Computations in Equivariant Homotopy and Algebraic K-Theory

获奖：DMS 1207774，首席研究员：Michael A. hill这个项目旨在通过某些映射的普世性来扩大我们对球同伦群的等变计算的认识。等变同伦的计算是出了名的困难，以一种美丽的，有时甚至是神秘的方式混合了表示理论和经典稳定同伦理论。最近，主要研究者Hopkins和Ravenel对Kervaire不变量1问题的解决方案引入了一些新的工具和技术来研究等变同伦，巩固了早期的同伦工作，并推广了自然发生的过滤。在这个项目中，PI打算使用这些令人兴奋的新工具继续在等变同伦方面取得计算上的进展。有两种主要的方法：计算PI、Hopkins和Ravenel引入的谱族的等变同伦群，以及重新概念化代数k理论的部分环切迹方法。前者使用切片过滤技术来处理像霍普金斯-米勒光谱这样的光谱，允许从非传统的方法直接攻击球体的稳定同伦群。后者建立在规范机制的基础上，重写代数k理论等变方法中的经典结构，以计算上更易于接受的形式重新铸造它们。该项目直接解决了代数拓扑的核心：计算数字和不变量来理解空间。代数拓扑的目标是系统地建立代数对象（如数）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，所产生的数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原本相互关联的问题。例如，使用等变方法，首席研究员Hopkins和Ravenel解决了Kervaire Invariant One问题，这是代数拓扑中最古老的突出问题，其根源可以追溯到20世纪30年代。这反过来又给了我们如何用球体这样简单的空间来构建空间的信息。该项目旨在以解决方案中开发的技术为基础，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.