Recherche en théorie de l'homotopie

本提案的主要主题是同伦理论的研究。我的计划将探索两个主要方向：(I)拓扑复杂性和l - s -范畴；（II）非单连通空间。***(I)拓扑复杂性-一个人应该记住的图像是一个机器人手臂的位置固定在装配厂X的图像。问题是：是否有可能找到一个连续的路径，手臂将遵循任何两个给定的点之间的手臂可以到达，即，对角线映射d:X -> X X X X承认一个适当的连续截面？不幸的是，地图上唯一允许有全球部分的装配厂X是没有障碍物的（没有柱子支撑屋顶，甚至没有机器人本身！）因此，我们不得不接受下一个最好的选择：连续局部截面，即空间X的拓扑复杂度TC(X)是最小的自然数k，使得存在由k+1个开集U1，…，Uk+1构成的X X X的覆盖，允许与对角映射d相关的经典评价颤振XI->X X X X的连续局部截面si:Ui->XI。实际上TC(X)是Lusternick-Schnirelmann范畴论框架内颤振截面范畴的一个特例。这些l.s.型不变量的研究在过去的40年里吸引了大量的资源，并且直到今天仍然是一个非常有活力的学科，在同伦理论、微分几何、动力系统和工程中都有应用。这是理想的HQP在纯数学和/或应用数学。我未来五年的目标是继续研究这些不变量，并在我的研究中纳入至少一名博士生和两名硕士生。***(II)非单连通空间——Quillen和Sullivan构造理论对单连通空间的有理同伦型进行代数建模。不幸的是，两者都不能处理像RP2这样的空间。为了纠正这种情况，Bousfield和Kan提出了以下方法：设X是一个允许有普世性覆盖UX的空间，并考虑其相关的纤维UX->X->B(pi1(X))，应用纤维合理化得到一个新的纤维UXo->X'->B(pi1(X))，其中UXo是单连通空间UX的通常合理化。该纤维的总空间X‘是原始空间X合理化的模型。下一步是按照Quillen和Sullivan的精神找到一个代数范畴来表征X’。Gomez-Tato， Halperin和tanr<s:1>构建了一个局部系统的代数范畴，它承认极小模型和实现函子，这与Sullivan非常相似。他们的理论的主要缺点是它只适用于具有有限型普遍覆盖的空间。我的目标是通过使用Quillen的愿景来建模UXo并建立那些本地系统，将这些想法扩展到所有的通用覆盖范围。该方向最好有一名博士后和至少一名博士生。

The major theme of this proposal is research in homotopy theory. My program will explore two main directions: (I) topological complexity and L.-S.-category; and (II) non-simply connected spaces. ***(I) Topological complexity – The image that one should keep in mind is the one of a robotic arm positioned somewhere fixed in an assembly plant X. The problem is: is it possible to find a continuous path that the arm will follow between any two given points that the arm can reach, i.e., does the diagonal map d:X -> X x X admit an appropriate continuous section? Unfortunately the only assembly plant X for which the map d admits a global section is one with no obstruction (no pillard to hold up the roof, not even the robot itself!). Hence we have to settle for the next best thing: continuous local sections, i.e., the topological complexity, TC(X), of a space X is the least natural number k such that there exists a covering of X x X formed by k+1 open sets U1,…,Uk+1, admitting continuous local sections si:Ui->XI of the classical evaluation fibration XI->X x X associated to the diagonal map d. In fact TC(X) is a special case of the sectional category of a fibration within the Lusternick-Schnirelmann category theory framework. The study of these L.S.-type inavariants have attracted massive amount of resources over the last 40 years and continues to this date to be a very dynamic subject with applications to homotopy theory, differential geometry, dynamical systems and engineering. It is ideal for HQP's in both pure and/or applied mathematics. My goal for the next five years is to pursue the study of these invariants and to incorporate to my research at least one Ph.D. student and two Master's students. ***(II) Non-simply connected spaces - Quillen and Sullivan constructed theories to algebraically model the rational homotopy type of simply-connected spaces. Unfortunately, both can't handle spaces like RP2. To remedy this situation, Bousfield and Kan proposed the following: Let X be a space admitting a universal cover UX and consider the associated fibration UX->X->B(pi1(X)) and apply fibrewise rationalization to obtain a new fibration UXo->X'->B(pi1(X)), where UXo is the usual rationalization of the simply-connected space UX. The total space X' of that fibration is the model for the rationalization of the original space X. The next step is to find an algebraic category to characterize X' in the spirit of Quillen and Sullivan. Gomez-Tato, Halperin and Tanré constructed an algebraic category of local systems that admitted minimal models and realization functors very much like Sullivan. The main drawback of their theory is that it only applies to spaces having a finite type universal cover. My goal is to expand these ideas to all universal covers by using the Quillen vision to model UXo and build those local systems. This direction is ideal to incorporate a post-doctoral fellow and at least one Ph.D. student.