Collaborative Research: Algebraic K-Theory, Topological Periodic Cyclic Homology, and Noncommutative Algebraic Geometry

代数拓扑学起源于对几何对象在一定光滑变形下保持不变的代数不变量的研究。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环谱范畴（表示乘法上同伦理论的对象）的构造，它适合于直接类似于经典代数的构造。这一举动被证明是非常富有成效的，既提供了不变量，为老问题提供了新的思路，也提出了新的问题，这些问题与数学和物理的其他领域有着意想不到的联系。由该基金资助的项目在称为代数k理论的丰富不变量和相关理论称为拓扑Hochschild，循环和周期同调的背景下执行该计划。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。本研究继续了一项广泛的研究计划，旨在应用pi在代数k理论和迹方法方面的最新工作来研究数论、非交换代数几何和辛拓扑中的各种基本问题。它还包括一个项目，以发展等变衍生代数几何的基础，这已应用于组织在拓扑模形式的研究中观察到的计算现象。pi最近的工作导致了K(S)的同伦群的完整描述（根据其他已知的光谱），并通过state - poitou对偶的光谱提升对环切迹纤维的规范识别。pi有一个应用这项工作的计划，为Kummer-Vandiver猜想提供新的证据。如果成功，这将提供另一个从代数拓扑输入解决数论问题的例子。pi先前将他们的工作应用于环切迹纤维上，以解决p进Langlands程序中关于稳定同余子群（co）同调的猜想。pi描述了一系列将在p进朗兰兹程序研究中使用光纤同伦理论数据的项目。pi最近的其他工作建立了可对偶dg范畴的拓扑周期循环同调（TP）的Kunneth定理。这个结果已经在非交换代数几何中有了有趣的应用，这是将TP看作一种非交换的Weil上同调理论的结果。该基金包括一个建立这一观点并将TP应用于非交换代数几何的项目。基于与Abouzaid和Kragh的对话，pi已经开始通过包裹的Fukaya范畴探索代数k理论和TP在辛拓扑中的应用。pi描述了一系列项目，这些项目利用他们的专业知识和先前的成果来研究该领域的基本问题。PI Blumberg先前曾与Mike Hill合作开发等变交换环谱理论的基础。PI Mandell是研究拓扑Andre-Quillen同源性（TAQ）的权威学者之一。在与Basterra， Hill和Lawson的合作中，pi研究了等变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。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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 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 cohomology theories) 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 and related theories known as topological Hochschild, cyclic, and periodic homology. 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.This research continues a broad research program aimed at applying recent work of the PIs on algebraic K-theory and trace methods to study a wide variety of basic problems in number theory, noncommutative algebraic geometry, and symplectic topology. It also includes a project to develop the foundations of equivariant derived algebraic geometry, which has applications to organizing computational phenomena observed in the study of topological modular forms. The PIs' recent work has resulted in a complete description of the homotopy groups of K(S) (in terms of other known spectra) and a canonical identification of the fiber of the cyclotomic trace via a spectral lift of Tate-Poitou duality. The PIs have a program to apply this work to provide novel evidence for the Kummer-Vandiver conjecture. If successful, this would provide another example of input from algebraic topology addressing questions in number theory. The PIs previously applied their work on the fiber of the cyclotomic trace to resolve conjectures in the p-adic Langlands program about the (co)homology of stable congruence subgroups. The PIs describe a series of projects that would use homotopy theoretic data about the fiber in the study of the p-adic Langlands program. Other recent work of the PIs established a Kunneth theorem for topological periodic cyclic homology (TP) of dualizable dg categories. This result has already had interesting applications in noncommutative algebraic geometry, as a consequence of regarding TP as a kind of noncommutative Weil cohomology theory. The grant includes a project to establish this viewpoint and to apply TP in noncommutative algebraic geometry. Based on conversations with Abouzaid and Kragh, the PIs have started exploring applications of algebraic K-theory and TP to symplectic topology via the wrapped Fukaya category. The PIs describe a series of projects that leverage their expertise and prior results to study fundamental questions in this area. PI Blumberg has previously worked with Mike Hill to develop the foundations of the theory of equivariant commutative ring spectra. PI Mandell is one of the foremost experts on topological Andre-Quillen homology (TAQ). In collaboration with Basterra, Hill, and Lawson, the PIs 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.