Research in Geometry and Topology

几何与拓扑研究

• 批准号: 0502441
• 负责人: Mladen Bestvina
• 金额: --
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0502441&HistoricalAwards=false
• 关键词: Research; Geometry; Topology

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.

提案围绕三个主题展开。第一个问题是由逻辑学中的一阶谓词演算激发的群理论中的问题。这些问题涉及到，例如，理解集合的同态的一个固定组到一个自由组，扩展到另一个固定组。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。我们的想法是，看看是否可以使用适当选择的莫尔斯函数的模空间，并在商的Teichmuller空间的Torelli群，分别计算同调。莫尔斯函数是几何定义的--它是心脏收缩，即最短曲线的长度（或者在Torelli群的情况下是它的同调版本）。最后一个题目是关于直角Artin群的拟等距刚性。在过去的15年里，这门学科已经取得了一些巨大的进展，但是这类特殊的群提出了许多以前没有涉及过的问题。这项提案的主题是研究自然出现在数学中的几个对象的拓扑和几何性质。例如，考虑具有固定数量的孔的表面的所有可能的“形状”的集合。这个集合形成了一个空间，称为曲线的模空间。这个空间的“洞数”的计算最近已经由atopological tour de force完成。这里提出了一种不同的计算方法，这种方法也可以提供额外的信息。为了描述这种方法，考虑下面的类比。当水慢慢地倒入一个正常的玻璃杯时，玻璃杯中的水的表面将永远是圆的。相比之下，如果把水倒进一个有凹柄的玻璃杯里，一旦水到达把手，水面将由两个圆盘组成。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，例如曲线的模空间，可以类似地分析。这里提出了一种特殊的“倒水”方式。