Proposal for a Conference on the Geometry and Topology of Manifolds

奖项：dms -0406314首席研究员：Crichton L. ogle该奖项为俄亥俄州立大学“流形的几何和拓扑”会议的参与者提供部分支持。会议的主要主题是在平方可积上同调、k理论和几何群论中出现的流形不变量。受邀的演讲者和其他参与者包括一些女数学家、刚获得博士学位的学生和研究生。流形是一个几何空间，其中每个点都有可以与某维欧几里得空间识别的邻域。例如，一个二维球体表面上的每个点都位于一个开放的半球上，这个半球可以通过地图投影被平展到欧几里得平面上。高维流形的一个例子是三个粒子在平面上运动而不发生碰撞的物理坐标空间：每个粒子的动力学数据由两个位置和两个速度坐标组成，因此三个粒子中的每一个都需要四个数字来记录它的动力学，同时报告所有三个运动的粒子需要十二个坐标。流形的“不变量”是可计算的对象（不一定是数字），它帮助我们区分一个空间和另一个空间，一些已知的不变量可以通过微积分方法（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其相关的代数对象的非常长的范围行为（几何群论）来构建。

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).