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: 2052702
• 负责人: Michael Hill
• 金额: $ 10.06万
• 依托单位: University of California-Los Angeles
• 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=2052702&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数学长期计划的思想，技术和见解：剪刀一致性和代数K理论。 剪刀的一致性起源于希尔伯特的第三个问题，该问题询问何时在三维空间中的两个Polyhedra是“剪刀的一致”，这意味着一个可以通过将其切成较小的Polyhedra来获得一个，并以不同的方式重新分配。 这个问题以及Dehn的解决方案启动了广泛的研究计划。在过去的120年中，这些想法已经发展起来，现在几乎连接到几何分支。开创性的最近工作提供了该计划与代数K理论之间的基本联系，该计划本身就是一个深刻而快速发展的研究领域。 代数K理论交织了三个主要数学领域：拓扑，代数几何和数理论。发展剪刀一致性与代数K理论之间的联系将大大提高两者的研究。 这项工作还提供了触动新的研究途径的平台，该途径将带来有关广泛几何问题的现代代数K理论研究的工具和技术。 该项目还包括许多努力，以支持该领域的学生和新研究人员，扩大并扩大对这些创新思想的访问。这本广泛的新计划开发了组合或“剪切和paste”的基础，即代数K理论，将这些新的新工具应用来解决杰出的几何问题，并扩大了组合型应用程序的范围。 它将代数K理论的现代技术带入了新兴的K理论方法来剪切和剪切不变，并将这种方法应用于代数拓扑，差异拓扑和代数几何学中的各种问题。由于痕量方法的发明，代数K理论在过去30年中看到了一场令人惊叹的革命，但是这些工具尚未针对组合K理论开发，该项目希望这一缺陷能够补救。这需要开发这种新理论的基础，并利用与均值同义理论的联系。 组合K理论的新计算和分析工具将导致各种几何问题的进展，包括应用于多种多样和可逆TQFT，品种和动机措施以及固定点理论的应用。 这些领域的许多问题在剪切不变的方面都有自然的解释。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响标准，被认为值得通过评估来获得支持。