A quantitative Steinitz' theorem.
Tropical convexity.
Classification of six-point metrics.
The tropical Grassmannian.
Toric hyperkähler varieties.
Alexander duality in subdivisions of Lawrence polytopes.
Matroid polytopes, nested sets and Bergman fans.
Gröbner bases and Stanley decompositions of determinantal rings.
Viro's theorem for complete intersections
Gröbner bases of lattices, corner polyhedra, and integer programming.
How to draw tropical planes.
Product formulas for resultants and Chow forms.
All 11 3 and 12 3-configurations are rational.
An Infinite Family of Minor-Minimal Nonrealizable 3-Chirotopes.
On the Coordinatization of Oriented Matroids.
Stanley decompositions of the Bracket ring.
Sagbi bases of Cox–Nagata rings
We degenerate Cox–Nagata rings to toric algebras by means of sagbi bases induced by configurations over the rational function field. For del Pezzo surfaces, this degeneration implies the Batyrev–Popov conjecture that these rings are presented by ideals of quadrics. For the blow-up of projective -space at points, sagbi bases of Cox–Nagata rings establish a link between the Verlinde formula and phylogenetic algebraic geometry, and we use this to answer questions due to D’Cruz–Iarrobino and Buczyńska–Wiśniewski....
Binomial residues
A binomial residue is a rational function defined by a hypergeometric integral whose kernel is singular along binomial divisors. Binomial residues provide an integral representation for rational solutions of -hypergeometric systems of Lawrence type. The space of binomial residues of a given degree, modulo those which are polynomial in some variable, has dimension equal to the Euler characteristic of the matroid associated with .
Brownian motion tree models are toric
Felsenstein's classical model for Gaussian distributions on a phylogenetic tree is shown to be a toric variety in the space of concentration matrices. We present an exact semialgebraic characterization of this model, and we demonstrate how the toric structure leads to exact methods for maximum likelihood estimation. Our results also give new insights into the geometry of ultrametric matrices.
Uniform Oriented Matroids Without the Isotopy Property.
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