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

这项研究汇集了两个由来已久的数学项目：剪刀同余和代数K理论的想法、技术和见解。剪刀同余起源于希尔伯特的第三个问题，即当三维空间中的两个多面体是“剪刀同余”时，意味着一个可以通过将其切割成更小的多面体并以不同的方式重新组装而从另一个获得。这个问题，连同德恩的解决方案，启动了一项广泛的研究计划。在过去的120年里，这些思想得到了发展，现在几乎与几何学的每一个分支都有联系。最近开创性的工作提供了这个程序和代数K理论之间的基本联系，代数K理论本身就是一个深入和快速发展的研究领域。代数K-理论交织在数学的三个主要领域：拓扑学、代数几何和数论。发展剪刀同余与代数K-理论之间的联系将极大地促进这两个领域的研究。这项工作也为寻找新的研究途径提供了平台，这些研究途径将带来现代代数K理论研究广泛几何问题的工具和技术。这个项目还包括一些努力，以支持该领域的学生和新的研究人员，扩大和拓宽这些创新思想的途径。这一广泛的新研究计划发展了组合代数K-理论的基础，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到新兴的K-理论方法中来剪切和粘贴不变量，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-理论在过去的三十年里经历了一场令人震惊的革命，但这些工具还没有被开发出来用于组合K-理论，这是本项目希望弥补的一个缺陷。这需要发展这一新理论的基础，并探索与等变同伦理论的联系。组合K-理论的新的计算和分析工具将导致在各种几何问题上的进展，包括在流形和可逆TQFT、簇和动机测度以及不动点理论中的应用。这些领域中的许多问题在剪切和粘贴不变量方面都有自然的解释。这个奖项反映了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。

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.