Manifolds, Moduli Spaces and Homotopy Theory

拟议的研究将继续发展一个相对较新的数学领域，其中同伦理论被用来研究各种模空间。 一个重要的早期结果，这方面的解决方案，由马德森和韦斯，一个猜想的芒福德。他们的结果涉及封闭的二维流形的对称群，或者等价地，它涉及这种流形的模空间。 一个主要部分加拉休斯提出的活动将研究扩展这一理论的流形更高的层面。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、辛几何和可能的理论物理中有预期的应用。加拉修斯的研究将涉及流形及其对称性的研究。 流形是空间的一种广义形式，在数学和科学中随处可见。 流形的定义性质是它由有限个真实的参数局部参数化（例如，地球表面由两个参数局部参数化，经度和纬度）。 一个经典的主题代数拓扑，加拉修斯的研究领域，是分类的流形：如何决定是否两个流形是同构的，以及如何写一个列表的所有可能的流形。 一个相关的经典问题是流形对称性的分类（例如，绕轴旋转是球面的对称性）。 加拉太的研究将采用新的方法来研究这些经典问题。

The proposed research 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. Their result concerns the group of symmetries of closed two-dimensional manifolds or, equivalently, it concerns the moduli space of such manifolds. A major part of Galatius' proposed activity will study the extension of this theory to manifolds of higher dimension. 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.Galatius' research will concern the study of manifolds and their symmetries. A manifold is a generalized versions 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 real parameters (for example, the surface of the earth is locally parametrized by two parameters, longitude and latitude). A classic 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. A related classical problem is the classification of symmetries of manifolds (for example, rotation about an axis is a symmetry of the sphere). Galatius' research will apply new methods to study these classic questions.