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

代数拓扑学起源于研究几何对象在一定的光滑变形下保持不变的代数不变量。渐渐地，人们意识到被称为上同调理论的代数不变量本身可以由被称为谱的几何对象来表示。现代同伦理论的一个中心胜利是构造了适合于进行类似于经典代数的构造的环谱范畴。事实证明，这是卓有成效的，因为它提供了不变量，这为老问题提供了新的线索。此外，它还提出了一些新问题，这些问题与数学和物理的其他领域有着意想不到的联系。这个项目是在一个称为代数K-理论的不变量及其相关理论的背景下进行的。该项目研究这些理论在数论、代数几何和几何拓扑以及代数拓扑本身的广泛问题中的应用。该奖项为从事部分研究的学生提供支持。这项拨款资助了一个广泛的研究项目，旨在应用PI在代数K-理论、迹方法和等变稳定同伦理论方面的最新工作来研究同伦理论中的各种问题。PI以前的工作研究了代数数环的球谱的代数K-理论和割圆迹的纤维。本文在Tate-Poitou对偶的意义下，扩展了对更一般的代数p-整数环上的割圆迹的纤维的研究，并对其他类型的Artin对偶进行了更一般的K-理论问题的研究。该项目探索了几何灵魂嵌入和在PI之前的工作中发现的Kummer-Vandiver猜想之间的联系。PI的前人的工作也给出了一个分裂，它至少在正则环的背景下与代数K-理论上的p-adad插值Adams运算的存在是一致的，并给出了证据。该项目调查了对这一问题的具体猜测和方法。该项目提出了一种新的方法来处理Hatcher-Waldhausen映射，这将在几何、微分和辛拓扑中产生影响。在等变稳定同伦理论中，给出了正维紧李群的乘法范数映射的构造方法。这就需要为非等变因式分解同调建立一个新的基础，并有望为正维紧李群构造一个真正的等变因式分解同调理论。该项目包括PIS与Basterra、Hill和Lawson合作研究等变Taq，作为开发等变衍生代数几何基础的更广泛计划的一部分。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优势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.