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，询问哪些形状可以被切割并重新排列成另一个形状。例如，以250平方尺的地毯来说，是否总可以用它来铺250平方尺的房间呢？有人可能认为这取决于房间的形状，但实际上并非如此：假设地毯和房间都有直边，这总是可能的。然而，在三维情况下，这不再是真的：如果你有一个100立方英寸的盒子要装满泡沫塑料，而我给你100立方英寸的泡沫橡胶，你可能无法装满盒子，因为橡胶可能形状错误。将这些问题推广到其他维度和几何是非常困难的，而且对答案知之甚少。第二种影响更具哲学性：作为数学家，我们经常通过将问题“分割”成更小的问题，解决每一个更小的问题，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：弄清楚是否有一种独特的方式来重组一个大问题的解决方案，或者是否有很多种。目前的剪刀一致性项目通过构建一个框架来同时解决这些问题，该框架可以将不同类型的“对象”“剪切”和“粘贴”在一起，无论它们是形状还是数学对象。这个框架使我们能够一起分析所有这样的问题，并更多地了解重组解决方案时出现的困难。此外，它在数学的许多子领域都有应用，包括代数几何、逻辑、数论和范畴理论，为统一不同的问题提供了一个新的观点。剪刀同余问题是将某些对象(如可定义的集合、簇或多面体)分类到分解和同构的问题。使用以前开发的将剪刀同余问题转化为谱的技术，该项目继续通过稳定同伦理论的透镜来分析剪刀同余问题。这个项目有三个总体目标：(1)利用剪刀同余谱中存在的更高的同伦信息来分析变种的Grothendieck环；(2)将Goncharov将混合动机与球剪刀同余群联系起来的结果推广到剪刀同余谱；(3)探索利用剪刀同余谱进行谱值基元积分的可能性。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不变量和代数不变量，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.