Generalized scissors congruence

剪刀同余的研究有两个主要影响。第一个是纯粹的几何问题，询问哪些形状可以切割并重新排列成另一个形状。例如，给定250平方英尺的地毯，是否总是可以用它铺满250平方英尺的房间？有人可能会认为这取决于房间的形状，但实际上并非如此：假设地毯和房间都是直的，这总是可能的。然而，这在三维空间中就不成立了：如果你有一个100立方厘米的盒子要装满泡沫橡胶，我给你100立方厘米的泡沫橡胶，你可能无法装满这个盒子，因为橡胶的形状可能是错误的。将这些问题推广到其他维度和几何是非常困难的，而且我们对答案知之甚少。第二个影响更具有哲学意义：作为数学家，我们经常通过将问题“切割”成更小的问题，解决每个更小的问题，然后重新组合解决方案来解决问题。然而，总是有一个重要的最后一步：弄清楚是否有一个独特的方法来重新组合一个大问题的解决方案，或者是否有很多。目前关于剪刀一致性的项目通过构建一个框架来“剪切”和“粘贴”不同类型的“对象”，无论是形状还是数学对象，同时解决了这些问题。这个框架使我们能够一起分析所有这些问题，并更多地了解重新组合解决方案时出现的困难。此外，它还应用于数学的许多不同的子领域，包括代数几何、逻辑、数论和范畴论，产生了一种统一不同问题的新观点。剪刀同余问题是将某些对象（如可定义的集合、变种或多面体）分类到分解和同构的问题。利用先前开发的将剪刀同余问题转化为谱的技术，本项目继续通过稳定同伦理论的透镜分析剪刀同余问题。本项目有三个总体目标：(1)利用剪刀同余谱中的高同列信息分析品种的Grothendieck环；(2)将Goncharov关于混合Tate动机与球形剪刀同余群的结果推广到剪刀同余谱；(3)探索利用剪刀同余谱发展谱值动机积分的可能性。通过将剪刀同余问题之间的经典映射推广到剪刀同余谱，我们希望产生新的几何和代数不变量，从而扩展对这些问题的理解。

The study of scissors congruence has two major influences. The first is purely geometric, asking which shapes can be cut up and rearranged into one another. For example, given 250 sq ft of carpet, is it always possible to carpet a 250 sq ft room with it? One may think that this depends on the shape of the room, but it actually does not: assuming that both the carpet and the room have straight sides, it is always possible. However, this is no longer true with three dimensions: if you have a 100 cu in box to fill with foam rubber and I give you 100 cu in of foam rubber, you may not be able to fill up the box because the rubber may be the wrong shape. Generalizing these problems to other dimensions and geometries is very difficult, and very little is known about the answers. The second influence is more philosophical: as mathematicians, we often address problems by "cutting" them up into smaller problems, solving each of the smaller problems and the reassembling the solutions. However, there is always an important last step: figuring out whether there is a unique way to reassemble a solution to the large problem, or whether there are many. The current project on scissors congruence addresses these issues simultaneously by constructing a framework for "cutting" and "pasting" together different kinds of "objects," be they shapes or mathematical objects. This framework allows us to analyze all such questions together and learn more about the difficulties that arise when reassembling solutions. In addition, it has applications in many different subfields of mathematics, including algebraic geometry, logic, number theory and category theory, producing a novel viewpoint from which to unify different problems.A scissors congruence problem is the problem of classifying certain objects (such as definable sets, varieties, or polytopes) up to decomposition and isomorphism. Using previously developed techniques for turning a scissors congruence problem into a spectrum, this project continues analyzing scissors congruence problems through the lens of stable homotopy theory. This project has three general objectives: (1) analyzing the Grothendieck ring of varieties using the higher homotopical information present in the scissors congruence spectrum, (2) extending results of Goncharov relating mixed Tate motives to spherical scissors congruence groups to scissors congruence spectra, and (3) exploring the possibility of using scissors congruence spectra for developing spectrum-valued motivic integration. By generalizing classical maps between scissors congruence problems to scissors congruence spectra we hope to produce new geometric and algebraic invariants which will extend understanding of these problems.