Algebraic invariants of structured ring spectra, arithmetic, and geometry

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。这一建议描述了与结构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的中心猜想的方法，该猜想是通过色滤、相关算术和流形几何来描述球谱的K理论的程序的基础。在拓扑Hochschild同调中，该建议描述了一种实现Hesselholt的愿景的方法，该愿景是建立一个“可加的”基元谱序列，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，该建议描述了一个项目，该项目利用了一些发展起来的研究代数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迅速发展的联系有关。在取向理论中，该建议描述了一系列与环谱单位、取向和转移之间的关系有关的项目，特别是在等变环境下。这些项目在数学物理中具有潜在的应用。代数拓扑学起源于研究几何对象在一定光滑变形下保持的代数不变量。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，即所谓的谱。现代同伦理论的一个中心胜利是构造了环谱范畴(表示乘法上同调理论的对象)，这些范畴适合于进行直接类似于经典代数的构造。事实证明，这一举措取得了令人难以置信的成果，既提供了揭示旧问题的不变量，也提出了与数学和物理的其他领域有着意想不到的联系的新问题。由这笔拨款资助的项目在一个称为代数K-理论的丰富不变量的背景下执行这一计划，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、代数几何和弦拓扑中的广泛问题。

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.