Research

We introduce twisted triple crossing diagram maps, collections of points in projective space associated to bipartite graphs on the cylinder, and use them to provide geometric realizations of the cluster integrable systems of Goncharov and Kenyon constructed from toric dimer models. Using this notion, we provide geometric proofs that the pentagram map and the cross-ratio dynamics integrable systems are cluster integrable systems. We show that in appropriate coordinates, cross-ratio dynamics is described by geometric R-matrices, which solves the open question of finding a cluster algebra structure describing cross-ratio dynamics.

We prove a correspondence between Ising models in a torus and the algebro-geometric data of a Harnack curve with a certain symmetry and a point in the real part of its Prym variety, extending the correspondence between dimer models and Harnack curves and their Jacobians due to Kenyon and Okounkov.

Cactus networks were introduced by Lam as a generalization of planar electrical networks. He defined a map from these networks to the Grassmannian Gr($n+1,2n$) and showed that the image of this map, \(\mathcal X_n\) lies inside the totally nonnegative part of this Grassmannian. In this paper, we show that \(\mathcal X_n\) is exactly the elements of \(\operatorname{Gr}(n+1,2n)\) that are both totally nonnegative and isotropic for a particular skew-symmetric bilinear form. For certain classes of cactus networks, we also explicitly describe how to turn response matrices and effective resistance matrices into points of \(\operatorname{Gr}(n+1,2n)\) given by Lam's map. Finally, we discuss how our work relates to earlier studies of total positivity for Lagrangian Grassmannians.

We determine which bipartite graphs embedded in a torus are move-reduced. In addition, we classify equivalence classes of such move-reduced graphs under square/spider moves. This extends the class of minimal graphs on a torus studied by Goncharov--Kenyon, and gives a toric analog of Postnikov's results on a disk.

We construct an electrical-network version of the twist map for the positive Grassmannian, and use it to solve the inverse problem of recovering conductances from the response matrix. Each conductance is expressed as a biratio of Pfaffians as in the inverse map of Kenyon and Wilson; however, our Pfaffians are the more canonical \\$B\\$ variables instead of their tripod variables, and are coordinates on the positive orthogonal Grassmannian studied by Henriques and Speyer.

A biperiodic planar resistor network is a pair $(G,c)$ where $G$ is a graph embedded on the torus and $c$ is a function from the edges of $G$ to non-zero complex numbers. Associated with the discrete Laplacian on a biperiodic planar network is its spectral data: a triple $(C,S,\nu)$, where $C$ is a curve, and $S$ is a divisor on it, which we show is a point in the Prym variety of $C$. We give a complete classification of networks (modulo a natural equivalence) in terms of their spectral data. The space of networks has a large group of cluster automorphisms arising from the Y-$\Delta$ transformation, giving discrete cluster integrable systems. We show that these automorphisms are integrable in the algebro-geometric sense: under the spectral transform, they become translations in the Prym variety.

In 2015, Vladimir Fock proved that the spectral transform, associating to an element of a dimer cluster integrable system its spectral data, is birational by constructing an inverse map using theta functions on Jacobians of spectral curves. We provide an alternate construction of the inverse map that involves only rational functions in the spectral data.

Cluster integrable systems are a broad class of integrable systems modelled on bipartite dimer models on the torus. Many discrete integrable dynamics arise by applying sequences of local transformations, which form the cluster modular group of the cluster integrable system. This cluster modular group was recently characterized by George and Inchiostro. There exist some discrete integrable dynamics that make use of non-local transformations associated with geometric \(R\)-matrices. In this article we characterize the generalized cluster modular group -- which includes both local and non-local transformations -- in terms of extended affine symmetric groups. We also describe the action of the generalized cluster modular group on the spectral data associated with cluster integrable systems.

Associated to a convex integral polygon \(N\) is a cluster integrable system \( \mathcal X_N \) constructed from the dimer model. We compute the group \(G_N\) of symmetries of \(\mathcal X_N\), called the (2-2) cluster modular group, showing that it is a certain abelian group conjectured by Fock and Marshakov. Combinatorially, non-torsion elements of \(G_N\) are ways of shuffling the underlying bipartite graph, generalizing domino-shuffling. Algebro-geometrically, \(G_N\) is a subgroup of the Picard group of a certain algebraic surface associated to \(N\).

Groves are spanning forests of a finite region of the triangular lattice that are in bijection with Laurent monomials that arise in solutions of the cube recurrence. We introduce a large class of probability measures on groves for which we can compute exact generating functions for edge probabilities. Using the machinery of asymptotics of multivariate generating functions, this lets us explicitly compute arctic curves, generalizing the arctic circle theorem of Petersen and Speyer. Our class of probability measures is sufficiently general that the limit shapes exhibit all solid and gaseous phases expected from the classification of ergodic Gibbs measures in the resistor network model.

Associated to a convex integral polygon \(N\) in the plane are two integrable systems: the cluster integrable system of Goncharov and Kenyon constructed from the planar dimer model, and the Beauville integrable system, associated with the toric surface of \(N\). There is a birational map, called the spectral transform, between the phase spaces of the two integrable systems. When \(N\) is the triangle \(\text{Conv}\{(0,0),(d,0),(0,d)\}\), we show that the spectral transform is a birational isomorphism of integrable systems.

We introduce triple crossing diagram (TCD) maps, which encode projective configurations of points and lines, as a unified framework for constructions arising in various areas of geometry, such as discrete differential geometry, discrete geometric dynamics and hyperbolic geometry. We define two types of local moves for TCD maps, one of which is governed by the discrete Schwarzian KP (dSKP) equation, and establish their multi-dimensional consistency. We construct two distinct cluster structures on the space of TCD maps, called projective and affine cluster structures, and show that they are related via an operation called section. This framework organizes and unifies a wide range of examples, including Q-nets, Darboux maps, line complexes, T-graphs, t-embeddings, triangulations and geometric discrete integrable systems such as the pentagram map and cross-ratio dynamics, which are further developed in companion papers.