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，有时是相当遥远的数学领域借用思想和技术，帮助证明了似乎遥不可及的长期存在的假设。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，它们会影响新的和以前存在的研究分支的发展，增加数学作为一个整体的力量，并增强其美感。* 对无限群的现代研究将数学的几个领域与群论联系起来。我的研究主题是无限离散群理论、数理逻辑、几何学和拓扑学之间的相互作用。我提出的研究计划的目的是技术的发展，研究相对双曲群的基本理论和相关问题。当群作为代数对象被引入时，数理逻辑从其基本理论的角度研究群;也就是说，重点是在给定群中为真的群论语言中的所有一阶句子的集合。一个自然的问题是在何种程度上的基本理论允许一个区分代数性质的团体。从几何学和拓扑学中借用工具被证明在回答这类问题时非常有用。我对这个研究领域的兴趣是由几个著名的开放性问题引起的，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、数理逻辑和纯数学的相关领域的发展产生了很大的影响。

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.**