Extending Pappus' Theorem
dc.contributor.author | Hoersch, Florian | |
dc.date.accessioned | 2017-12-22T14:34:23Z | |
dc.date.available | 2017-12-22T14:34:23Z | |
dc.date.issued | 2017-12-22 | |
dc.date.submitted | 2017-12-18 | |
dc.description.abstract | Let $M_1$ and $M_2$ be matroids such that $M_2$ arises from $M_1$ by relaxing a circuit-hyperplane. We will prove that if $M_1$ and $M_2$ are both representable over some finite field $GF(q)$, then $M_1$ and $M_2$ have highly structured representations. Roughly speaking, $M_1$ and $M_2$ have representations that can be partitioned into a bounded number of blocks each of which is \enquote{triangular}, a property we call weakly block-triangular. Geelen, Gerards and Whittle have announced that, under the hypotheses above, the matroids $M_1$ and $M_2$ both have pathwidth bounded by some constant depending only on $q$. That result plays a significant role in their announced proof of Rota's Conjecture. Bounding the pathwidth of $M_1$ and $M_2$ is currently the single most complicated part in the proof of Rota's Conjecture. Our result is intended as a step toward simplifying this part. A matroid $N$ is said to be a fragile minor of another matroid $M$ if $M/C\backslash D = N$ for some $C,D \subseteq E(M)$, but $M/C'\backslash D' \neq N$ whenever $C \neq C'$ or $D \neq D'$. As a second result, we will prove that, given a $GF(q)$-representable matroid $N$, every $GF(q)$-representable matroid $M$ having $N$ as a fragile minor has a representation which is weakly block-triangular. | en |
dc.identifier.uri | http://hdl.handle.net/10012/12784 | |
dc.language.iso | en | en |
dc.pending | false | |
dc.publisher | University of Waterloo | en |
dc.title | Extending Pappus' Theorem | en |
dc.type | Master Thesis | en |
uws-etd.degree | Master of Mathematics | en |
uws-etd.degree.department | Combinatorics and Optimization | en |
uws-etd.degree.discipline | Combinatorics and Optimization | en |
uws-etd.degree.grantor | University of Waterloo | en |
uws.contributor.advisor | Geelen, Jim | |
uws.contributor.affiliation1 | Faculty of Mathematics | en |
uws.peerReviewStatus | Unreviewed | en |
uws.published.city | Waterloo | en |
uws.published.country | Canada | en |
uws.published.province | Ontario | en |
uws.scholarLevel | Graduate | en |
uws.typeOfResource | Text | en |