Research in Geometry and Topology

该提案围绕3个主题展开。第一个是关于一阶谓词演算不逻辑引发的群论问题。这些问题涉及到，例如，理解一个固定群到一个自由群的同态集合，这个自由群可以扩展到另一个固定群。第二个主题是关于曲线的模空间和映射类群的Torelli子群的拓扑。其思想是看看Torelli群是否可以分别在模空间和TeichMuller空间的商上使用适当选择的Morse函数来计算同调。Morse函数是几何定义的--它是收缩，即最短曲线的长度(在Torelli群的情况下是这个的同调版本)。最后讨论直角Artin群的拟等距刚性。在过去的大约15年里，这门学科取得了一些重大进展，但这类特殊的群提出了许多以前没有讨论过的问题。该提议的主题是研究几个自然出现在数学中的对象的拓扑和几何性质。例如，考虑具有固定数量的孔数的曲面的所有可能的“形状”的集合。这个集合形成了一个空间，称为曲线模空间。这个空间的“洞的数量”的计算最近已经由拓扑学环游完成。这里提出了一种不同的计算方法，这种方法也会给出额外的信息。为了描述这种方法，请考虑下面的类比。当水慢慢倒入异形玻璃中时，玻璃中的水的表面总是不稳定的。相比之下，如果将水倒入带有全息手柄的玻璃杯中，一旦水到达手柄，水面就会由两个圆盘组成。因此，通过检查水面的形状，我们可以判断玻璃杯是否有把手。拓扑空间，如曲线的模空间，也可以进行类似的分析。提出了一种特殊的“浇水”方式。

The proposal centers around 3 topics. The first is about questions ingroup theory motivated by the first order predicate calculus inlogic. The questions involve e.g. understanding the sets ofhomomorphisms of a fixed group into a free group that extend toanother fixed group. The second topic concerns the topology of modulispace of curves and of the Torelli subgroup of a mapping classgroup. The idea is to see if homology can be computed using a suitablychosen Morse function on the moduli space, and on the quotient ofTeichmuller space by the Torelli group, respectively. The Morsefunction is geometrically defined -- it is the systole, i.e. thelength of the shortest curve (or the homological version of this inthe case of Torelli groups). The last topic is on the quasi-isometricrigidity of right-angled Artin groups. The subject has seen somegreat advances in the last 15 years or so, but this particular classof groups raises many issues not addressed before.The subject of the proposal is the study of topological and geometricproperties of several objects that naturally appear inmathematics. For example, consider the collection of all possible"shapes" of a surface with a fixed number of holes. This collectionforms a space, called moduli space of curves. The calculation of the"number of holes" for this space has been recently completed by atopological tour de force. A different method for this calculation isproposed here, one that would also give additionalinformation. To describe the method, consider the followinganalogy. When water is slowly poured into anormal glass, the surface of water in the glass will always be adisk. By contrast, if water is poured into a glass with a hollowhandle, the surface of water will consist of two disks once waterreaches the handle. Thus, by examining the shape of the surface ofwater we can tell if a glass has a handle or not. Topological spaces,such as moduli space of curves, can be analyzed similarly. Aparticular way of "pouring water" is proposed here.