Homotopy theoretic methods in the study of moduli spaces

模空间研究中的同伦理论方法

• 批准号: 0505740
• 负责人: Soren Galatius
• 金额: $ 10.12万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505740&HistoricalAwards=false
• 关键词: Homotopy; theoretic; methods; study; moduli

The proposed activity will continue the development of a relatively newarea of mathematics in which algebraic topology is used to study modulispaces of Riemann surfaces and related objects. The main result so far inthis area is the solution, by Madsen and Weiss, of a generalization of aconjecture of Mumford. This subject lies in the overlap between algebraictopology and other branches of mathematics, with expected applications inalgebraic geometry, symplectic geometry, and possibly theoretical physics.The proposed activity will further investigate and develop the result ofMadsen and Weiss. Specifically, the proposed activity will define ad-dimensional cobordism category C^d and determine the homotopy type ofits classifying space. This will imply the result of Madsen and Weiss,but is a much more general result, opening for many new possibleapplications. At the same time, it will be conceptually simpler andallows for a much simpler proof. This is one project of the proposal. The remaining four proposed projects are related.The theorem of Madsen and Weiss is a very important theorem, that hasalready recieved much interest. It sheds a completely new light on themoduli spaces M_g of Riemann surfaces of genus g and on the 2-dimensionalcobordism category. These objects has been studied and used for longtime, and in many different areas of mathematics. On the other hand it isclear that the result is not definitive, for example it is restricted todimension d=2. The development of the proper generalization should be ofgreat interest and importance for algebraic topology as well as forrelated fields.

拟议的活动将继续开发相对较数学的新区，其中代数拓扑用于研究Riemann表面和相关对象的模块空间。 到目前为止，这一地区的主要结果是Madsen和Weiss的解决方案是对Mumford的概括的概括。 该主题在于代数科学与数学的其他分支之间的重叠，预期的应用Inalgebraic几何形状，合成性几何形状以及可能的理论物理学。拟议的活动将进一步研究和发展MADSEN和WEISS的结果。 具体而言，所提出的活动将定义AD尺寸的共同体类别C^d，并确定对空间进行分类的同型类型。 这将暗示Madsen和Weiss的结果，但对于许多新的可能性开放，这是一个更普遍的结果。 同时，从概念上讲，这将是更简单的Andallows，以提供更简单的证明。 这是提案的一个项目。 其余四个拟议的项目是相关的。Madsen和Weiss的定理是一个非常重要的定理，已经收到了很大的兴趣。 它为属G属的Riemann表面和2维bobordism类别的riemann表面的themoduli空间提供了全新的灯光。 这些对象已被研究并用于长期以及数学的许多不同领域。 另一方面，结果不是确定的，例如，它限制了dimension d = 2。 适当的概括的发展对于代数拓扑以及相关领域应该具有很大的兴趣和重要性。