Geometric group theory, topology, mathematical logic and algorithms

几何群论、拓扑、数理逻辑与算法

• 批准号: RGPIN-2016-06154
• 负责人: Bumagin, Inna
• 金额: $ 1.09万
• 依托单位: Carleton University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675553
• 关键词: Geometric; group; theory; topology; mathematical

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.**

纯数学的最新发展证明了跨学科方法解决问题的力量。借鉴不同的、有时相当遥远的数学领域的思想和技术，有助于证明长期以来看似遥不可及的猜想。然而，更重要的是，当各种数学传统和思维方式融合在一起时，它们会影响新的和先前存在的研究分支的发展，增强数学整体的力量并增强其美感。 ***对无限群的现代研究将数学的多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、几何和拓扑之间的相互作用。***我提出的研究计划旨在开发研究相对双曲群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，重点是群论语言中在给定群中成立的所有一阶句子的集合。一个自然的问题是，基本理论在多大程度上允许人们区分群的代数性质。事实证明，借用几何和拓扑的工具对于回答此类问题非常有用。****我对这一研究领域的兴趣是由几个关于群论、几何和逻辑之间的联系的著名开放问题激发的。该方向的研究对几何群论、数理逻辑及纯数学相关领域的发展具有重大影响。**