On the Bellows conjecture.

다국어 초록 (Multilingual Abstract)

This work is a report on the Bellows Conjecture. It summarizes the results that are known so far, gives a number of new results, and raises several open questions.
In 1813 A. L. Cauchy proved that convex polyhedra are rigid. However, there are flexible non-convex polyhedra, and the Bellows Conjecture arose naturally from the study of these three dimensional flexible polyhedral surfaces. The surfaces are given by their vertices and incidence relations, and they can be continuously deformed while keeping the faces congruent. It was observed that the volume enclosed by the surfaces stays constant during the deformation, and this lead to the following general conjecture: “<italic>The generalized volume of a flexible polyhedron in R<super>n</super> stays constant during a continuous flex</italic>.&rdquo.
I. Sabitov was the first to give a proof of the Conjecture for <italic> n</italic> = 3 in 1995, and the subsequent research has been based on his ideas. Instead of directly proving the Bellows Conjecture, we are focusing on proving the so-called Integrality Conjecture: “<italic>The volume of a flexible polyhedron in R<super>n</super> is integral over the ring generated by the squared edge lengths of the polyhedron over the rational numbers Q </italic>”. We show that the Integrality Conjecture implies the Bellows Conjecture.
After carefully defining the terms and objects, we give a detailed exposition of the algebraic methods that are involved. We discuss the theories of places and valuations, and how they can used to prove integrality. After stating the geometric lemmata involved, we give a proof of the Integrality Conjecture for dimension <italic>n</italic> = 3, as well as partial results for higher dimensions, carefully pointing out the difficulties that arise. We also present examples for non-trivial flexible polyhedra in dimension <italic>n </italic> = 4.