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

在这个项目中，我们打算扩展我们的方法（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Tarski关于自由群的一阶理论的猜想中发展到几个方向：负曲率和直角Artin群下的模型理论和代数几何；lambda -树、lambda -双曲测地线空间和立方体复形上的群作用；分组和淘汰过程的算法问题；各种群和代数中的方程。解方程自古以来就是数学的主题之一。方程提供了一种通用语言来描述各种各样的科学问题。多年来（实际上是几个世纪以来），方程及其解变得非常复杂，因此要理解它们隐藏的结构，必须使用代数和几何中非常复杂的技术。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中起着基本的作用。今天，当数学变得越来越复杂，方程式本身不能描述科学现象的微妙之处；新的问题需要更强大的手段和更强大的语言。在日常数学实践中出现的物体的大多数基本属性都可以用一种非常特殊但通用的语言来描述，即所谓的一阶逻辑。在这个项目中，我们研究群的性质，在所有他们的整体，可以用一阶逻辑来描述。在此过程中，我们希望能阐明群论和代数中几个基本的开放性问题。

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.