Geometric group theory, topology, mathematical logic and algorithms

纯数学的最新发展证明了跨学科方法解决问题的力量。从不同的数学领域借鉴思想和技术，有时是相当遥远的，帮助证明了似乎遥不可及的长期猜测。然而，更重要的是，当各种数学传统和思维方式结合在一起时，它们影响着新的和以前存在的研究分支的发展，增加了整个数学的力量，增强了它的美。 对无限群的现代研究使数学的几个领域与群论产生了联系。我研究的主题是无限离散群理论、数理逻辑、几何和拓扑学之间的相互作用。 我提出的研究计划旨在发展研究相对双曲群的基本理论和相关问题的技术。虽然群是作为代数对象引入的，但数理逻辑是从群的基本理论的角度来研究群的，也就是说，重点是研究在给定群中成立的群论语言中所有一阶句子的集合。一个自然的问题是，初等理论在多大程度上允许人们区分群的代数性质。借用几何学和拓扑学的工具回答这类问题被证明是非常有用的。 我对这一研究领域的兴趣是由几个著名的开放问题激发的，这些问题涉及群论、几何和逻辑之间的联系。这一方向的研究对几何群论、数理逻辑以及纯数学相关领域的发展都有很大的影响。

Recent developments in pure mathematics demonstrate the power of an interdisciplinary approach to problem solving. Borrowing ideas and techniques from different, sometimes fairly distant, areas of mathematics helped prove long-standing conjectures that seemed to be out of reach. Yet, even more importantly, when various mathematical traditions and ways of thinking are brought together, they influence the development of both new and previously existing branches of research, increase the power of mathematics as a whole and enhance its beauty. The modern study of infinite groups brings several areas of mathematics into contact with group theory. The theme of my research is the interplay between the theory of infinite discrete groups, mathematical logic, geometry and topology. My proposed research program is aimed at the development of techniques to study the elementary theory of relatively hyperbolic groups and related questions. While groups were introduced as algebraic objects, mathematical logic studies groups from the point of view of their elementary theory; that is, the focus is on the set of all first order sentences in the language of group theory which are true in a given group. A natural question is to what extent the elementary theory allows one to distinguish algebraic properties of groups. Borrowing tools from geometry and topology proved extremely useful in answering questions of this kind. My interest in this field of research has been motivated by several famous open questions asking about connections between group theory, geometry and logic. Research in this direction has a strong influence on the development of geometric group theory, mathematical logic and related areas of pure mathematics.