Mathematical Sciences: Homotopy Theory and Its Applications

小行星9422413 Neisendorfer的项目是计算某些空间的同调，这些空间是逆极限，例如，无限多个半径为零的球体的联合，并且有一个共同点。 利用连通自由李代数的逆极限是自由的这一新的代数结果，可以应用有理同伦理论的方法。 更深层次的结果似乎可以获得使用几何方法的Pontrjagin和托姆。 拉文埃尔正在研究与望远镜猜想有关的稳定同伦理论问题。 这似乎是理解有限复形的稳定同伦群的最佳方法，这是代数拓扑学中最古老的问题之一。 代数拓扑学（英语：Algebraic topology）是对几何问题（通常在三维以上）的定性而非定量研究，对结构进行分类和计数，而不是测量它们的大小。 例如，二维平面和三维空间之间的一个本质区别是，删除一条线将前者分开，而不是将后者分成两个独立的部分。 拓扑学家已经设计出一种叫做同伦和同调的方法，用来系统地跟踪非常复杂的几何对象（例如，可以作为常微分方程组的解的空间出现的对象）中的这些性质。 Neisendorfer和Ravenel的研究关注于改进和扩展这些方法。 ***

9422413 Ravenel Neisendorfer's project is to compute the homology of certain spaces that are inverse limits, for example, a union of infinitely many spheres of radii converging to zero and with one point in common. Using the new algebraic result that an inverse limit of connected free Lie algebras is free, methods of rational homotopy theory can be applied. Deeper results seem to be obtainable using the geometric methods of Pontrjagin and Thom. Ravenel is studying problems in stable homotopy theory related to the telescope conjecture. This appears to be the best approach to understanding the stable homotopy groups of finite complexes, which is one of the oldest problems in algebraic topology. In more general terms, algebraic topology is the study of geometric problems (often in more than 3 dimensions) in their qualitative rather than quantitative aspects, classifying and counting structures rather than measuring their sizes. For example an essential difference between a 2-dimensional plane and a 3-dimensional space is that removing a line separates the former but not the latter into two separate pieces. Topologists have devised methods called homotopy and homology for keeping track of such properties systematically in very complicated geometric objects (objects than can arise, for example, as the spaces of solutions to sytems of ordinary or differential equations). Neisendorfer and Ravenel's research is concerned with refining and extending these methods. ***