Computer-assisted proofs in geometry and physics
Name
867870770-MIT.pdf
Description
Full printable version
Size
8.69 MB
Format
Adobe PDF
Checksum (MD5)
b6d07f0d7a56fc9ab47e3c55b803c0e7
Author(s)
Minton, Gregory T. (Gregory Thomas)
Advisor(s)
Abhinav Kumar.
Date Issued
2013
Publisher
Massachusetts Institute of Technology
Abstract
In this dissertation we apply computer-assisted proof techniques to two problems, one in discrete geometry and one in celestial mechanics. Our main tool is an effective inverse function theorem which shows that, in favorable conditions, the existence of an approximate solution to a system of equations implies the existence of an exact solution nearby. This allows us to leverage approximate computational techniques for finding solutions into rigorous computational techniques for proving the existence of solutions. Our first application is to tight codes in compact spaces, i.e., optimal codes whose optimality follows from linear programming bounds. In particular, we show the existence of many hitherto unknown tight regular simplices in quaternionic projective spaces and in the octonionic projective plane. We also consider regular simplices in real Grassmannians. The second application is to gravitational choreographies, i.e., periodic trajectories of point particles under Newtonian gravity such that all of the particles follow the same curve. Many numerical examples of choreographies, but few existence proofs, were previously known. We present a method for computer-assisted proof of existence and demonstrate its effectiveness by applying it to a wide-ranging set of choreographies.
Description
Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013.
Cataloged from PDF version of thesis.
Includes bibliographical references.
Subjects
Mathematics.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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