尔等价关系的研究）的问题，拓扑和模型理论。首先，他将使用描述集理论的技术，通过来自模型论的工具来研究一个重要的拓扑空间--伪弧的结构。第二，他将应用描述集合论的方法来研究与可数完备理论相关的典型群-例如，他将研究一个特定的拓扑空间-一个遗传上不可分解的连续体，称为伪弧。这些类型的拓扑空间最早是在20世纪25年代作为极其复杂的曲线的奇怪例子引入数学的。今天，我们知道它们在许多数学背景下自然出现，例如，在流体动力学中，在欧氏空间中的光滑动力系统中，在拓扑群中，在连续函数的研究中，等等。Solecki打算通过使用数理逻辑的分支-它最初是为了有效地研究抽象集合的大小而发展起来的，人们希望通过集合论的应用，跨越抽象学各个子领域的界限，在分析和拓扑学的方法应该导致新的和实质性的见解。

Solecki's research interests lie in applications ofmathematical logic (model theory, set theory and, primarily,descriptive set theory) to analysis and topology. The projectconsists of several parts. For example, Solecki proposes toinvestigate a problem of automatic continuity of universallymeasurable homomorphisms on Polish groups. A possible solution tothis problem will certainly involve descriptive set theory ofPolish groups. However, it is also likely to rely heavily, on theone hand, on set theoretic assumptions (like the ContinuumHypothesis) and, on the other hand, on algebraic properties of theunderlying group (amenability, freeness). Indications of theconnections between this problem and set theory and algebra can befound in earlier work of Christensen, Mokobodzki and Solecki. Inanother part of the project, Solecki will apply methods developedfirst in set theory (in the study of Borel equivalence relations)to problems in topology and model theory. First, he will usedescriptive set theoretic techniques augmented by tools comingfrom model theory to investigate the structure of an importanttopological space---the pseudo-arc. Second, he will apply methodsfrom descriptive set theory to study groups associated canonicallywith countable complete theories---Lascar's Galois groups.In the project, Solecki will apply techniques and notionsdeveloped in mathematical logic to problems in other areas ofmathematics. For example, he will investigate a certaintopological space---a hereditarily indecomposable continuum calledthe pseudo-arc. These types of topological spaces were firstintroduced in mathematics in the first quarter of the twentiethcentury as curious examples of extremely complicated curves.Today, we know that they appear naturally in many mathematicalcontexts, e.g., in fluid dynamics, in smooth dynamical systemsliving in Euclidean spaces, among topological groups, in the studyof continuous functions, etc. Solecki intends to gain a betterunderstanding of the pseudo-arc by using methods from descriptiveset theory, a branch of mathematical logic, which were originallydeveloped to study in an effective way the size of abstract sets.It is hoped that crossing boundaries of subfields of mathematicsby applying set theoretic methods in analysis and topology shouldlead to new and substantial insights.