Manifolds and Moduli Spaces

在这项拨款下进行的部分研究将涉及流形及其模的研究。流形是空间的广义概念，在数学和科学中随处可见。流形的定义性质是它被有限数量的参数局部参数化(例如，地球表面可以被两个参数局部参数化，即经度和纬度)。代数拓扑学的一个经典主题，加拉修斯的研究领域，是流形的分类：如何确定两个流形是否同构，以及如何写出所有可能的流形的列表？这里进行的研究涉及到分类流形在家族中如何变化的相关问题：它们的对称性是什么，它们的变形是什么。在这项资助下进行的研究将应用新的方法来研究这些经典和重要的问题。在这笔拨款下进行的另一项研究涉及由数论产生的对象的变形，使用同伦理论的方法进行研究。拟议的活动将继续发展一个相对较新的数学领域，其中同伦理论被用于研究各种模空间。这一领域的一个重要早期结果是马德森和韦斯对芒福德猜想的解答。自那时以来，发生了许多重大的新发展，其中许多都要归功于首席研究员，比如1995年哈彻猜想的解。这门学科位于代数拓扑学和其他数学分支之间的重叠部分，在代数几何、辛几何以及可能的理论物理中都有预期的应用。该项目的另一部分涉及同伦理论思想在数论中的应用。在派生环境中工作是同伦理论中一个非常熟悉的概念，并经常被成功地引入到代数等领域。最近，使用派生的设置来研究变形受到了很大的关注。该项目将开发并将这些技术应用于伽罗瓦表示法，这是数论中最重要的对象。

Parts of the research conducted under this grant will concern the study of manifolds and their moduli. Manifolds are a generalized concepts of space, appearing all over mathematics and science. The defining property of a manifold is that it is locally parametrized by a finite number of parameters (for example, the surface of the earth can locally be parametrized by two parameters, namely longitude and latitude). A classical topic in algebraic topology, Galatius' field of research, is the classification of manifolds: How does one decide whether two manifolds are isomorphic, and how does one write a list of all possible manifolds? The research conducted here concerns the related question of classifying how manifolds may vary in families: what are their symmetries and what are their deformations. The research conducted under this grant will apply new methods to study these classical and important questions. Another part of the research conducted under this grant concerns deformations of objects arising from number theory, studied using methods from homotopy theory.The proposed activity will continue the development of a relatively new area of mathematics in which homotopy theory is used to study various moduli spaces. An important early result this area is the solution, by Madsen and Weiss, of a conjecture of Mumford. Significant new developments have happened since then, many of which, such as the solution to a 1995 conjecture of Hatcher, are due to the principal investigator. This subject lies in the overlap between algebraic topology and other branches of mathematics, with expected applications in algebraic geometry, symplectic geometry, and possibly theoretical physics. Another part of the project concerns applications of homotopy theoretic ideas in number theory. Working in a derived setting is a very familiar idea in homotopy theory, and has often been successfully imported into algebra and other fields. Recently, the use of a derived setting for the study of deformations has received much attention. The project will develop and apply these techniques to Galois representations, objects of fundamental importance in number theory.