# Farbod Shokrieh - Research and Publications

My MathSciNet profile
My Google Scholar profile

## Publications and Preprints

• The monodromy pairing and discrete logarithm on the Jacobian of finite graphs. Journal of Mathematical Cryptology, Volume 4, Issue 1, Pages 43-56, 2010.
• Journal version, arXiv version

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, Jacobian group, and Picard group. The construction of this group closely mirrors the construction of the Jacobian variety of an algebraic curve. Motivated by this analogy, it was recently suggested by Norman Biggs that the critical group of a finite graph is a good candidate for doing discrete logarithm based cryptography. In this paper, we study a bilinear pairing on this group and show how to compute it. Then we use this pairing to find the discrete logarithm efficiently, thus showing that the associated cryptographic schemes are not secure. Our approach resembles the MOV attack on elliptic curves.

• Improved simulation of nondeterministic Turing machines. Theoretical Computer Science, Pages 66-73, 2010.
(with S. Kalyanasundaram, R. J. Lipton, K. W. Regan)
• Journal version

The standard simulation of a nondeterministic Turing machine (NTM) by a deterministic one essentially searches a large bounded-degree graph whose size is exponential in the running time of the NTM. The graph is the natural one defined by the configurations of the NTM. All methods in the literature have required time linear in the size $S$ of this graph. This paper presents a new simulation method that runs in time $\tilde{O}(\sqrt{S})$. The search savings exploit the one-dimensional nature of Turing machine tapes. In addition, we remove most of the time dependence on nondeterministic choices of states and tape head movements.

• Chip-firing games, potential theory on graphs, and spanning trees. Journal of Combinatorial Theory, Series A, Volume 120, Issue 1, Pages 164-182, 2013.
(with Matthew Baker)
• Journal version, arXiv version

We study the interplay between chip-firing games and potential theory on graphs, characterizing reduced divisors ($G$-parking functions) on graphs as the solution to an energy (or potential) minimization problem and providing an algorithm to efficiently compute reduced divisors. Applications include an efficient bijective proof of Kirchhoff's matrix-tree theorem and a new algorithm for finding random spanning trees. The running times of our algorithms are analyzed using potential theory, and we show that the bounds thus obtained generalize and improve upon several previous results in the literature.

• Divisors on graphs, connected flags, and syzygies. International Mathematics Research Notices (IMRN), no. 24, Pages 6839-6905, 2014.
• Journal version, arXiv version, extended abstract (appeared in FPSAC '13)

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 minimal Gröbner basis for each higher syzygy module. In each case the given minimal Gröbner basis is also a minimal generating set. The Betti numbers of the binomial ideal and its natural initial ideal coincide and they correspond to the number of connected flags in the graph. In particular the Betti numbers are independent of the characteristic of the base field. For complete graphs the problem was previously studied by Postnikov and Shapiro and by Manjunath and Sturmfel. The case of a general graph was stated as an open problem.

• Canonical representatives for divisor classes on tropical curves and the Matrix-Tree Theorem. Forum of Mathematics, Sigma, Volume 2, 25 pages, 2014.
(with Yang An, Matthew Baker, and Greg Kuperberg)
• Journal version, arXiv version

Let $\Gamma$ be a compact tropical curve (or metric graph) of genus $g$. Using the theory of tropical theta functions, Mikhalkin and Zharkov proved that there is a canonical effective representative for each linear equivalence class of divisors of degree $g$ on $\Gamma$. We characterize these canonical effective representatives as the set of break divisors on $\Gamma$ and present a new combinatorial proof that there is a unique break divisor in each equivalence class. We also establish the closely related fact that for fixed $q$ in $\Gamma$, every divisor of degree $g-1$ is linearly equivalent to a unique $q$-orientable divisor. For both of these results, we also prove discrete versions for finite unweighted graphs which do not follow from the results of Mikhalkin-Zharkov. As an application of the theory of break divisors, we provide a "geometric proof" of (a dual version of) Kirchhoff's celebrated Matrix-Tree Theorem. Indeed, we show that each weighted graph model $G$ for $\Gamma$ gives rise to a canonical polyhedral decomposition of the $g$-dimensional real torus ${\rm Pic}^g(\Gamma)$ into parallelotopes $C_T$, one for each spanning tree $T$ of $G$, and the dual Kirchhoff theorem becomes the statement that the volume of ${\rm Pic}^g(\Gamma)$ is the sum of the volumes of the cells in the decomposition.

