New Approaches to Global Homotopy Theory

小行星9704761 调查员的一般领域的专业化是代数拓扑。 他最近的工作是为了实现更好的 理解同伦理论的整体结构。 这项工作结合了两个迄今为止不相交的方法，“色的方法”和“演算函子”的方法。 在与M。Mahowald研究了不稳定同伦理论的Goodwillie导数，并利用它们证明了关于球面不稳定同伦的一个新结果。 在其他工作中，调查员 研究泰勒多项式（与导数相反） 的不稳定同伦理论，并提供了一个明确的空间层次的描述。 调查人员的一个关键因素是 他与Mahowald的联合工作的贡献是一个组合 分析的偏序集的分区的有限集，这把 对球体的色过滤产生影响。 进一步 对所涉及的组合学的研究使研究人员发现了看似新的和神秘的递归模式。 他 认为同伦中的某些深层周期现象 理论可以通过这种组合周期性来研究。 在最近的工作中，研究人员利用这种组合 周期性地在导数上构造某些“自映射” 不稳定同伦理论，这意味着，除其他外， 存在有限复形，其上同调实现某些 Steenrod代数的有限子代数 研究者认为 这些有限的复合物将用于定位， 周期性 在研究领域的代数拓扑，他感兴趣的是如何分析空间的连续功能之间的两个拓扑空间。 事实证明，泰勒级数理论的一个类似物可以发展，人们可以通过其“多项式逼近”来研究连续映射的空间（这是T。Goodwillie）。 该investigato r结合这一理论与更经典的方法在拓扑结构，他用它（在联合工作与M。Mahowald）证明了一个关于全局结构的新结果，这是一个重要的代数工具，即所谓的球面不稳定同伦。 他现在打算使用上述多项式近似来构造具有有趣（上同调）性质的某些空间，这些空间的存在本身就应该导致对拓扑学的进一步认识。 他的工作有很强的组合风格（具体来说，他工作中的一个主要小工具是有限集的分区格），他希望他的一些结果会引起组合学家的兴趣。 ***

9704761 Arone The investigator's general area of specialization is algebraic topology. His recent work has been concerned with achieving a better understanding of the global structure of homotopy theory. This work combines two hitherto disjoint approaches, the "chromatic approach," and the "calculus of functors" approach. In joint work with M. Mahowald, the investigator examined the Goodwillie derivatives of unstable homotopy theory and used them to prove a new result about unstable homotopy of spheres. In other work, the investigator studied the Taylor polynomials (as opposed to derivatives) of unstable homotopy theory and provided an explicit space-level description of these. One of the key ingredients of the investigator's contribution to his joint work with Mahowald was a combinatorial analysis of the poset of partitions of a finite set, which turned out to have consequences for the chromatic filtration of spheres. Further investigation of the combinatorics involved has led the investigator to discover seemingly new and mysterious recurrence patterns. He believes that some of the deep periodic phenomena in homotopy theory can be studied via this combinatorial periodicity. In very recent work, the investigator utilized this combinatorial periodicity to construct certain "self maps" on the derivatives of unstable homotopy theory, which imply, among other things, the existence of finite complexes whose cohomology realizes certain finite subalgebras of the Steenrod algebra. The investigator believes that these finite complexes will be useful for localization and periodicity. In the investigator's area of Algebraic Topology, he is interested in how to analyze the space of continuous functions between two topological spaces. It turns out that an analogue of the theory of Taylor series can be developed, and one can study the space of continuous maps via its "polynomial approximations" (an idea due to T. Goodwillie). The investigato r combined this theory with more classical methods in topology, and he used it (in joint work with M. Mahowald) to prove a new result on the global structure of an important algebraic tool of the trade, the so-called unstable homotopy of spheres. He now intends to use the aformentioned polynomial approximations to construct certain spaces with interesting (cohomological) properties, spaces whose mere existence should lead to further insights in topology. His work has a strong combinatorial flavor (specifically, one of the main gadgets in his work is the lattice of partitions of a finite set), and he hopes that some of his results will be of interest to combinatorialists. ***