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

合作研究：代数K理论、拓扑周期循环同调和非交换代数几何

基本信息

批准号：
1812064
负责人：
Andrew Blumberg
金额：
$ 27.53万
依托单位：
University of Texas at Austin
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2022-09-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1812064&HistoricalAwards=false
关键词：
Collaborative Research Algebraic Theory Topological

项目摘要

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.

代数拓扑始于对几何物体的代数不变式的研究，这些对象在某些平滑的变形下保存。逐渐地，人们意识到，这些代数不变的（称为共同论理论）本身可以用几何对象（称为谱）表示。现代同义理论的中心胜利是构造环光谱类别（代表乘法共同体学理论的对象），这些谱图适用于直接类似于经典代数的构造。事实证明，这一举动是令人难以置信的卓有成效的，无论是提供了不变的，从旧问题和提出与其他数学和物理学领域有意外联系的新问题中提供了新的启示。这项赠款资助的项目在一个称为代数K理论的丰富不变式和相关理论的情况下进行了该计划，称为拓扑霍奇柴尔德，周期性和周期性同源性。 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.它还包括一个项目，以开发衍生的代数几何形状的基础，该几何形状在组织拓扑模块化形式的研究中的组织计算现象中有应用。 PIS的最新工作使K（S）的同质群（根据其他已知光谱）进行了完整描述，并通过泰特 - poitou二元性的频谱提升，对循环痕迹的纤维进行了规范鉴定。 PI有一个计划应用这项工作，为Kummer-vandiver猜想提供新的证据。如果成功，这将为代数拓扑的投入提供另一个示例，以解决数字理论中的问题。 PI先前在p-Adic Langlands计划中将其工作应用于环形迹线的纤维上，以解决有关稳定一致性亚组的（CO）同源性的猜想。 PI描述了一系列项目，这些项目将在P-ADIC Langlands计划的研究中使用有关纤维的同源理论数据。 PIS的其他最新工作还建立了可划分DG类别的拓扑周期性循环同源性（TP）的Kunneth定理。由于TP是一种非共同的Weil共同体学理论，因此该结果已经在非共同代数几何形状中具有有趣的应用。该赠款包括一个项目，以建立此观点并将TP应用于非共同代数几何形状。根据与Abouzaid和Kragh的对话，PIS已开始通过包装的Fukaya类别探索代数K理论和TP对符号拓扑的应用。 PI描述了一系列项目，这些项目利用其专业知识和先前的结果来研究该领域的基本问题。 Pi Blumberg以前曾与Mike Hill合作，开发了均衡式通勤环光谱理论的基础。 Pi Mandell是拓扑Andre-Quillen同源性（TAQ）的最重要的专家之一。与Basterra，Hill和Lawson合作，PIS研究Edoivariant Taq是一项更广泛的计划的一部分，以开发均等的代数几何形状的基础。如果成功的话，该计划将为来自拓扑模块化形式的作品提供的现象学数据提供组织原则。该奖项反映了NSF的法定任务，并被认为是使用基金会的知识分子优点和更广泛的影响评估标准的评估值得支持的。