Algebraic invariants of structured ring spectra, arithmetic, and geometry

结构化环谱、算术和几何的代数不变量

• 批准号: 0906105
• 负责人: Andrew Blumberg
• 金额: $ 14.66万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0906105&HistoricalAwards=false
• 关键词: Algebraic; invariants; structured; ring; spectra

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). This proposal describes several major research thrusts related to algebraic invariants of structured ring spectra. In algebraic K-theory, the PI describes an approach to resolving the central conjectures of Waldhausen and Rognes that underpin a program to describe the K-theory of the sphere spectrum via a chromatic filtration, relating arithmetic and manifold geometry. In topological Hochschild homology, the proposal describes an approach to realize a vision of Hesselholt to build an ``additive'' motivic spectral sequence in which the deRham-Witt complex and TR-theory play the respective roles of Milnor K-theory and algebraic K-theory. In string topology, the proposal describes an project which utilizes some of the technology developed to study algebraic K-theory to describe algebraic invariants of ``brane'' categories; this is related to the burgeoning connections between string topology and symplectic topology. In orientation theory, the proposal describes a series of projects associated to the relationship between units of ring spectra, orientations, and transfers, particularly in the equivariant setting.These latter projects have potential applications to mathematical physics.Algebraic topology began as the study of algebraic invariants of geometric objects which are preserved under certain smooth deformations. Gradually, it was realized that these algebraic invariants (called cohomology theories) could themselves be represented by geometric objects, known as spectra. A central triumph of modern homotopy theory has been the construction of categories of ring spectra (representing objects for multiplicative cohomologytheories) which are suitable for performing constructions directly analogous to those of classical algebra. This move has turned out to be incredibly fruitful, both by providing invariants which shed new light on old questions as well as by raising new questions which have unexpected connections to other areas of mathematics and physics. The project funded by this grant carries out this program in the setting of a rich invariant called algebraic K-theory, studying foundational properties of this theory from the perspective of homotopy theory and applying the results to a broad range of questions in manifold geometry, algebraic geometry, and string topology.

该奖项根据 2009 年美国复苏和再投资法案（公法 111-5）提供资金。该提案描述了与结构化环谱的代数不变量相关的几个主要研究方向。 在代数 K 理论中，PI 描述了一种解决 Waldhausen 和 Rognes 中心猜想的方法，该猜想支持通过色过滤、关联算术和流形几何来描述球谱 K 理论的程序。 在拓扑 Hochschild 同调中，该提案描述了一种实现 Hesselholt 愿景的方法，即构建“加性”动机谱序列，其中 deRham-Witt 复形和 TR 理论分别扮演 Milnor K 理论和代数 K 的角色-理论。 在弦拓扑中，该提案描述了一个项目，该项目利用一些为研究代数 K 理论而开发的技术来描述“膜”类别的代数不变量；这与弦拓扑和辛拓扑之间的新兴联系有关。 在取向理论中，该提案描述了一系列与环谱单位、取向和传递之间的关系相关的项目，特别是在等变设置中。这些项目在数学物理中具有潜在的应用。代数拓扑开始于代数的研究在某些平滑变形下保留的几何对象的不变量。 逐渐地，人们意识到这些代数不变量（称为上同调理论）本身可以用几何对象（称为谱）来表示。 现代同伦理论的一个主要胜利是环谱范畴的构造（代表乘法上同调理论的对象），它适合于执行与经典代数直接类似的构造。 事实证明，这一举措取得了令人难以置信的成果，既提供了为旧问题提供新思路的不变量，又提出了与数学和物理其他领域有着意想不到的联系的新问题。 该项目资助的项目在称为代数 K 理论的丰富不变量的背景下开展该项目，从同伦理论的角度研究该理论的基本性质，并将结果应用于流形几何、代数等领域的广泛问题。几何和弦拓扑。