Proposal for a Conference on the Geometry and Topology of Manifolds

摘要奖：DMS-0406314首席研究员：克莱顿·L·奥格尔该奖项为俄亥俄州立大学举行的一次关于流形的几何和拓扑的会议的参与者提供部分费用支持。会议的主要主题是平方可积上同调流形的不变量、K理论和几何群论。受邀的演讲者和其他参与者包括一些女数学家、新近毕业的博士和研究生。流形是一个几何空间，其中每个点都有邻域，可以与某个维度的欧几里德空间相一致。例如，二维球体表面上的每个点都位于一个开放的半球内，可以通过地图投影将其展平到欧几里得平面上。高维流形的一个例子是三个粒子在平面上无碰撞运动的物理坐标空间：每个粒子的动力学数据由两个位置和两个速度坐标组成，因此这三个粒子中的每个需要四个数字来记录其动力学，并同时报告所有三个运动粒子需要12个坐标。流形的“不变量”是帮助我们区分空间的可计算对象(不一定是数字)，一些已知的不变量可以通过微积分的方法(如平方可积上同调)、代数工具(K理论)以及通过观察流形或与之相关的代数对象的非常长的范围行为(几何群论)来构造。

AbstractAward: DMS-0406314Principal Investigator: Crichton L. OgleThis award provides partial support for participant costs of aconference on the "Geometry and Topology of Manifolds" at OhioState University. The main themes of the meeting are invariantsof manifolds arising in square-integrable cohomology, K-theory,and geometric group theory. Invited speakers and otherparticipants include a number of female mathematicians, recentPhDs, and graduate students.A manifold is a geometric space in which every point hasneighborhoods that can be identified with Euclidean space of somedimension. For example, every point on the surface of a2-dimensional sphere lies in an open hemisphere that can beflattened onto the Euclidean plane by a map projection. Anexample of a higher dimensional manifold would be the space ofphysical coordinates for three particles moving in the planewithout collisions: each particle's dynamical data consists oftwo position and two velocity coordinates, thus each of the threeparticles requires four numbers to record its dynamics and toreport all three moving particles simultaneously calls for twelvecoordinates. "Invariants" of manifolds are computable objects(not necessarily numbers) that help us tell one space fromanother, and some of the known invariants can be constructed bymethods of calculus (such as square-integrable cohomology),algebraic tools (K-theory), and by looking at very long-rangebehavior of the manifold or of algebraic objects associated to it(geometric group theory).