Ph.D. Recipients and their Thesis Abstracts

Geometry

Algebra, Analysis, Combinatorics, Differential Equations / Dynamical Systems, Differential Geometry, Geometry, Interdisciplinary, Lie Groups, Logic, Mathematical Physics, PDE / Numerical Analysis, Probability, Statistics, Topology

A Diagrammatic Formal System for Euclidean Geometry

Abstract: It has long been commonly assumed that geometric diagrams can only be used as aids to human intuition and cannot be used in rigorous proofs of theorems of Euclidean geometry. This work gives a formal system FG whose basic syntactic objects are geometric diagrams and which is strong enough to formalize most if not all of what is contained in the first several books of Euclid's Elements. This formal system is much more natural than other formalizations of geometry have been. Most correct informal geometric proofs using diagrams can be translated fairly easily into this system, and formal proofs in this system are not significantly harder to understand than the corresponding informal proofs. It has also been adapted into a computer system called CDEG (Computerized Diagrammatic Euclidean Geometry) for giving formal geometric proofs using diagrams. The formal system FG is used here to prove meta mathematical and complexity theoretic results about the logical structure of Euclidean geometry and the uses of diagrams in geometry.

On the Bellows Conjecture

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 led to the following general conjecture: "The generalized volume of a flexible polyhedron in R^n stays constant during a continuous flex."

I.~Sabitov was the first to give a proof of the conjecture for n = 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: "The volume of a flexible polyhedron in R^n is integral over the ring generated by the squared edge lengths of the polyhedron over the rational numbers Q." 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 be used to prove integrality. After stating the geometric lemmata involved, we give a proof of the Integrality Conjecture for dimension n = 3, as well as partial results for higher dimensions, carefully pointing out the difficulties that arise. We also present examples for nontrivial flexible polyhedra in dimension n = 4.

Last modified: April 7, 2003