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

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。由这笔赠款资助的项目在一个称为代数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)的背景下执行这一计划。该项目研究这些理论在数论、代数几何和几何拓扑中的广泛应用，以及代数拓扑本身。这项研究继续广泛的研究计划，旨在应用PI关于代数K-理论和迹方法的最新工作来研究数论、非对易代数几何和辛拓扑中的各种基本问题。它还包括一个发展等变衍生代数几何基础的项目，该基础应用于组织在拓扑模形式研究中观察到的计算现象。PI最近的工作得到了K(S)的同伦群的完整描述(根据其他已知的谱)，并通过Tate-Poitou对偶的谱提升对割圆迹的纤维进行了典范识别。PI有一个程序来应用这项工作，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学解决数论问题的输入例子。PI以前将他们的工作应用于割圆迹的纤维上，以解决p-adic朗兰兹程序中关于稳定同余子群的(上)同调的猜想。PI描述了一系列项目，这些项目将使用关于光纤的同伦理论数据来研究p-adic朗兰兹计划。PIS最近的其他工作建立了可对偶化dG范畴的拓扑周期循环同调(TP)的kunneth定理。这一结果在非对易代数几何中已经有了有趣的应用，这是将TP视为一种非对易Weil上同调理论的结果。这项拨款包括一个建立这一观点的项目，并将TP应用于非对易代数几何。基于与Abouzaid和Kragh的对话，PI已经开始探索代数K-理论和TP在通过包裹Fukaya范畴的辛拓扑中的应用。投资促进机构描述了一系列项目，这些项目利用他们的专业知识和先前的成果来研究这一领域的基本问题。皮布伦伯格之前曾与迈克·希尔合作，发展了等变对易环谱理论的基础。Pi Mandell是拓扑Andre-Quillen同调(Taq)的最重要的专家之一。在与Basterra、Hill和Lawson的合作中，PI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 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.