FRG: Collaborative Research: Trace Methods and Applications for Cut-and-Paste K-Theory

FRG：协作研究：剪切粘贴 K 理论的追踪方法和应用

• 批准号: 2052970
• 负责人: Andrew Blumberg
• 金额: $ 18.08万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2052970&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Trace; Methods

This research brings together ideas, techniques, and insights from two long-standing programs in mathematics: scissors congruence and algebraic K-theory. Scissors congruence originated in Hilbert's 3rd Problem, which asks when two polyhedra in three-dimensional space are "scissors congruent," meaning one can be obtained from the other by cutting it into smaller polyhedra and reassembling in a different way. This question, together with its solution by Dehn, initiated an extensive program of research. Over the past 120 years these ideas have grown and now connect to almost every branch of geometry. Ground-breaking recent work provides a fundamental link between this program and algebraic K-theory, which is itself a deep and rapidly developing area of research. Algebraic K-theory intertwines three major fields of mathematics: topology, algebraic geometry, and number theory. Developing the connection between scissors congruence and algebraic K-theory will significantly advance research in both. This work also provides the platform for striking new research avenues that will bring to bear the tools and techniques of modern algebraic K-theory research on a wide range of geometric questions. This project additionally includes a number of efforts to support students and new researchers in the field, expanding and broadening access to these innovative ideas.This broad new program of research develops the foundations of combinatorial, or "cut-and-paste," algebraic K-theory, applies these new tools to resolve outstanding geometric questions, and expands the scope of combinatorial K-theory to new applications. It brings modern techniques in algebraic K-theory to the emerging K-theoretic approach to cut-and-paste invariants, and applies this approach to a variety of problems in algebraic topology, differential topology, and algebraic geometry. Algebraic K-theory has seen a stunning revolution in the last thirty years due to the invention of trace methods, but these tools have not yet been developed for combinatorial K-theory, a deficiency that this project hopes to remedy. This requires developing the foundations of this new theory and exploiting connections to equivariant homotopy theory. New computational and analytic tools for combinatorial K-theory will lead to progress on a wide variety of geometric problems, including applications to manifolds and invertible TQFTs, varieties and motivic measures, and fixed point theory. Many questions in these fields have natural interpretations in terms of cut-and-paste invariants.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.

这项研究汇集了两个长期存在的数学项目的思想、技术和见解：剪刀同余和代数 K 理论。 剪刀全等起源于希尔伯特第三个问题，该问题询问三维空间中的两个多面体何时是“剪刀全等”，这意味着可以通过将另一个多面体切割成更小的多面体并以不同的方式重新组装来获得一个多面体。 这个问题以及德恩的解决方案引发了一项广泛的研究计划。在过去的 120 年里，这些想法不断发展，现在几乎与几何学的每个分支都有联系。最近的突破性工作提供了该程序与代数 K 理论之间的基本联系，代数 K 理论本身就是一个深入且快速发展的研究领域。 代数 K 理论交织着数学的三个主要领域：拓扑、代数几何和数论。发展剪刀同余和代数 K 理论之间的联系将显着推进这两个领域的研究。 这项工作还为引人注目的新研究途径提供了平台，将现代代数 K 理论研究的工具和技术应用于广泛的几何问题。 该项目还包括许多支持该领域的学生和新研究人员的努力，扩大和拓宽对这些创新思想的获取。这一广泛的新研究计划开发了组合或“剪切和粘贴”代数 K 的基础-理论，应用这些新工具来解决突出的几何问题，并将组合 K 理论的范围扩展到新的应用。 它将代数 K 理论中的现代技术引入新兴的用于剪切和粘贴不变量的 K 理论方法，并将这种方法应用于代数拓扑、微分拓扑和代数几何中的各种问题。由于迹方法的发明，代数 K 理论在过去三十年中经历了一场惊人的革命，但这些工具尚未开发用于组合 K 理论，这是该项目希望弥补的缺陷。这需要发展这一新理论的基础并利用与等变同伦理论的联系。 组合 K 理论的新计算和分析工具将在各种几何问题上取得进展，包括流形和可逆 TQFT、簇和动机测度以及不动点理论的应用。 这些领域的许多问题在剪切和粘贴不变量方面都有自然的解释。该奖项反映了 NSF 的法定使命，并且通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。