Geoff Whittle turned 60 in 2010. To mark this occasion, this special issue (guestedited by Dillon Mayhew and Charles Semple) aims to recognize and celebrate his mathematical contributions. Perhaps surprisingly, we can confidently say that Geoff’s best work lies ahead of him, for some of the most significant results from his longstanding collaboration with Jim Geelen and Bert Gerards have been a...

The matroids that are representable over GF (3) and some other fields depend on the choice of field. This paper gives matrix characterisations of the classes that arise. These characterisations are analogues of the characterisation of regular matroids as the ones that can be represented over the rationals by a totally-unimodular matrix. Some consequences of the theory are as follows. A matroid ...

A structure M is pregeometric if the algebraic closure is a pregeometry in all M ′ elementarily equivalent to M . We define a generalisation: structures with an existential matroid. The main examples are superstable groups of U-rank a power of ω and d-minimal expansion of fields. Ultraproducts of pregeometric structures expanding a field, while not pregeometric in general, do have an unique exi...

We discuss the relationship between the notion of independence as defined in lattice theory, matroid theory, and Boolean algebras. Collections of Boolean sub-algebras {At} are indeed endowed with the following independence relation (IB) ∩At 6= ∧ with ∧ the initial element of the Boolean algebra they belong to. However, those collections can be given several algebraic interpretations in terms of...

In a secret-sharing scheme, a secret value is distributed among a set of parties by giving each party a share. The requirement is that only predefined subsets of parties can recover the secret from their shares. The family of the predefined authorized subsets is called the access structure. An access structure is ideal if there exists a secret-sharing scheme realizing it in which the shares hav...

In this paper we consider the problem of finding the densest subset subject to co-matroid constraints. We are given a monotone supermodular set function f defined over a universe U , and the density of a subset S is defined to be f(S)/|S|. This generalizes the concept of graph density. Co-matroid constraints are the following: given matroid M a set S is feasible, iff the complement of S is inde...

The families of matroid independence complexes and pure shifted simplicial complexes are two remarkable classes of simplicial complexes that share several structural properties, but for which theorems are typically proved in different ways. In a groundbreaking paper Kook, Reiner and Stanton [4] proved that matroid complexes are Laplacian integral. Afterwards Duval and Reiner [1] proved the anal...

Several attempts have been made to extend the theory of matroids (here referred to as ordinary or classical matroids) to theories of more general objects, in particular the Coxeter matroids of Borovik, Gelfand and White ([7], first introduced as WP-matroids in [10]), and the ∆-matroids and (equivalent but for notation) symmetric matroids of Bouchet (see, for example, [8]). The special cases of ...

Given a matroid M on the ground set E, the Bergman fan B̃(M), or space of M -ultrametrics, is a polyhedral complex in RE which arises in several different areas, such as tropical algebraic geometry, dynamical systems, and phylogenetics. Motivated by the phylogenetic situation, we study the following problem: Given a point ω in RE , we wish to find an M -ultrametric which is closest to it in the ...