Mathematical Sciences: Geometry and Low-Dimensional Topology in Group Theory

小行星9703756 Zlil Sela打算完成他的论文序列的同构问题的双曲群和序列的论文自同构的自由群。 他进一步计划完成写作他的工作结构的一套解决方案的系统方程在一个自由组，并对家庭的这种系统取决于一组定义参数。 这项工作是基于借用概念和思想，从代数几何和低维拓扑和实施它们在群论。 Sela和E.里普斯（希伯来大学）长期从事基本代数结构的研究，这些结构被称为“群”，它们抽象并抓住了对称的本质。 该领域的三个关键问题之一是同构问题，即判断两个以不同方式呈现的组是否真的相同的问题。 众所周知，不可能有解决这个问题的通用配方，一个将适用于完全任意的一对群体。 出于这个原因，人们一直在寻找在大类有趣的群中工作的方法，而塞拉正在接近（格罗莫夫）双曲群类的方法，这些群的兴趣来自于被称为流形的几何对象，包括三维的任意流形和任何维度的负弯曲流形。 在这项工作中发展的思想和技术导致了其他问题和意想不到的答案，所有轴承之间的深刻和密切的联系代数群和几何流形。 在过去的一年中，塞拉已经制定了一个几何方法的解决方案的方程组在一个所谓的自由组。 这种方法大量借用了代数几何。 最近，他推动这种几何方法进一步研究指数家庭的一套解决方案的方程组。 塞拉和里普斯现在预计，他们开发的结构理论将在群论和数理逻辑中找到许多应用。 特别是，他们相信他们手中的工具将为逻辑学家阿尔弗雷德·塔斯基（Alfred Tarski）在1950年左右提出的几个问题提供肯定的答案。 由于他们的技术借用大量的低维拓扑，他们希望他们的结构理论也将找到应用的基本问题，低维拓扑。 ***

9703756 Sela Zlil Sela intends to complete his sequence of papers on the isomorphism problem for hyperbolic groups and the sequence of papers on automorphisms of free groups. He further plans to complete writing his work on the structure of sets of solutions to systems of equations in a free group, and on families of such systems depending on a set of defining parameters. This work is based on borrowing notions and ideas from algebraic geometry and low dimensional topology and implementing them in group theory. Sela and E. Rips (Hebrew University) are engaged in a long-standing investigation of fundamental algebraic structures known as ``groups,'' structures which abstract and capture the essence of symmetry. One of three key problem of the field is the isomorphism problem, the problem of telling when two groups, presented differently, are really the same. It is known that there can be no general recipe for solving this problem, one that will work for a completely arbitrary pair of groups. For this reason, recipes have been sought that work within large classes of interesting groups, and Sela is closing in on such a recipe for the class of (Gromov) hyperbolic groups, groups whose interest derives from being attached to geometric objects known as manifolds, both arbitrary manifolds of dimension three and negatively curved ones of any dimension. The ideas and techniques evolved in this work have led to other questions and to unexpected answers, all bearing on the deep and intimate connection between the algebra of groups and the geometry of manifolds. During the last year, Sela has developed a geometric approach to sets of solutions of equations in a so-called free group. The approach borrows heavily from algebraic geometry. More recently he has pushed this geometric approach further in order to study indexed families of sets of solutions of equations in groups. Sela and Rips now expect that the structure theory they have developed will find many applications in group theory and in mathematical logic. In particular, they believe that the tools they have in hand will provide positive answers to several questions posed by the logician Alfred Tarski around 1950. Since their techniques borrow heavily from low dimensional topology, they hope that their structure theory will also find applications to basic questions in low-d topology. ***