PhD Work

• (with J. D. Clifton, Z. Walsh) The unbreakable quasi-graphic matroids

  Abstract

  A matroid M is unbreakable if it is connected and M/F is connected for every flat F of M. Oxley and Pfeil characterized the unbreakable graphic matroids, and Fife, Mayhew, Oxley, and Semple characterized the graphs underlying 3-connected unbreakable frame matroids. We extend the latter result by giving a complete characterization of the 3-connected unbreakable quasi-graphic matroids. As a special case we obtain a characterization of the 3-connected lifted-graphic matroids.

  

Masters Thesis

Abstract

This project extends calculus from Euclidean spaces to higher-dimensional objects known as manifolds. We develop a coordinate-free approach to calculus on \(\mathbb{R}^n\) using differential forms and derivations. After rigorously defining topological and smooth manifolds, we explore smooth maps, tangent spaces, and the differential as a generalization of the derivative. Key results include the inverse function theorem and criteria for submanifolds. We also study immersions, submersions, embeddings, and introduce differential 1-forms. Finally, we present partitions of unity, which connect local and global analysis, enabling smooth constructions across charts on manifolds.

Masters Thesis PDF