Matroids and Canonical Forms: Theory and Applications

This thesis proposes a combinatorial generalization of a nilpotent operator
on a vector space. The resulting object is highly natural, with basic
connections to a variety of fields in pure mathematics, engineering, and the
sciences. For the purpose of exposition we focus the discussion of applications
on homological algebra and computation, with additional remarks in lattice
theory, linear algebra, and abelian categories. For motivation, we recall that
the methods of algebraic topology have driven remarkable progress in the
qualitative study of large, noisy bodies of data over the past 15 years. A
primary tool in Topological Data Analysis [TDA] is the homological persistence
module, which leverages categorical structure to compare algebraic shape
descriptors across multiple scales of measurement. Our principle application to
computation is a novel algorithm to calculate persistent homology which, in
certain cases, improves the state of the art by several orders of magnitude.
Included are novel results in discrete, spectral, and algebraic Morse theory,
and on the strong maps of matroid theory. The defining theme throughout is
interplay between the combinatorial theory matroids and the algebraic theory of
categories. The nature of these interactions is remarkably simple, but their
consequences in homological algebra, quiver theory, and combinatorial
optimization represent new and widely open fields for interaction between the
disciplines.