Federico Ardila

The Bergman complex of a matroid and phylogenetic trees

We study the Bergman complex B(M) of a matroid M: a polyhedral complex which arises in algebraic geometry, but which we describe purely combinatorially. We prove that a natural subdivision of the Bergman complex of M is a geometric realization of the order complex of the proper part of its lattice of flats. In addition, we show that the Bergman fan B˜(Kn) of the graphical matroid of the complete graph Kn is homeomorphic to the space of phylogenetic trees Tn × R. This leads to a proof that the link of the origin in Tn is homeomorphic to the order complex of the proper part of the partition lattice Πn.

Matroid Polytopes and their Volumes

We express the matroid polytope PM of a matroid M as a signed Minkowski sum of simplices, and obtain a formula for the volume of PM. This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian Grk,n. We then derive analogous results for the independent set polytope and the underlying flag matroid polytope of M. Our proofs are based on a natural extension of Postnikov’s theory of generalized permutohedra.

Root Polytopes and Growth Series of Root Lattices

The convex hull of the roots of a classical root lattice is called a root polytope. We determine explicit unimodular triangulations of the boundaries of the root polytopes associated to the root lattices

Gelfand-Tsetlin polytopes and Feigin-Fourier-Littelmann-Vinberg polytopes as marked poset polytopes

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Valuations for Matroid Polytope Subdivisions

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Positively oriented matroids are realizable

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Pruning processes and a new characterization of convex geometries

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Correction to “Combinatorics and geometry of power ideals”: Two counterexamples for power ideals of hyperplane arrangements

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Pamela Harris: The Mathematical Rise and Social Contribution of a Dreamer

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Algebraic Structures on Polytopes

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