In: Systematic Biology, 2020, vol. 69, no. 3, p. 557–565
We answer two questions raised by Bryant, Francis, and Steel in their work on consensus methods in phylogenetics. Consensus methods apply to every practical instance where it is desired to aggregate a set of given phylogenetic trees (say, gene evolution trees) into a resulting, “consensus” tree (say, a species tree). Various stability criteria have been explored in this context, seeking...
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In: European Journal of Combinatorics, 2020, vol. 87, p. 103098
We tackle the problem of a combinatorial classification of finite metric spaces via their fundamental polytopes, as suggested by Vershik (2010).In this paper we consider a hyperplane arrangement associated to every split pseudometric and, for tree-like metrics, we study the combinatorics of its underlying matroid.- We give explicit formulas for the face numbers of fundamental polytopes and ...
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In: Transactions of the American Mathematical Society, 2019, vol. 373, no. 3, p. 1909–1940
We give an explicit presentation for the integral cohomology ring of the complement of any arrangement of level sets of characters in a complex torus (alias ``toric arrangement''). Our description parallels the one given by Orlik and Solomon for arrangements of hyperplanes and builds on De Concini and Procesi's work on the rational cohomology of unimodular toric arrangements. As a byproduct...
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In: Advances in Applied Mathematics, 2018, vol. 95, p. 199–270
We initiate the study of group actions on (possibly infinite) semimatroids and geometric semilattices. To every such action is naturally associated an orbit-counting function, a two-variable “Tutte” polynomial and a poset which, in the representable case, coincides with the poset of connected components of intersections of the associated toric arrangement.In this structural framework we...
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In: Advances in Mathematics, 2017, vol. 313, p. 746–802
We compute the cohomology ring of the complement of a toric arrangement with integer coefficients and investigate its dependency from the arrangement's combinatorial data. To this end, we study a morphism of spectral sequences associated to certain combinatorially defined subcomplexes of the toric Salvetti category in the complexified case, and use a technical argument in order to extend the...
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In: Proceedings of the American Mathematical Society, 2017, vol. 145, no. 3, p. 955–970
We define a partial ordering on the set $ \mathcal {Q}=\mathcal {Q}(\mathsf {M})$ of pairs of topes of an oriented matroid $ \mathsf {M}$, and show the geometric realization $ \vert\mathcal {Q}\vert$ of the order complex of $ \mathcal {Q}$ has the same homotopy type as the Salvetti complex of $ \mathsf {M}$. For any element $ e$ of the ground set, the complex $ \vert\mathcal {Q}_e\vert$...
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In: Journal of the European Mathematical Society, 2015, vol. 17, no. 3, p. 483–521
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.
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