New Approaches to Global Homotopy Theory

9704761阿隆研究员的一般专门化领域是代数拓扑学。他最近的工作是更好地理解同伦理论的全局结构。这项工作结合了两种迄今不相交的方法，“色法”和“函子演算”方法。在与M.Mahowald的合作中，研究人员研究了不稳定同伦理论的Goodwillie导数，并用它们证明了关于球面不稳定同伦的一个新结果。在另一项工作中，研究人员研究了不稳定同伦理论的泰勒多项式(相对于导数)，并给出了这些多项式的显式空间级描述。这位研究人员在他与Mahowald的联合工作中做出的贡献的关键成分之一是对有限集的划分的偏序集的组合分析，这被证明是对球体的色过滤的结果。对所涉及的组合学的进一步研究使研究人员发现了看似新的神秘的复发模式。他认为，同伦理论中的一些深周期现象可以通过这种组合周期来研究。在最近的工作中，研究者利用这种组合周期性在不稳定同伦理论的导数上构造了某些“自映射”，这意味着有限复形的存在，其中，有限复形的上同调实现了Steenrod代数的某些有限子代数。研究人员认为，这些有限的复合体将有助于局部化和周期性。在代数拓扑学的研究领域，他感兴趣的是如何分析两个拓扑空间之间的连续函数空间。事实证明，泰勒级数理论可以发展成一个类似的理论，人们可以通过它的“多项式逼近”(T·古德威利的一个想法)来研究连续映射的空间。这位研究者将这一理论与拓扑学中更经典的方法相结合，并利用它(与M.Mahowald共同工作)证明了该行业重要的代数工具--球面的不稳定同伦的全局结构的一个新结果。他现在打算使用前述多项式近似来构造某些具有有趣(上同调)性质的空间，这些空间的存在应该会导致对拓扑学的进一步洞察。他的工作具有强烈的组合色彩(具体地说，他工作中的主要小工具之一是有限集的划分格)，他希望他的一些结果能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. ***