109 citations · 289 across the 23 of their papers we have counts for
24 papers
On determinants of resistance matrices
Harry Richman, Farbod Shokrieh, Chenxi Wu
We prove a new combinatorial identity for the determinant of the resistance matrix of a finite graph, which involves counts of spanning trees and forests. This generalizes a result…
Integral torsion points on abelian varieties over function fields
Robin de Jong, Nicole Looper, Farbod Shokrieh
We prove an analogue, over global function fields, of a conjecture due to Su-Ion Ih concerning the non-Zariski density of torsion points on abelian varieties that are integral with…
Tropical Poincaré bundle, Fourier-Mukai transform, and a generalized Poincaré formula
Soham Ghosh, Farbod Shokrieh
We construct a tropical analogue of the Poincaré bundle and prove a (cohomological) Fourier-Mukai transform for real tori with integral structures. We then prove a tropical analogu…
Principal minors of tree distance matrices
Harry Richman, Farbod Shokrieh, Chenxi Wu
We prove that the principal minors of the distance matrix of a tree satisfy a combinatorial expression involving counts of rooted spanning forests of the underlying tree. This gene…
Canonical local heights and Berkovich skeleta
Robin de Jong, Farbod Shokrieh
We discuss canonical local heights on abelian varieties over non-archimedean fields from the point of view of Berkovich analytic spaces. Our main result is a refinement of Néron's…
Counting two-forests and random cut size via potential theory
Harry Richman, Farbod Shokrieh, Chenxi Wu
We prove a lower bound on the number of spanning two-forests in a graph, in terms of the number of vertices, edges, and spanning trees. This implies an upper bound on the average c…
Jumps in the height of the Ceresa cycle
Robin de Jong, Farbod Shokrieh
We study the jumps in the archimedean height of the Ceresa cycle, as introduced by R. Hain in his work on normal functions on moduli spaces of curves, and as further analyzed by P.…
Torsor Structures on Spanning Trees
Farbod Shokrieh, Cameron Wright
We study two actions of the (degree 0) Picard group on the set of the spanning trees of a finite ribbon graph. It is known that these two actions, denoted and respectiv…
Tautological cycles on tropical Jacobians
Andreas Gross, Farbod Shokrieh
The classical Poincaré formula relates the rational homology classes of tautological cycles on a Jacobian to powers of the class of Riemann theta divisor. We prove a tropical analo…
Cycles, cocycles, and duality on tropical manifolds
Andreas Gross, Farbod Shokrieh
We prove a Poincaré duality for the Chow rings of smooth fans whose support are tropical linear spaces. As a consequence, we show that cycles and cocycles on tropical manifolds are…
A sheaf-theoretic approach to tropical homology
Andreas Gross, Farbod Shokrieh
We introduce a sheaf-theoretic approach to tropical homology, especially for tropical homology with potentially non-compact supports. Our setup is suited to study the functorial pr…
Tropical moments of tropical Jacobians
Robin de Jong, Farbod Shokrieh
Each metric graph has canonically associated to it a polarized real torus called its tropical Jacobian. A fundamental real-valued invariant associated to each polarized real torus…
Metric graphs, cross ratios, and Rayleigh's laws
Robin de Jong, Farbod Shokrieh
We study a notion of cross ratios on metric graphs and electrical networks. We show that several known results immediately follow from the basic properties of cross ratios. We show…
Faltings height and Néron-Tate height of a theta divisor
Robin de Jong, Farbod Shokrieh
We prove a formula, which, given a principally polarized abelian variety over the field of algebraic numbers, relates the stable Faltings height of with the Néron--Tate…
Cosmology of gravity
Sante Carloni, João Luís Rosa, José P. S. Lemos
Using dynamical system analysis, we explore the cosmology of theories of order up to eight order of the form . The phase space of these cosmology reveals that higher-…
Effective divisor classes on metric graphs
Andreas Gross, Farbod Shokrieh, Lilla Tóthmérész
We introduce the notion of semibreak divisors on metric graphs (tropical curves) and prove that every effective divisor class (of degree at most the genus) has a semibreak divisor…
Canonical measures on metric graphs and a Kazhdan's theorem
Farbod Shokrieh, Chenxi Wu
We extend the notion of canonical measures to all (possibly non-compact) metric graphs. This will allow us to introduce a notion of "hyperbolic measures" on universal covers of met…
Non-Archimedean and tropical theta functions
Tyler Foster, Joseph Rabinoff, Farbod Shokrieh +1
We define a tropicalization procedure for theta functions on abelian varieties over a non-Archimedean field. We show that the tropicalization of a non-Archimedean theta function is…
Divisors on graphs, binomial and monomial ideals, and cellular resolutions
Fatemeh Mohammadi, Farbod Shokrieh
We study various binomial and monomial ideals arising in the theory of divisors, orientations, and matroids on graphs. We use ideas from potential theory on graphs and from the the…
Canonical representatives for divisor classes on tropical curves and the Matrix-Tree Theorem
Yang An, Matthew Baker, Greg Kuperberg +1
Let be a compact tropical curve (or metric graph) of genus . Using the theory of tropical theta functions, Mikhalkin and Zharkov proved that there is a canonical effective r…
Divisors on graphs, Connected flags, and Syzygies
Fatemeh Mohammadi, Farbod Shokrieh
We study the binomial and monomial ideals arising from linear equivalence of divisors on graphs from the point of view of Gröbner theory. We give an explicit description of a minim…
Chip-firing games, potential theory on graphs, and spanning trees
Matthew Baker, Farbod Shokrieh
We study the interplay between chip-firing games and potential theory on graphs, characterizing reduced divisors (-parking functions) on graphs as the solution to an energy (or…
The monodromy pairing and discrete logarithm on the Jacobian of finite graphs
Farbod Shokrieh
Every graph has a canonical finite abelian group attached to it. This group has appeared in the literature under a variety of names including the sandpile group, critical group, Ja…
Chip-Firing Games, -Parking Functions, and an Efficient Bijective Proof of the Matrix-Tree Theorem
Farbod Shokrieh
Kirchhoff's matrix-tree theorem states that the number of spanning trees of a graph G is equal to the value of the determinant of the reduced Laplacian of . We outline an effici…