Manifolds, Moduli Spaces and Homotopy Theory

提出的研究将继续发展一个相对新的数学领域，其中同伦理论被用来研究各种模空间。这一领域一个重要的早期成果是Madsen和Weiss对Mumford猜想的解答。他们的结果涉及到封闭二维流形的对称群，或者等价地，它涉及到这种流形的模空间。加拉提乌斯提议的活动的主要部分是研究将这一理论扩展到更高维度的流形。这门学科存在于代数拓扑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。加拉提乌斯的研究将涉及流形及其对称性的研究。流形是空间的广义形式，在数学和科学中随处可见。流形的定义性质是由有限数量的实参数局部参数化（例如，地球表面由经度和纬度两个参数局部参数化）。代数拓扑学的一个经典课题是流形的分类，也是加拉提乌斯的研究领域：如何决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。一个相关的经典问题是流形对称的分类（例如，绕轴旋转是球体的对称）。加拉提乌斯的研究将应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.