Recherche en théorie de l'homotopie

该提案的主题是同伦理论的研究。我的计划将探索两个主要方向：（一）拓扑复杂性和L.-S.类别； (II) 非简连通空间。 ***(I) 拓扑复杂性 – 应该记住的图像是固定在装配厂 X 中某处的机械臂。问题是：是否有可能在机械臂可以到达的任意两个给定点之间找到机械臂将遵循的连续路径，即对角线图 d:X -> X x X 是否承认适当的连续部分？不幸的是，地图 d 承认其全局截面的唯一装配厂 X 是一个没有障碍物的装配厂（没有柱子支撑屋顶，甚至连机器人本身也没有！）。因此，我们必须满足于下一个最好的事情：连续局部截面，即空间 X 的拓扑复杂度 TC(X) 是最小自然数 k，使得存在由 k 1 个开集 U1,…,Uk 1 形成的 X x X 的覆盖，承认与对角图 d 相关的经典评估纤维 XI->X x X 的连续局部截面 si:Ui->XI。事实上，TC(X) 是 Lusternick-Schnirelmann 范畴论框架内纤维化截面范畴的一个特例。这些 L.S. 型不变量的研究在过去 40 年中吸引了大量资源，并且至今仍然是一个非常活跃的学科，应用于同伦理论、微分几何、动力系统和工程。它是纯数学和/或应用数学领域 HQP 的理想选择。我未来五年的目标是继续研究这些不变量，并将至少一名博士学位纳入我的研究中。学生和两名硕士生。 ***(II) 非单连通空间 - Quillen 和 Sullivan 构建了理论来对单连通空间的有理同伦类型进行代数建模。不幸的是，两者都不能处理像 RP2 这样的空格。为了弥补这种情况，Bousfield 和 Kan 提出了以下建议：令 X 为承认通用覆盖 UX 的空间，并考虑相关的纤维化 UX->X->B(pi1(X)) 并应用纤维合理化以获得新的纤维化 UXo->X'->B(pi1(X))，其中 UXo 是简单连通空间 UX 的通常有理化。该纤维化的总空间 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 Tanré 构建了局部系统的代数范畴，该范畴与沙利文非常相似，允许最小模型和实现函子。他们的理论的主要缺点是它仅适用于具有有限类型通用覆盖的空间。我的目标是通过使用 Quillen 愿景来建模 UXo 并构建这些本地系统，将这些想法扩展到所有通用领域。这个方向非常适合吸收一名博士后研究员和至少一名博士。学生。

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.