Collaborative Research: Algebraic K-Theory, Arithmetic, and Equivariant Stable Homotopy Theory

代数拓扑开始于研究在某些平滑变形下保留的几何对象的代数不变量。 逐渐地，人们意识到代数不变量（称为上同调理论）本身可以用称为谱的几何对象来表示。 现代同伦理论的一个主要胜利是构建了适合执行类似于经典代数的构造的环谱类别。 事实证明，通过提供不变量，这是卓有成效的，这为老问题提供了新的线索。此外，它还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。 该项目在称为代数 K 理论和相关理论的不变量的设置下进行。 该项目研究这些理论在数论、代数几何、几何拓扑以及代数拓扑本身的广泛问题中的应用。该奖项为将参与部分研究的学生提供支持。该赠款资助一项广泛的研究计划，旨在应用 PI 最近在代数 K 理论、迹方法和等变稳定同伦理论方面的工作来研究同伦理论中的各种问题。 PI 之前的工作研究了球谱的代数 K 理论和代数数环的分圆迹的光纤。 当前的项目将研究扩展到更一般的环上代数 p 整数上的分圆迹的纤维，根据泰特-普瓦图对偶性和更普遍的其他类型 Artin 对偶性的相关 K 理论问题。 该项目探索了几何 Soule 嵌入与 PI 先前工作中发现的 Kummer-Vandiver 猜想之间的联系。 PI 之前的工作还给出了一种分裂，该分裂与代数 K 理论上的 p 进插值 Adams 运算的存在相一致，并提供了证据，至少在正则环的背景下。 该项目研究了解决这个问题的具体猜想和方法。 该项目提出了一种新的 Hatcher-Waldhausen 映射方法，该方法将对几何、微分和辛拓扑产生影响。 该项目提出了在正维紧李群的等变稳定同伦理论中构建乘法范数映射。这需要非等变因式分解同调的新基础，并有望为正维紧李群构建真正的等变因式分解同调理论。 该项目包括 PI 与 Basterra、Hill 和 Lawson 的合作，研究等变 TAQ，作为更广泛计划的一部分，为等变派生代数几何奠定基础。 如果成功，该计划将为来自拓扑模块形式工作的现象学数据提供组织原则。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Algebraic topology began as the study of those algebraic invariants of geometric objects that are preserved under certain smooth deformations. Gradually, it was realized that the 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 that are suitable for performing constructions analogous to those of classical algebra. This has proved fruitful by providing invariants, which shed new light on old questions. In addition, it has raised new questions that have unexpected connections to other areas of mathematics and physics. This project works in the setting of an invariant called algebraic K-theory and related theories. The project studies applications of these theories to a broad range of questions in number theory, algebraic geometry, and geometric topology, as well as algebraic topology itself. The award provides support for students who will be engaged in parts of this research.This grant funds a broad research program aimed at applying recent work of the PIs on algebraic K-theory, trace methods, and equivariant stable homotopy theory to study a wide variety of problems in homotopy theory. Prior work of the PIs studied the algebraic K-theory of the sphere spectrum and the fiber of the cyclotomic trace for algebraic number rings. The current project expands the investigation to the fiber of the cyclotomic trace on more general rings over algebraic p-integers in terms of Tate-Poitou duality and a related K-theory question more generally for other kinds of Artin duality. The project explores a connection between the geometric Soule embedding and the Kummer-Vandiver conjecture discovered in the PIs' prior work. The PIs' prior work also gives a splitting that is consistent with and gives evidence for the existence of p-adically interpolated Adams operations on the algebraic K-theory at least in the context of regular rings. The project investigates specific conjectures and approaches to this problem. The project advances a new approach to the the Hatcher-Waldhausen map that would have implications in geometric, differential, and symplectic topology. The project proposes a construction of multiplicative norm maps in equivariant stable homotopy theory for positive dimensional compact Lie groups. This requires a new foundation for non-equivariant factorization homology and holds the promise of constructing a genuine equivariant factorization homology theory for positive dimensional compact Lie groups. The project includes a collaboration of the PIs with Basterra, Hill, and Lawson to study equivariant TAQ as part of a broader program to develop the foundations for equivariant derived algebraic geometry. If successful, this program will provide an organizing principle for phenomenological data coming from work on topological modular forms.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.