SGER: Midwest Topology Network

中西部地区有一个庞大的代数拓扑学活跃研究人员网络，分布在数十所大学和学院中。定期会议不断激发新的想法和潜在的新的研究方向。这项拨款的目的是为中西部地区不同大学的拓扑学家之间建立新的合作和新的研究项目提供手段，否则这些项目将无法进行。这项资助将支持三种主要类型的活动：研究合作旅行、研究新手旅行以及与其他网络的交流。代数拓扑学试图将空间和几何问题简化为代数问题。通常，这涉及到将某种代数对象（如向量空间或数值不变量）分配给几何对象或空间，以捕获或测量某些内在特征。这些不变量可以用来区分不同的空间。所研究的不变量通常是离散的，因此它们在连续变形或“同伦”下不会改变。这种自由通常允许拓扑学家将复杂的问题变形为具有相同不变量的更简单的问题，这样更容易计算。由于较简单的问题通常保留了原始数学对象的重要信息，因此代数拓扑为广泛的数学问题提供了有效的工具。

The midwest contains a large network of active researchers in algebraic topology spread out through dozens of universities and colleges. Regular conferences continually spark new ideas and potential new research directions. The purpose of this grant is to provide the means to foster the new collaborations and new research projects among midwest topologists at different universities that would otherwise go unpursued. This grant would support three key types of activity: Travel for research collaboration, travel for beginning researchers, and exchange with other networks.Algebraic topology attempts to reduce questions about space and geometry to questions in algebra. Often this involves assigning some kind of algebraic object like a vector space or numerical invariant to a geometric object or space to capture or measure some intrinsic feature. These invariants can then potentially be used to distinguish between different spaces. The invariants studied are typically discrete, and so they do not change under continuous deformation or "homotopy". This freedom often allows topologists to deform complicated problems into simpler ones that have the same invariants, which are then easier to compute. Since the simpler problems often retain important information about the original mathematical objects, algebraic topology provides effective tools for a wide range of mathematical problems.