• Divisors on graphs, binomial and monomial ideals, and cellular resolutions. Mathematische Zeitschrift, Volume 283, Issue 1, Pages 59-102, 2016.
• Journal version, arXiv version

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 theory of Delaunay decompositions for lattices to describe their minimal polyhedral cellular free resolutions. We show that the resolutions of all these ideals are closely related and that their ${\mathbb Z}$-graded Betti tables coincide. As corollaries, we give conceptual proofs of conjectures and questions posed by Postnikov and Shapiro, by Manjunath and Sturmfels, and by Perkinson, Perlman, and Wilmes. Various other results related to the theory of chip-firing games on graphs also follow from our general techniques and results.

• Non-Archimedean and tropical theta functions. Mathematische Annalen, 2018.
(with Tyler Foster, Joseph Rabinoff, and Alejandro Soto)
• Journal version, arXiv version, free view-only journal version

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 a tropical theta function, and that the tropicalization of a non-Archimedean Riemann theta function is a tropical Riemann theta function, up to scaling and an additive constant. We apply these results to the construction of rational functions with prescribed behavior on the skeleton of a principally polarized abelian variety. We work with the Raynaud--Bosch--Lütkebohmert theory of non-Archimedean theta functions for abelian varieties with semi-abelian reduction.

• Canonical measures on metric graphs and a Kazhdan's theorem. 36 pages, 2017.
(with Chenxi Wu)
• arXiv version
Here is a link to my short note on basic von Neumann algebra theory, written mainly for my own benefit.

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 metric graphs. Kazhdan's theorem for Riemann surfaces describes the limiting behavior of canonical (Arakelov) measures on finite covers in relation to the hyperbolic measure. We will prove a generalized version of this theorem for metric graphs, allowing any infinite Galois cover to replace the universal cover. We will show all such limiting measures satisfy a version of Gauss-Bonnet formula which, using the theory of von Neumann dimensions, can be interpreted as a trace formula. In the special case where the infinite cover is the universal cover, we will provide explicit methods to compute the corresponding limiting (hyperbolic) measure. Our ideas are motivated by non-Archimedean analytic and tropical geometry.

• Effective divisor classes on metric graphs. 29 pages, 2018.
(with Andreas Gross and Lilla Tóthmérész)
• arXiv version

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 representative. This appropriately generalizes the notion of break divisors (in degree equal to genus). Our method of proof is new, even for the special case of break divisors. We provide an algorithm to efficiently compute such semibreak representatives. Semibreak divisors provide the tool to establish some basic properties of effective loci inside Picard groups of metric graphs. We prove that effective loci are pure-dimensional polyhedral sets. We also prove that a 'generic' divisor class (in degree at most the genus) has rank zero, and that the Abel-Jacobi map is 'birational' onto its image. These are analogues of classical results for Riemann surfaces.

• Limits of canonical forms on towers of Riemann surfaces. 19 pages, 2018.
(with Hyungryul Baik and Chenxi Wu)
• arXiv version

We prove a generalized version of Kazhdan's theorem for canonical forms on Riemann surfaces. In the classical version, one starts with an ascending sequence $\{S_n \rightarrow S\}$ of finite Galois covers of a hyperbolic Riemann Surface $S$, converging to the universal cover. The theorem states that the sequence of forms on $S$ inherited from the canonical forms on $S_n$'s converges uniformly to (a multiple of) the hyperbolic form. We prove a generalized version of this theorem, where the universal cover is replaced with any infinite Galois cover. Along the way, we also prove a Gauss--Bonnet type theorem in the context of arbitrary infinite Galois covers.

 email: farbod@math.cornell.edu or farbod@math.ku.dk

 Disclaimer: This is not a publication of Cornell University and Cornell University has not edited or examined the content.