Research in geometric group theory

在几何群论领域提出了三个相当雄心勃勃的项目。第一个灵感来自兹利尔·塞拉对塔尔斯基逻辑问题的几何解。PI和Mladen Bestvina计划完成他们的程序，用几何方法解决Mal‘cev的两个四十年的问题。他们还建议推广他们的方法，并给出Tarski问题的另一种证明。与Michael Handel合作的第二个项目是解决自由群F的外自同构群OUT(F)的共轭问题。这一领域的许多研究都是由线性群、曲面映射类群和OUT(F)之间的相似性驱动的。对于这另外两类，共轭问题已得到解决。最后一个项目，再次与Mladen Bestvina合作，是为了更好地理解自由群的外自同构群的几何。特别地，提出了映射类群环境中类似于曲线复形的复数是字双曲线的结论。几何群论是一个相对年轻的数学分支，它将数学中其他领域的问题表述出来，然后用几何术语来解决。这种方法在有时被视为远离几何学的领域取得了成功。例如，Zlil Sela用几何方法解决了Tarski的一个古老的逻辑问题。受塞拉方法的启发，PI和Mladen Bestvina提出了解决Mal‘cev的两个老逻辑问题。群论是这些方法特别成功的另一个领域。群体在数学和科学中无处不在。分子的对称性就是一个例子。有一类重要的群被称为“自由群”，所有其他群都可以从它构造出来。自由群F的对称集是另一个重要的群，记为(F)，它一直是当前很感兴趣的主题。PI提出了另外两个项目，重点是Out(F)。在一篇文章中，他和Michael Handel提出解决Out(F)上的一个古老而基本的问题，即它的共轭问题是可解的。在另一个项目中，PI和Bestvina建议更好地理解Out(F)的内在几何。

Three rather ambitious projects are proposed in the area of geometricgroup theory. The first is inspired by Zlil Sela's geometric solutionof Tarski's logic problem. The PI and Mladen Bestvina plan to completetheir program to solve by geometric means two forty-year-old problemsof Mal'cev. They also propose to generalize their methods and providean alternate proof to the Tarski problem. The second project, jointwith Michael Handel, is to solve the conjugacy problem for the outerautomorphism group Out(F) of a free group F. Much of the research inthis area is driven by the similarities between linear groups, surfacemapping class groups, and Out(F). The conjugacy problem has beensolved for these other two classes. The final project, again jointwith Mladen Bestvina, is to better understand the geometry of outerautomorphism groups of free groups. In particular, it is proposed to showthat a complex analogous to the curve complex in the mapping classgroup setting is word hyperbolic.Three projects are proposed in the area of geometric grouptheory. Geometric group theory is a relatively young branch ofmathematics in which problems from other areas of mathematics arereformulated and then solved in geometric terms. This approach hasbeen successful in areas that are sometimes viewed as distant fromgeometry. For example, Zlil Sela used geometric methods to solve anold logic problem of Tarski. Inspired by Sela's methods, the PI andMladen Bestvina propose to solve two old logic problems ofMal'cev. Group theory is another area where these methods have beenparticularly successful. Groups are ubiquitous in math and thesciences. The set of symmetries of a molecule is an example. There isan important class of groups called "free groups" from which all othergroups can be constructed. The set of symmetries of a free group F isanother important group, denoted Out(F), which has been the subject ofmuch current interest. The PI proposes two other projects focusing onOut(F). In one, he and Michael Handel propose to solve an old andfundamental problem on Out(F), namely that its conjugacy problem issolvable. In the other project, the PI and Bestvina propose to betterunderstand the intrinsic geometry of Out(F).