Homotopy theoretic methods in the study of moduli spaces

提出的活动将继续发展一个相对较新的数学领域，其中代数拓扑被用来研究黎曼曲面和相关对象的模空间。到目前为止，这一领域的主要成果是Madsen和Weiss对Mumford猜想的一般化的解决方案。这门学科是代数拓扑学和其他数学分支之间的重叠，在代数几何、辛几何以及可能的理论物理中有预期的应用。拟议的活动将进一步调查和发展madsen和Weiss的结果。具体来说，所提出的活动将定义ad维协同范畴C^d，并确定其分类空间的同伦类型。这将暗示Madsen和Weiss的结果，但这是一个更普遍的结果，为许多新的可能的应用打开了大门。同时，它将在概念上更简单，并且允许更简单的证明。这是提案中的一个项目。其余四个拟议项目是相互关联的。Madsen和Weiss的定理是一个非常重要的定理，已经引起了人们的极大兴趣。它对g属的黎曼曲面的模空间M_g和二维酒精范畴有了全新的认识。这些对象已经在数学的许多不同领域被研究和使用了很长时间。另一方面，很明显，结果不是确定的，例如，它被限制在维度d=2。对代数拓扑学及其相关领域进行适当的推广是一个非常重要的研究课题。

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.