Computations in Equivariant Homotopy and Algebraic K-Theory

摘要奖：DMS 1207774，主要研究者：Michael A.希尔这个项目旨在扩大我们的知识等变计算，并通过普遍性的某些地图，同伦群体的领域。等变同伦计算是出了名的困难，它以一种美丽的，有时甚至是神秘的方式混合了表示论和经典稳定同伦理论。最近的解决方案由首席研究员，霍普金斯，和拉文埃尔的凯维尔不变一个问题介绍了几个新的工具和技术，等变同伦，僵化早期同伦工作和推广自然发生的过滤。在这个项目中，PI打算使用这些新的，令人兴奋的工具来继续在等变同伦中进行计算。主要有两种方法：计算由PI、霍普金斯和Ravenel引入的谱族的等变同伦群，并重新概念化代数K理论的分圆迹方法的一部分。前者使用切片过滤技术来处理谱，如Hopkins-Miller谱，允许从非传统方法直接攻击球面的稳定同伦群。后者建立在规范机制的基础上，以代数K理论的等变方法重写经典结构，将它们重新塑造成计算上更适合的形式。该项目直接解决了代数拓扑的核心问题：计算数字和不变量以理解空间。代数拓扑的目标是系统地建立代数对象（如数字）和几何对象（如空间）之间的联系。等变代数拓扑将空间中固有的对称性的集合作为数据的一部分，系统地将具有相同对称性的空间分组，并且产生的数字和不变量必须反映这一点。记住额外的结构会使计算更丰富，但也更复杂，它使我们能够梳理出其他相互关联的问题。例如，使用等变方法，主要研究者霍普金斯和拉文埃尔解决了Kervaire Invariant One问题，这是代数拓扑学中最古老的突出问题，其根源可以追溯到20世纪30年代。这反过来又提供了关于我们如何从更简单的空间（如球体）中构建空间的信息。该项目旨在建立在解决方案中开发的技术基础上，解决代数和拓扑中的其他计算问题。

AbstractAward: DMS 1207774, Principal Investigator: Michael A. HillThis project seeks to broaden our knowledge of equivariant computations and, by universality of certain maps, of the homotopy groups of spheres. Equivariant homotopy calculations are notoriously difficult, hybridizing representation theory and classical stable homotopy theory in beautiful, and sometimes mysterious, ways. The recent solution by the principal investigator, Hopkins, and Ravenel to the Kervaire Invariant One problem introduced several new tools and techniques to equivariant homotopy, rigidifying earlier homotopical work and generalizing naturally occurring filtrations. In this project, the PI intends to use these new, exciting tools to continue making computational inroads in equivariant homotopy. There are two main approaches: computing equivariant homotopy groups of the families of spectra introduced by the PI, Hopkins, and Ravenel, and reconceptualizing parts of the cyclotomic trace approaches to algebraic K-theory. The former uses slice filtration techniques to tackle spectra like the Hopkins-Miller spectra, allowing a direct attack on the stable homotopy groups of spheres from a non-traditional approach. The latter builds on the norm machinery to rewrite the classical constructions in the equivariant approaches to algebraic K-theory, recasting them in computationally more amenable forms.The project addresses directly the heart of algebraic topology: computing numbers and invariants to understand spaces. The goal of algebraic topology is to systematically build a connection between algebraic objects like numbers and geometric objects like spaces. Equivariant algebraic topology remembers a collection of symmetries inherent in a space as part of the data, systematically grouping spaces with the same symmetries, and the numbers and invariants produced must reflect this. Remembering the extra structure makes richer, but more complicated, computations, and it allows us to tease apart otherwise interconnected problems. For example, using equivariant methods, the principal investigator, Hopkins, and Ravenel solved the Kervaire Invariant One problem, the oldest outstanding problem in algebraic topology with roots dating back to the 1930s. This in turn gave information about how we can build spaces out of simpler ones like spheres. This project aims to build on the techniques developed in the solution, tackling other computational problems in algebra and topology.