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

这项研究汇集了思想，技术和见解，从两个长期的计划在数学：剪刀全等和代数K理论。 剪刀全等（英语：Scissors congruence）起源于希尔伯特第三问题，该问题询问三维空间中的两个多面体何时是“剪刀全等”的，这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，连同它的解决方案的德恩，开始了广泛的研究计划。在过去的120年里，这些思想不断发展，现在几乎与几何学的每一个分支都有联系。最近的突破性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而迅速发展的研究领域。 代数K理论交织着数学的三个主要领域：拓扑学、代数几何和数论。发展剪刀同余和代数K-理论之间的联系将大大推进两者的研究。 这项工作还提供了平台，打击新的研究途径，将承担现代代数K-理论研究的工具和技术，对广泛的几何问题。 该项目还包括一系列努力，以支持学生和新的研究人员在该领域，扩大和拓宽访问这些创新的想法。这个广泛的新的研究计划发展的基础组合，或“剪切和粘贴”，代数K理论，应用这些新的工具来解决突出的几何问题，并扩大组合K理论的范围，以新的应用。 它带来了现代技术在代数K理论的新兴K理论的方法剪切和粘贴不变量，并适用于这种方法的各种问题，代数拓扑，微分拓扑和代数几何。在过去的30年里，由于迹方法的发明，代数K-理论发生了惊人的革命，但这些工具还没有被开发用于组合K-理论，这是本项目希望弥补的不足。这需要发展这个新理论的基础，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导致各种几何问题的进展，包括流形和可逆TQFT，品种和motivic措施，以及不动点理论的应用。 这个奖项反映了NSF的法定使命，并通过使用基金会的知识价值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.