CAREER: Constructing K-Theoretic Invariants for Geometric Objects

解决问题的一种常用方法是将问题分解为较小的子问题，解决每个较小的问题，然后将答案组合成原始问题的解决方案。最后一步通常非常困难，因为有多种方法可以将解决方案的各个部分粘合在一起。k理论的数学领域研究将这些解重新组合在一起的不同方法，以及不同组合部分背后的关系。由k理论构造的不变量可以在许多领域中找到，从数论到代数几何到拓扑学。这个国家科学基金会奖通过k理论的视角，通过研究几何物体，被称为多面体、变种和流形，如何被切割和重新组装，来研究领域之间的新联系。通过使用在代数中发现的传统k理论技术分析这些几何剪切和粘贴问题，该项目希望能够阐明与几何和代数有关的长期猜想。公团还计划建立面向所有地区儿童的K-12数学圈，并组织面向3-5年级的地区数学俱乐部。目前，学校的数学俱乐部在很大程度上依赖于家长的参与，因此只有在感兴趣的家长有学生在学校时才会成立；在一个中央组织下组织这样的俱乐部将有助于机构记忆，并使更多的学生可以使用它。本研究项目由三部分组成。第一部分是深入探讨多面体的剪子同余，因为它涉及到实数和复数的代数k理论。利用Rognes的稳定秩滤波，PI和合作者Jonathan Campbell打算在Rognes滤波的滤波部分和得到的多面体剪子同余之间构造映射；这个项目的长期目标是研究剪刀同余群和Beilinson- Soule猜想之间的联系。本项目第二部分研究了品种格罗滕迪克谱的衍生动力测度的构建。动机测度主要有两种：上同调测度，它涉及到变量上同调上的结构；厄米测度，它涉及到充实欧拉数。这些措施在研究品种的格罗滕迪克环方面被证明是非常富有成效的；本项目拟将其提升到品种的格罗滕迪克谱，并用它们来研究品种剪切粘贴问题的高不变量。上同源项目是乔纳森·坎贝尔和杰西·沃尔夫森的联合工作；hermite项目是和Kirsten Wickelgren合作的。这个项目的第三部分是对“平方k理论”的研究，它使用四项（而不是三项）关系。这样的关系——例如，包含-排除原理[P] + [Q] = [P u Q] + [P n Q]——经常出现在几何剪切和粘贴问题中，如sk不变量，Bittner的Grothendieck环的变体，以及McMullen的多面体代数的定义。该项目建议将k理论的定义扩展到这样的关系，从而允许为这些（和类似的）问题构造更高的派生不变量。该奖项反映了美国国家科学基金会的法定使命，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。

A common approach to problem-solving is to split a problem into smaller sub-problems, solve each of the smaller problems, and assemble the answers into a solution to the original problem. This last step is often very difficult, as there are multiple ways of gluing the pieces of the solution together. The mathematical area of K-theory studies different ways of putting such solutions back together, as well as the relations behind differently-assembled pieces. The invariants constructed by K-theory can be found in many fields, from number theory to algebraic geometry to topology. This National Science Foundation award investigates novel connections between fields through a K-theoretic perspective, by studying how geometric objects, known as polytopes, varieties and manifolds, can be cut apart and reassembled. By analyzing these geometric cut-and-paste problems using traditional K-theoretic techniques found in algebra, the project hopes to shed light on longstanding conjectures relating geometry and algebra. The PI also intends to establish a K-12 math circle available to all local children, and the organization of a district-wide math club for 3-5 graders. Currently, math clubs at schools are heavily dependent on parental involvement and are therefore only available when interested parents have students at the school; organizing such clubs under a central organization would help with institutional memory and make it available to more students. The research project consists of three parts. The first is an in-depth exploration of the scissors congruence of polytopes as it relates to the algebraic K-theory of the real and complex numbers. Using Rognes' stable rank filtration the PI and collaborator Jonathan Campbell intend to construct maps between the filtered parts of Rognes' filtration and the derived scissors congruence of polytopes; the long-term goal of this project is to investigate the connections between scissors congruence groups and the Beilinson--Soule conjecture. The second part of this project investigates the construction of derived motivic measures on the Grothendieck spectrum of varieties. There are two main flavors of motivic measure: the cohomological measures, which involve structures on the cohomology of the variety, and the Hermitian measures, which involve enriching the Euler number. These measures have proven extremely fruitful in studying the Grothendieck ring of varieties; the project proposes to lift these to the Grothendieck spectrum of varieties and use them to study the higher invariants of cut-and-paste problems on varieties. The cohomological project is joint work with Jonathan Campbell and Jesse Wolfson; the Hermitian project is joint with Kirsten Wickelgren. The third part of this project is an investigation of "squares K-theory", which uses four-term (instead of three-term) relations. Such relations---for example, the principle of inclusion-exclusion [P] + [Q] = [P u Q] + [P n Q]---appear frequently in geometric cut-and-paste problems such as SK-invariants, Bittner's presentation of the Grothendieck ring of varieties, and the definitions of McMullen's polytope algebra. The project proposes to extend the definition of K-theory to such relations, thus allowing the construction of higher derived invariants for these (and similar) problems.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.