Categorical Invariants of Graphs and Matroids
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Date
2024-01-09
Authors
Miyata, Dane
Journal Title
Journal ISSN
Volume Title
Publisher
University of Oregon
Abstract
Graphs and matroids are two of the most important objects in combinatorics.We study invariants of graphs and matroids that behave well with respect to
certain morphisms by realizing these invariants as functors from a category of
graphs (resp. matroids).
For graphs, we study invariants that respect deletions and contractions ofedges. For an integer $g > 0$, we define a category of $\mathcal{G}^{op}_g$ of graphs of genus at most
g where morphisms correspond to deletions and contractions. We prove that this
category is locally Noetherian and show that many graph invariants form finitely
generated modules over the category $\mathcal{G}^{op}_g$. This fact allows us to exihibit many
stabilization properties of these invariants. In particular we show that the torsion
that can occur in the homologies of the unordered configuration space of n points
in a graph and the matching complex of a graph are uniform over the entire family
of graphs with genus $g$.
For matroids, we study valuative invariants of matroids. Given a matroid,one can define a corresponding polytope called the base polytope. Often, the base
polytope of a matroid can be decomposed into a cell complex made up of base
polytopes of other matroids. A valuative invariant of matroids is an invariant that
respects these polytope decompositions. We define a category $\mathcal{M}^{\wedge}_{id}$ of matroids
whose morphisms correspond to containment of base polytopes. We then define the
notion of a categorical matroid invariant which categorifies the notion of a valuative
invariant. Finally, we prove that the functor sending a matroid to its Orlik-Solomon
algebra is a categorical valuative invariant. This allows us to derive relations among
the Orlik-Solomon algebras of a matroid and matroids that decompose its base
polytope viewed as representations of any group $\Gamma$ whose action is compatible with
the polytope decomposition.
This dissertation includes previously unpublished co-authored material.