89 research outputs found
Towards a matroid-minor structure theory
This paper surveys recent work that is aimed at generalising the results and techniques of the Graph Minors Project of Robertson and Seymour to matroids
Matroid theory for algebraic geometers
This article is a survey of matroid theory aimed at algebraic geometers.
Matroids are combinatorial abstractions of linear subspaces and hyperplane
arrangements. Not all matroids come from linear subspaces; those that do are
said to be representable. Still, one may apply linear algebraic constructions
to non-representable matroids. There are a number of different definitions of
matroids, a phenomenon known as cryptomorphism. In this survey, we begin by
reviewing the classical definitions of matroids, develop operations in matroid
theory, summarize some results in representability, and construct polynomial
invariants of matroids. Afterwards, we focus on matroid polytopes, introduced
by Gelfand-Goresky-MacPherson-Serganova, which give a cryptomorphic definition
of matroids. We explain certain locally closed subsets of the Grassmannian,
thin Schubert cells, which are labeled by matroids, and which have applications
to representability, moduli problems, and invariants of matroids following
Fink-Speyer. We explain how matroids can be thought of as cohomology classes in
a particular toric variety, the permutohedral variety, by means of Bergman
fans, and apply this description to give an exposition of the proof of
log-concavity of the characteristic polynomial of representable matroids due to
the author with Huh.Comment: 74 page
Projective geometries in exponentially dense matroids. II
We show for each positive integer that, if is a
minor-closed class of matroids not containing all rank- uniform
matroids, then there exists an integer such that either every rank-
matroid in can be covered by at most rank- sets, or
contains the GF-representable matroids for some prime power
and every rank- matroid in can be covered by at most
rank- sets. In the latter case, this determines the maximum density
of matroids in up to a constant factor
On excluded minors for real-representability
AbstractWe show that for any infinite field K and any K-representable matroid N there is an excluded minor for K-representability that has N as a minor
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