### A combinatorial curvature flow for compact 3-manifolds with boundary.

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In this paper we present factorization theorems for strong maps between matroids of arbitrary cardinality. Moreover, we present a new way to prove the factorization theorem for strong maps between finite matroids.

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 $R=\mathbb{Z}$, 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...

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.

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...

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.

For a real central arrangement $\mathcal{A}$, Salvetti introduced a construction of a finite complex Sal$\left(\mathcal{A}\right)$ which is homotopy equivalent to the complement of the complexified arrangement in [Sal87]. For the braid arrangement ${\mathcal{A}}_{k-1}$, the Salvetti complex Sal$\left({\mathcal{A}}_{k-1}\right)$ serves as a good combinatorial model for the homotopy type of the configuration space $F(\u2102,k)$ of $k$ points in $C$, which is homotopy equivalent to the space ${C}_{2}\left(k\right)$ of k little $2$-cubes. Motivated by the importance of little cubes in homotopy theory, especially in the study of...