Algebraic Geometry of Partially Commutative Groups

求解方程并描述其解一直是数学的核心部分。这里我们有以群为框架的方程，我们以群的形式为其寻找解。我们知道如何描述阿贝尔群上的方程的解以及自由群中一个变量的方程的解。这个项目考虑了一类群，它包含自由群和阿贝尔群，但却很容易描述和服从分析。我们的最终目标是理解这些群上的方程的解，但为了取得进展，我们将注意力限制在单变量方程上。我们都打算对群论(与方程有关的)领域作一些初步的调查，在这些领域中有一些我们认为我们可能有一些技巧要解决的难题。

Solving equations and describing their solutions has always been a central part of mathematics. Here we have equations which are framed in terms of groups and for which we seek solutions in groups. We know how to describe solutions to equations over Abelian groups and to equations in one variiable in free groups. This project considers a class of groups that contains free and Abelian groups but yet is simple to describe and which is amenable to analysis. Our eventual aim is to understand solutions to equations over these groups, but to make progress we are restricting attention to equations of one variable. We allso intend to make some preliminary investigations into areas of group theory (related to equations) where there are some difficult problems that we think we may have some techniques to solve.