Hyperplane Neural Codes and the Polar Complex

Hyperplane codes are a class of convex codes that arise as the output of a one layer feed-forward neural network. Here we establish several natural properties of stable hyperplane codes in terms of the {\it polar complex} of the code, a simplicial complex associated to any combinatorial code. We prove that the polar complex of a stable hyperplane code is shellable and show that most currently known properties of the hyperplane codes follow from the shellability of the appropriate polar complex.Comment: 23 pages, 5 figures. To appear in Proceedings of the Abel Symposiu

Triangulations of Grassmannians and flag manifolds

MacPherson conjectured that the Grassmannian $\mathrm{Gr}(2, \mathbb{R}^n)$ has the same homeomorphism type as the combinatorial Grassmannian \|\mbox{MacP}(2,n)\|, while Babson proved that the spaces $\mathrm{Gr}(2,\mathbb{R}^n)$ and $\mathrm{Gr}(1,2,\mathbb{R}^n)$ are homotopy equivalent to their combinatorial analogs $\|\mathrm{MacP}(2,n)\|$ and \|\mbox{MacP}(1,2,n)\| respectively. We will prove that $\mathrm{Gr}(2, \mathbb{R}^n)$ and $\mathrm{Gr}(1,2, \mathbb{R}^n)$ are homeomorphic to \|\mbox{MacP}(2,n)\| and \|\mbox{MacP}(1,2,n)\| respectively.Comment: 29 page

Modular flats of oriented matroids and poset quasi-fibrations

We study the combinatorics of modular flats of oriented matroids and the topological consequences for their Salvetti complexes. We show that the natural map to the localized Salvetti complex at a modular flat of corank one is what we call a poset quasi-fibration -- a notion derived from Quillen's fundamental Theorem B from algebraic $K$-theory. As a direct consequence, the Salvetti complex of an oriented matroid whose geometric lattice is supersolvable is a $K(\pi,1)$-space -- a generalization of the classical result for supersolvable hyperplane arrangements due to Falk, Randell and Terao. Furthermore, the fundamental group of the Salvetti complex of a supersolvable oriented matroid is an iterated semidirect product of finitely generated free groups -- analogous to the realizable case. Our main tools are discrete Morse theory, the shellability of certain subcomplexes of the covector complex of an oriented matroid, a nice combinatorial decomposition of poset fibers of the localization map, and an isomorphism of covector posets associated to modular elements. We provide a simple construction of supersolvable oriented matroids. This gives many non-realizable supersolvable oriented matroids and by our main result aspherical CW-complexes.Comment: 27 paper, 7 figure

The Fundamental Group of the Complement of the Complexification of a Real Arrangement of Hyperplanes

AbstractLet A be an arrangement of hyperplanes (i.e., a finite set of 1-codimension vector subspaces) in Rd. We say that the linear ordering of the hyperplanes A,H1≺H2≺···≺Hn, is ashellabilityorder of A, if there is an oriented affine linelcrossing the hyperplanes of A on the given linear order. The intersection latticeL(A) is the set of all intersections of the hyperplanes of A partially ordered by reversed inclusion. Set M(Ac)≔Cd\⋃{H⊗C:H∈A}. We will prove:Suppose that there are shellability orders H1≺H2≺···≺Hnand H′1≺′H′2≺′···≺′H′n, respectively, ofAandA′,such that the bijective map Hi→H′i, i∈[n]determines an isomorphism of the intersection lattices L(A)and L(A′).Then the fundamental groupsπ1(M(Ac))andπ1(M(A′c)) are isomorphic