New Approaches to Global Homotopy Theory

9704761 Arone研究者的一般专业领域是代数拓扑学。他最近的工作一直致力于更好地理解同伦理论的整体结构。这项工作结合了两种迄今为止互不相干的方法，即“色方法”和“函子演算”方法。在与M. Mahowald的合作中，研究者检验了不稳定同伦理论的Goodwillie导数，并用它们证明了一个关于球不稳定同伦的新结果。在其他工作中，研究者研究了不稳定同伦理论的泰勒多项式（与导数相反），并提供了这些多项式的明确的空间级描述。在他与Mahowald的联合研究中，研究者的一个关键贡献是对有限集合的分区偏序集的组合分析，这对球体的色光过滤产生了影响。对所涉及的组合学的进一步调查使研究者发现了看似新的和神秘的递归模式。他认为，利用这种组合周期性可以研究同伦理论中的一些深周期现象。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”，这暗示了有限复的存在，其上同调实现了Steenrod代数的某些有限子代数。研究者认为，这些有限配合物将有助于定位和周期性。在代数拓扑的研究领域，他对如何分析两个拓扑空间之间的连续函数空间感兴趣。事实证明，可以发展出一种类似于泰勒级数的理论，并且可以通过其“多项式近似”（T. Goodwillie提出的想法）来研究连续映射的空间。研究者将这一理论与拓扑学中更经典的方法结合起来，并用它（与M. Mahowald合作）证明了一个重要的代数工具的全局结构的新结果，即所谓的球体不稳定同伦。他现在打算用前面提到的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了解。他的工作具有强烈的组合风格（具体来说，他工作中的主要工具之一是有限集合的分区格），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. ***