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

在这个项目中，我们建议扩展我们的方法(Makanin-Razborov的机械，群上的代数几何，非标准树上的自由作用，极限群的结构等)。在证明塔斯基关于自由群的一阶理论的猜想时，发展成几个方向：负曲率和直角Artin群的模型理论和代数几何；Lambda-树、Lambda-双曲测地空间和立方体复形上的群作用；群的算法问题和消去过程；各种群和代数中的方程。求解方程从古代起就是数学的主要主题之一。方程式提供了一种通用的语言来描述各种科学问题。多年来(实际上是几百年)，方程及其解变得非常复杂，因此要理解它们的隐藏结构，人们必须使用代数和几何中非常精细的技术。特别是，群被用来描述数学对象的对称性，它们在研究方程的解方面起着基础性的作用。如今，当数学变得越来越复杂时，仅靠方程无法描述科学现象的微妙之处；新的问题需要更强大的方法和更强大的语言。日常数学实践中出现的大多数对象的基本性质都可以用一种非常特殊但普遍的语言来描述，即所谓的一阶逻辑。在这个项目中，我们研究可以用一阶逻辑描述的群的全部性质。在此过程中，我们希望对群论和代数中的几个基本开放问题有所了解。

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.