Research in Geometry and Topology

该提案围绕3个主题展开。第一部分是关于逻辑中一阶谓词演算引发的群内论问题。这些问题包括理解一个固定群的同态集合到一个扩展到另一个固定群的自由群。第二个主题涉及曲线的模空间和映射类群的Torelli子群的拓扑结构。这个想法是看看是否可以用一个适当选择的莫尔斯函数分别在模空间和Torelli群的teichmuller空间的商上计算同调。摩尔函数是几何上定义的——它是收缩，即最短曲线的长度（或在托雷利群的情况下的同调版本）。最后一个题目是直角Artin群的准等距刚性。在过去15年左右的时间里，这一学科取得了一些巨大的进步，但这一类特殊的群体提出了许多以前没有解决的问题。该提案的主题是研究自然出现在数学中的几种对象的拓扑和几何特性。例如，考虑具有固定数量孔的表面的所有可能“形状”的集合。这些集合形成了一个空间，称为曲线的模空间。这个空间的“孔数”的计算最近已经由拓扑力作完成。这里提出了一种不同的计算方法，这种方法也会提供额外的信息。为了描述这种方法，请考虑下面的类比。当水慢慢倒入普通玻璃杯时，玻璃杯中的水表面总是呈圆盘状。相反，如果把水倒进一个空心把手的杯子里，一旦水到达把手，水的表面就会由两个圆盘组成。因此，通过观察水面的形状，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，可以用类似的方法进行分析。这里提出了具体的“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.