Homotopy theoretic methods in the study of moduli spaces

拟议的活动将继续发展一个相对较新的数学领域，其中代数拓扑学用于研究黎曼曲面和相关物体的模空间。 到目前为止，这一领域的主要结果是Madsen和韦斯对Mumford猜想的一个推广的解。 本课题是代数拓扑学与其他数学分支的交叉，在代数几何、辛几何和可能的理论物理中具有预期的应用，将进一步研究和发展Madsen和韦斯的结果。 具体地说，建议的活动将定义ad维配边范畴C^d并确定其分类空间的同伦类型。 这将暗示Madsen和韦斯的结果，但这是一个更一般的结果，为许多新的可能的应用打开了大门。 同时，它在概念上更简单，并且允许更简单的证明。 这是提案的一个项目。 Madsen和韦斯定理是一个非常重要的定理，已经引起了人们的极大兴趣。 它为亏格g的Riemann曲面的模空间M_g和二维二边范畴提供了全新的认识。 这些对象已经被研究和使用了很长时间，并在许多不同的数学领域。 另一方面，很明显，结果不是确定的，例如，它被限制为维度d=2。 适当推广的发展对代数拓扑学及其相关领域具有重要意义。

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.