CAREER: Constructing K-Theoretic Invariants for Geometric Objects

解决问题的一种常见方法是将问题分解为更小的子问题，解决每个更小的问题，并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，以及不同组合件背后的关系。 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理论的视角研究了场之间的新联系，通过研究几何对象，称为多面体，品种和流形，可以被分割和重新组装。 通过使用代数中的传统K理论技术分析这些几何剪切和粘贴问题，该项目希望揭示与几何和代数相关的长期存在的问题。 PI还打算为所有当地儿童建立一个K-12数学圈，并为3-5年级学生组织一个全区数学俱乐部。 目前，学校的数学俱乐部在很大程度上依赖于家长的参与，因此只有当感兴趣的家长在学校有学生时才可以参加;在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参加。 本研究项目由三部分组成。第一个是深入探索的剪同余多面体，因为它涉及到代数K理论的真实的和复数。 利用罗涅尼斯的稳定秩过滤，PI和合作者Jonathan坎贝尔打算在罗涅尼斯过滤的过滤部分和导出的多面体的剪刀同余之间构建映射;该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。 本项目的第二部分研究衍生动机措施的Grothendieck频谱品种的建设。 动机测度主要有两种类型：上同调测度，涉及簇的上同调结构;埃尔米特测度，涉及充实欧拉数。 这些措施已被证明是非常富有成效的研究格罗滕迪克环的品种;该项目建议提升这些品种的格罗滕迪克频谱，并使用它们来研究更高的不变量的剪切和粘贴问题的品种。 上同调项目是与乔纳森坎贝尔和杰西沃尔夫森联合工作;埃尔米特项目是与克尔斯滕Wickelgren联合。 这个项目的第三部分是“平方K理论”的调查，它使用四项（而不是三项）的关系。 这样的关系-例如，原则的包容排除[P] + [Q] = [P u Q] + [P n Q]-经常出现在几何剪切和粘贴问题，如SK-不变，比特纳的介绍Grothendieck环的品种，和定义的麦克马伦的多面体代数。 该项目建议将K理论的定义扩展到这些关系，从而允许为这些（和类似）问题构建更高的派生不变量。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.