Topics in Applications of Set Theory

索莱茨基的研究兴趣在于数理逻辑（模型论、集合论，主要是描述性集合论）在分析和拓扑学中的应用。这个项目由几个部分组成。例如，Solecki提议研究波兰群上普遍可测同态的自动连续性问题。这个问题的一个可能的解决方案肯定会涉及到波兰群体的描述性集合理论。然而，一方面，它也可能严重依赖于集合论假设（如连续假设），另一方面，依赖于基础群的代数性质（可调节性，自由性）。在Christensen， Mokobodzki和Solecki的早期工作中可以找到这个问题与集合论和代数之间联系的迹象。在项目的另一部分，Solecki将首先在集合论（研究Borel等价关系）中开发的方法应用于拓扑和模型论问题。首先，他将使用描述性集合论技术和模型论的工具来研究一个重要的拓扑空间——伪弧的结构。其次，他将运用描述性集合论的方法来研究与可数完备理论相关的群——拉斯卡的伽罗瓦群。在这个项目中，索莱茨基将把数理逻辑中的技术和概念应用到其他数学领域的问题中。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在20世纪的前25年首次被引入数学，作为极其复杂曲线的奇特例子。今天，我们知道它们自然地出现在许多数学环境中，例如，在流体动力学中，在欧几里得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打算通过使用描述集理论的方法来更好地理解伪弧，描述集理论是数学逻辑的一个分支，最初是为了有效地研究抽象集的大小而发展起来的。希望通过在分析和拓扑学中应用集合论方法来跨越数学子领域的界限，能够带来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.