Mathematical Sciences: Homotopy Theory and Its Applications

9422413拉韦内尔·内森多费尔的项目是计算某些空间的同调，这些空间是逆极限，例如，无限多个半径球体的并收敛到零并且有一个共同点。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。使用Pontrjagin和Thom的几何方法似乎可以得到更深层次的结果。拉弗内尔正在研究与望远镜猜想相关的稳定同伦理论中的问题。这似乎是理解有限复形的稳定同伦群的最好方法，这是代数拓扑学中最古老的问题之一。在更一般的术语中，代数拓扑学是从定性而不是定量的角度研究几何问题(通常在3维以上)，对结构进行分类和计数，而不是测量它们的大小。例如，2维平面和3维空间之间的一个本质区别是，删除一条线会将前者分割成两个单独的部分，但不会将后者分割成两个单独的部分。拓扑学家已经设计了称为同伦和同调的方法，用于在非常复杂的几何对象(例如，作为常微分方程组的解的空间而可能出现的对象)中系统地跟踪这些性质。内森多费尔和拉弗内尔的研究致力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. ***