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.

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。该建议描述了几个主要的研究推力相关的代数不变量的结构环光谱。 在代数K理论中，PI描述了一种解决瓦尔德豪森和罗涅涅中心结构的方法，该方法支持一个程序，通过色过滤来描述球谱的K理论，将算术和流形几何联系起来。 在拓扑Hochschild同调中，该建议描述了一种实现Hesselholt的愿景的方法，以建立一个“加法”motivic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理论的角色。 在弦拓扑中，该提案描述了一个项目，该项目利用了一些研究代数K理论的技术来描述“膜”范畴的代数不变量;这与弦拓扑和辛拓扑之间的新兴联系有关。 在定向理论中，该理论描述了一系列与环谱、定向和转移（特别是在等变条件下）的单位之间的关系相关的项目。这些项目在数学物理中有潜在的应用。代数拓扑学开始于研究几何对象在某些光滑变形下保持不变的代数不变量。 渐渐地，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来表示，称为谱。 现代同伦理论的一个中心胜利是构造了环谱范畴（代表乘法上同调理论的对象），这些范畴适合于直接类似于经典代数的构造。 这一举措已被证明是令人难以置信的富有成效的，无论是通过提供不变量，揭示了新的光在旧的问题，以及通过提出新的问题，有意想不到的连接到其他领域的数学和物理。 该项目由该基金资助，在一个丰富的不变量称为代数K理论的设置中执行该计划，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，代数几何和弦拓扑学中的广泛问题。