Generalized scissors congruence

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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.