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Discrete polymatroids.

Vlădoiu, Marius (2006)

Analele Ştiinţifice ale Universităţii “Ovidius" Constanţa. Seria: Matematică

How to draw tropical planes.

Herrmann, Sven, Jensen, Anders, Joswig, Michael, Sturmfels, Bernd (2009)

The Electronic Journal of Combinatorics [electronic only]

Matroids over a ring

Alex Fink, Luca Moci (2016)

Journal of the European Mathematical Society

We introduce the notion of a matroid M over a commutative ring R , assigning to every subset of the ground set an R -module according to some axioms. When R is a field, we recover matroids. When R = , and when R is a DVR, we get (structures which contain all the data of) quasi-arithmetic matroids, and valuated matroids, i.e. tropical linear spaces, respectively. More generally, whenever R is a Dedekind domain, we extend all the usual properties and operations holding for matroids (e.g., duality), and...

Minimality of toric arrangements

Giacomo d'Antonio, Emanuele Delucchi (2015)

Journal of the European Mathematical Society

We prove that the complement of a toric arrangement has the homotopy type of a minimal CW-complex. As a corollary we deduce that the integer cohomology of these spaces is torsionfree. We apply discrete Morse theory to the toric Salvetti complex, providing a sequence of cellular collapses that leads to a minimal complex.

Quaternionic geometry of matroids

Tamás Hausel (2005)

Open Mathematics

Building on a recent paper [8], here we argue that the combinatorics of matroids are intimately related to the geometry and topology of toric hyperkähler varieties. We show that just like toric varieties occupy a central role in Stanley’s proof for the necessity of McMullen’s conjecture (or g-inequalities) about the classification of face vectors of simplicial polytopes, the topology of toric hyperkähler varieties leads to new restrictions on face vectors of matroid complexes. Namely in this paper...

Symmetrized and continuous generalization of transversals

Martin Kochol (1996)

Mathematica Bohemica

The theorem of Edmonds and Fulkerson states that the partial transversals of a finite family of sets form a matroid. The aim of this paper is to present a symmetrized and continuous generalization of this theorem.

The Salvetti complex and the little cubes

Dai Tamaki (2012)

Journal of the European Mathematical Society

For a real central arrangement 𝒜 , Salvetti introduced a construction of a finite complex Sal ( 𝒜 ) which is homotopy equivalent to the complement of the complexified arrangement in [Sal87]. For the braid arrangement 𝒜 k - 1 , the Salvetti complex Sal ( 𝒜 k - 1 ) serves as a good combinatorial model for the homotopy type of the configuration space F ( , k ) of k points in C , which is homotopy equivalent to the space C 2 ( k ) of k little 2 -cubes. Motivated by the importance of little cubes in homotopy theory, especially in the study of...

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