Collaborative research: model theory and algebraic geometry in groups and algebras, non-standard actions, algorithmic problems

合作研究：群和代数中的模型理论和代数几何、非标准动作、算法问题

• 批准号: 1201379
• 负责人: Olga Kharlampovitch
• 金额: $ 13.58万
• 依托单位: CUNY Hunter College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201379&HistoricalAwards=false
• 关键词: Collaborative; research; model; theory; algebraic

In this project we propose to expand our methods (Makanin-Razborov's machinery, algebraic geometry over groups, free actions on non-standard trees, structure of limit groups, etc.) developed in the proof of the Tarski's conjectures on first-order theories of free groups into several directions: model theory and algebraic geometry in the presence of negative curvature and for right-angled Artin groups; group actions on Lambda-trees, Lambda-hyperbolic geodesic spaces, and cube complexes; algorithmic problems for groups and elimination processes; equations in a wide variety of groups and algebras.Solving equations is one of the main themes in mathematics from ancient times. Equations give a universal language to describe scientific problems in all their variety. Over the years (centuries, in fact) equations and their solutions became very complex, so to understand their hidden structure one has to use very elaborate techniques from algebra and geometry. In particular, groups are used to describe symmetries of mathematical objects, they play a fundamental role in studying solutions of equations. Nowadays, when mathematics gets ever more complex, equations alone cannot describe subtleties of scientific phenomena; new problems require much more powerful means and more powerful languages. Most of the essential properties of objects that occur in everyday mathematical practice can be described in a very particular but universal language, so called first-order logic. In this project we study properties of groups, in all their entirety, that can be described in the first-order logic. Along the way we hope to shed some light on several fundamental open problems in group theory and algebra.

In this project we propose to expand our methods (Makanin-Razborov's machinery, algebraic geometry over groups, free actions on non-standard trees, structure of limit groups, etc.) developed in the proof of the Tarski's conjectures on first-order theories of free groups into several directions: model theory and algebraic geometry in the presence of negative curvature and for right-angled Artin groups;小组对兰巴达树，兰巴达杂种大地测量空间和立方体综合体进行小组动作；小组和消除过程的算法问题；各种组和代数方程的方程式是远古时代的数学主题之一。 方程式提供了一种通用语言来描述其各种各样的科学问题。多年来（实际上几个世纪）方程式及其解决方案变得非常复杂，因此要了解其隐藏的结构，必须使用代数和几何形状的非常精心的技术。特别是，组用于描述数学对象的对称性，它们在研究方程解决方案方面起着基本作用。 如今，当数学变得越来越复杂时，仅方程就无法描述科学现象的微妙之处。新问题需要更强大的手段和更强大的语言。 日常数学实践中发生的对象的大多数基本属性可以用非常特殊但通用的语言描述，即所谓的一阶逻辑。 在这个项目中，我们可以全面研究小组的属性，可以在一阶逻辑中描述。一路上，我们希望能阐明小组理论和代数中的几个基本开放问题。