David E. Speyer

Tropical Linear Spaces

We define the tropical analogues of the notions of linear spaces and Plucker coordinates and study their combinatorics. We introduce tropical analogues of intersection and dualization and define a tropical linear space built by repeated dualization and transverse intersection to be constructible. Our main result is that all constructible tropical linear spaces have the same

Computing tropical varieties

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Reconstructing Trees from Subtree Weights

-vector and are “series-parallel”. We conjecture that this

Parameterizing tropical curves I: Curves of genus zero and one

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Weak separation and plabic graphs

-vector is maximal for all tropical linear spaces, with equality precisely for the series-parallel tropical linear spaces. We present many partial results towards this conjecture.

Powers of Coxeter elements in infinite groups are reduced

The tropical variety of a d-dimensional prime ideal in a polynomial ring with complex coefficients is a pure d-dimensional polyhedral fan. This fan is shown to be connected in codimension one. We present algorithmic tools for computing the tropical variety, and we discuss our implementation of these tools in the Grobner fan software Gfan. Every ideal is shown to have a finite tropical basis, and a sharp lower bound is given for the size of a tropical basis for an ideal of linear forms.

Grassmannians for scattering amplitudes in 4d

The tree-metric theorem provides a necessary and sufficient condition for a dissimilarity matrix to be a tree metric, and has served as the foundation for numerous distance-based reconstruction methods in phylogenetics. Our main result is an extension of the tree-metric theorem to more general dissimilarity maps. In particular, we show that a tree with n leaves is reconstructible from the weights of the m-leaf subtrees provided that n ≥ 2m - 1.

\mathcal{N}=4

In tropical geometry, given a curve in a toric variety, one defines a corresponding graph embedded in Euclidean space. We study the problem of reversing this process for curves of genus zero and one. Our methods focus on describing curves by parameterizations, not by their defining equations; we give parameterizations by rational functions in the genus zero case and by non-archimedean elliptic functions in the genus one case. For genus zero curves, those graphs which can be lifted can be characterized in a completely combinatorial manner. For genus one curves, show that certain conditions identified by Mikhalkin are sufficient and we also identify a new necessary condition.

SYM and 3d ABJM

Leclerc and Zelevinsky described quasicommuting families of quantum minors in terms of a certain combinatorial condition, called weak separation. They conjectured that all maximal by inclusion weakly separated collections of minors have the same cardinality, and that they can be related to each other by a sequence of mutations. On the other hand, Postnikov studied total positivity on the Grassmannian. He described a stratification of the totally nonnegative Grassmannian into positroid strata, and constructed their parametrization using plabic graphs. In this paper we link the study of weak separation to plabeic graphs. We extend the notion of weak separation to positroids. We generalize the conjectures of Leclerc and Zelevinsky, and related ones of Scott, and prove them. We show that the maximal weakly separated collections in a positroid are in bijective correspondence with the plabic graphs. This correspondence allows us to use the combinatorial techniques of positroids and plabic graphs to prove the (generalized) purity and mutation connectedness conjectures.

Sortable elements in infinite Coxeter groups

Let W be an infinite irreducible Coxeter group with (s 1 ,..., s n ) the simple generators. We give a short proof that the word s 1 s 2 ... s n s 1 s 2 ··· s n ···s 1 s 2 ...s n is reduced for any number of repetitions of s 1 s 2 ··· s n . This result was proved for simply laced, crystallographic groups by Kleiner and Pelley using methods from the theory of quiver representations. Our proof uses only basic facts about Coxeter groups and the geometry of root systems. We also prove that, in finite Coxeter groups, there is a reduced word for w 0 which is obtained from the semi-infinite word s 1 s 2 ···s n s 1 s 2 ··· s n ··· by interchanging commuting elements and taking a prefix.