Related papers: The tropical symplectic Grassmannian
We exhibit a faithful representation of the plactic monoid of every finite rank as a monoid of upper triangular matrices over the tropical semiring. This answers a question first posed by Izhakian and subsequently studied by several…
We consider functions $f:B\to\Rset$ that obey tropical analogs of classical Pl\"ucker relations on minors of a matrix. The most general set $B$ that we deal with in this paper is of the form $\{x\in \Zset^n\colon 0\le x\le a, m\le…
In symplectic topology one uses elliptic methods to prove rigidity results about symplectic manifolds and solutions of Hamiltonian equations on them, where the most basic example is given by geodesics on Riemannian manifolds. Harmonic maps…
We describe a family of tropical fans related to Grassmannian cluster algebras. These fans are related to the kinematic space of massless scattering processes in a number of ways. For each fan associated to the Grassmannian ${\rm Gr}(k,n)$…
We describe the symplectic structure and Hamiltonian dynamics for a class of Grassmannian manifolds. Using the two dimensional sphere ($S^2$) and disc ($D^2$) as illustrative cases, we write their path integral representations using…
Many important problems in extremal combinatorics can be stated as certifying polynomial inequalities in graph homomorphism numbers, and in particular, many ask to certify pure binomial inequalities. For a fixed collection of graphs…
In this paper we relate a well-known in symplectic geometry compactification of the space of symmetric bilinear forms considered as a chart of the Lagrangian Grassmannian to the specific compactifications of the space of electrical networks…
Tropicalization is a procedure for associating a polyhedral complex in Euclidean space to a subvariety of an algebraic torus. We study the question of which graphs arise from tropicalizing algebraic curves. By using Baker's specialization…
Tropical mathematics is used to establish a correspondence between certain microscopic and macroscopic objects in statistical models. Tropical algebra gives a common framework for macrosystems (subsets) and their elementary constituents…
Let $K$ be a real closed field with a nontrivial non-archimedean absolute value. We study a refined version of the tropicalization map, which we call real tropicalization map, that takes into account the signs on $K$. We study images of…
This paper concerns the action of linear symplectomorphisms on linear symplectic forms by conjugation in even dimensions. We prove that pfaffian and $-\frac{1}{2}\operatorname{tr}(JA)$ (sum function) of $A$ are invariants on the action. We…
We investigate special points on the Grassmannian which correspond to friezes with coefficients in the case of rank two. Using representations of arithmetic matroids we obtain a theorem on subpolygons of specializations of the coordinate…
A Laman graph $G$ is a minimally rigid graph in dimension two, and its realization number is its number of distinct embeddings with fixed generic edge lengths. While conjectured to grow exponentially in the number of vertices of $G$, the…
We consider a family of closed symplectic manifolds 4-manifolds which we call symplectic bielliptic surfaces and study its Lagrangian cobordism group of weakly-exact Lagrangian G-branes (that is, Lagrangians equipped with a grading, a Pin…
We introduce tropical analogues of the notion of volume of polytopes, leading to a tropical version of the (discrete) classical isoperimetric inequality. The planar case is elementary, but a higher-dimensional generalization leads to an…
We show that the asymptotic behavior of the two main competing notions of rank of a linear series on a tropical curve is governed by asymptotic invariants, closely paralleling the theory of volumes in algebraic geometry. We introduce and…
We develop a number of general techniques for comparing analytifications and tropicalizations of algebraic varieties. Our basic results include a projection formula for tropical multiplicities and a generalization of the Sturmfels-Tevelev…
In this paper, we study the interplay between tropical and analytic geometry for closed subschemes of toric varieties. Let $K$ be a complete non-Archimedean field, and let $X$ be a closed subscheme of a toric variety over $K$. We define the…
We prove the so-called Unitary Isotropy Theorem, a result on isotropy of a unitary involution. The analogous previously known results on isotropy of orthogonal and symplectic involutions as well as on hyperbolicity of orthogonal,…
Speyer recognized that matroids encode the same data as a special class of tropical linear spaces and Shaw interpreted tropically certain basic matroid constructions; additionally, Frenk developed the perspective of tropical linear spaces…