Related papers: A tropical intersection product in matroidal fans
In a previous work, we gave a construction of (not necessarily realizable) oriented matroids from a triangulation of a product of two simplices. In this follow-up paper, we use a variant of Viro's patchworking to derive a topological…
We revisit some papers by Eichler and Shimura in order to give an algebraic formulation (based on Farey symbols) for the intersection product on the space of modular symbols, as described by Pollack and Stevens. We define the period…
We first develop the local theory of functions on $\mathbb R^n$ defined by tropical Laurent polynomials. We study the structure of the semiring of functions, where two functions are identified when they coincide on a neighborhood of a fixed…
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 prove that the Chow ring of any simplicial fan is isomorphic to the middle degree part of the tropical cohomology ring of its canonical compactification. Using this result, we prove a tropical analogue of Kleiman's criterion of ampleness…
We construct families of exotic spin-1/2 chains using a procedure called ``hard rod deformation''. We treat both integrable and non-integrable examples. The models possess a large non-commutative symmetry algebra, which is generated by…
We define tropical analogues of the notions of linear space and Plucker coordinate and study their combinatorics. We introduce tropical analogues of intersection and dualization and define a tropical linear space built by repeated…
This paper is a continuation of my paper "Lattices of flats for symplectic matroids". We explore geometric constructions originating from the lattice of flats of ranked symplectic matroids. We observe that a ranked symplectic matroid always…
We study the tropicalization of intersections of plane curves, under the assumption that they have the same tropicalization. We show that the set of tropical divisors that arise in this manner is a pure dimensional balanced polyhedral…
We produce counterexamples to show that in the definition of the notion of intertwining operator for modules for a vertex operator algebra, the commutator formula cannot in general be used as a replacement axiom for the Jacobi identity. We…
This paper is the first in a series devoted to the development of a Hodge theory for tropical varieties. We introduce a notion of T-stability for tropical fans and prove that various geometric properties of tropical fans are T-stable. As a…
Given a tropical fan curve $\Sigma$ and a family of algebraic curves $X \rightarrow \mathbb{A}^k$ we define the realization locus $\mathop{Real}_\Sigma \subseteq \mathbb{A}^k$ as the set of fibers $X_a$ whose tropicalization is $\Sigma$. We…
In this paper we use the connections between tropical algebraic geometry and rigid analytic geometry in order to prove two main results. We use tropical methods to prove a theorem about the Newton polygon for convergent power series in…
We consider the mixing of a viscous fluid by the rotation of a pitched blade turbine inside an open, cylindrical tank, with air as the lighter fluid above. To examine the flow and interfacial dynamics, we utilise a highly-parallelised…
Let $ E $ be a possibly infinite set and let $ M $ and $ N $ be matroids defined on $ E $. We say that the pair $ \{ M,N \} $ has the Intersection property if $ M $ and $ N $ share an independent set $ I $ admitting a bipartition $…
We study in some generality intertwinings between $h$-transforms of Karlin-McGregor semigroups associated with one dimensional diffusion processes and those of their Siegmund duals. We obtain couplings so that the corresponding processes…
A classic problem connecting algebraic and geometric combinatorics is the realization problem: given a poset, determine whether there exists a polytope whose face lattice is the poset. In 1990s, Kapranov defined a poset as a hybrid between…
We introduce polar metrics on a product manifold, which have product and warped product metrics as special cases. We prove a de Rham-type theorem characterizing Riemannian manifolds that can be locally decomposed as a product manifold…
We explore a new variational principle for studying one-dimensional quantum systems in a trapping potential. We focus on the Fermi polaron problem, where a single distinguishable impurity interacts through a contact potential with a…
We design an atomtronic filter device that spatially separates the components of a two-component Fermi gas with repulsive contact interactions in a two-dimensional geometry. With the aid of density--potential functional theory (DPFT), which…