August 3-14, 2020

University of Michigan

"Dimers in Combinatorics and Cluster Algebras" will take place online in the weeks of August 3-7 and 10-14, 2020.

All talks will take place at https://umich.zoom.us/j/91300032965 . The password is the number of ways to tile a 2x5 rectangle with 2x1 dominos, as in the image below. The dominos should be treated as indistinguishable; in other (less fun) words, we are counting dimer covers of a 2x5 rectangle. The password should be written as a number, not spelled out with letters. You can also e-mail an organizer for the password.

We will have informal conversations at 1 PM on conference days. Our current plan is to do this using Gather. The Gather link is https://gather.town/IFS50ReKfPx3xXnl/DIMERS, the password is the same as for the Zoom meeting. We strongly recommend not being on Zoom and Gather at the same time. See this google document for more information.

Our schedule for each day is:

- 10:00-10:40 AM Expository talk
- 11:00-11:40 AM Research talk
- 12:00-12:40 PM Research talk
- 1:00 PM - ??? Informal discussions All times and dates are given in Ann Arbor Michigan, which is in the US Eastern Time Zone and will be observing Daylight Savings Time.

This conference is targeted at people who use dimers in a range of combinatorial areas, including exact enumerations of dimer covers, parametrizations of Grassmannians and positroid cells, cluster algebras, Schur processes and limit shapes, discrete integrable systems, dimer models in quantum field theory and non-commutative resolutions of toric singularities; we aim to welcome graduate students and young researchers begining work in these fields.

Expository: | Research: | ||||

Karin Baur | University of Leeds | Raf Bocklandt | University of Amsterdam | Greg Muller | University of Oklahoma |

Phillippe Di Francesco | University of Illinois at Urbana-Champaign | Mihai Ciucu | Indiana University | Matthew Pressland | University of Leeds |

Thomas Lam | University of Michigan | Sergey Fomin | University of Michigan | Jim Propp | UMass Lowell |

Alex Postnikov | MIT | Pavel Galashin | UCLA | Marianna Russkikh | MIT |

Helen Jenne | University of Oregon | Sibylle Schroll | Univerisity of Leicester | ||

Ray Karpman | Otterbein | Jeanne Scott | Universidad de los Andes | ||

Alastair King | University of Bath | Harold Williams | University of Southern California | ||

Tri Lai | University of Nebraska | Benjamin Young | University of Oregon |

To dimer models with boundary we associate dimer algebras (frozen Jacobian algebras). In certain cases, these algebras correspond to cluster-tilting objects in the cluster category associated to the Grassmannian introduced by Jensen, King and Su. We give various sources for such dimer models (surface triangulations, tilings, strand diagrams) and explain how they provide friezes. We introduce orbifold diagrams which as quotients by a rotational symmetry. We use these to give a construction of skew group categories over the Grassmannian cluster category. (sildes 1) (slides 2) (video 1) (video 2)

Raf Bocklandt

To a dimer model with weights on the vertices of the quiver, we assign a curved gentle A_infty algebra. The representation theory of this curved algebra can be used to model relative Fukaya categories of closed surfaces. We explain the details of this construction and its relation to specular duality for dimer models. (slides) (video)

Mihai Ciucu

In earlier work we showed that the correlation of gaps in dimer systems on the hexagonal lattice is governed, in the fine mesh limit, by Coulomb's law for 2D electrostatics. We also proved that the scaling limit of the average tile orientation is the electric field produced by a system of charges corresponding to the gaps. In this talk we show that the effect of microscopic gap displacements on the correlation of gaps is also controlled by the electric field of the corresponding system of charges. (slides) (video)

Phillippe Di Francesco

We use the T-system/Octahedron equation to describe the connections between cluster algebra, matrix representations, flat connections, non-intersecting paths on networks and dimer/domino tilings. We will first describe the toy model of the A1 T-system, and then generalize it.

In a second part we show how to use the cluster algebra formulation to access asymptotic properties of dimers such as the arctic circle phenomenon. Finally we will describe an alternative, seemingly unrelated method to compute arctic curves, the so-called Tangent Method, which has the advantage of applying to a wider range of systems, including interacting/osculating paths. (slides) (video 1) (video 2)

Sergey Fomin

We call a real plane algebraic curve C expressive if its defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of C. We give a necessary and sufficient criterion for expressivity (subject to a mild technical condition), describe several constructions that produce expressive curves, and relate their study to the combinatorics of plabic graphs, their quivers and links. This is joint work with E. Shustin. (slides) (video)

Pavel Galashin

The stochastic colored six-vertex model is a simple statistical mechanical model defined in terms of pipe dreams. We discover a new hidden symmetry of the model called flip-invariance, which generalizes recent shift-invariance results of Borodin-Gorin-Wheeler. Our proof relies on an equivalence between the stochastic colored six-vertex model and the Yang-Baxter basis of the Hecke algebra. We conclude by discussing the relationship of the model with Kazhdan-Lusztig polynomials and positroid varieties in the Grassmannian. (slides) (video) .

Helen Jenne

We will discuss a new result about the double-dimer model: the partition function for tripartite double-dimer configurations of a planar bipartite graph satisfies a recurrence related to Dodgson condensation (also called the Desnanot-Jacobi identity). A similar identity for the number of dimer configurations of a planar bipartite graph was established nearly 20 years ago by Kuo. Our work was motivated in part by its potential for applications. In the following talk, Ben Young will discuss an application to a problem in Pandharipande-Thomas and Donaldson-Thomas theory. (slides) (video)

Ray Karpman

A collection of k-element subsets of {1,2,…,m} is weakly separated if for any two members I and J of the collection, when the integers 1,2,…,m are arranged in a circle, there is a chord separating I∖J from J∖I. Oh, Postnikov and Speyer proved a correspondence between maximal weakly separated collections and Postnikov's plabic graphs, which give coordinate charts on the Grassmannian of k-planes in m-space. In this talk, we show that maximal "symmetric" weakly separated collections give coordinate charts on the Lagrangian Grassmannian of maximal isotropic subspaces with respect to a symplectic form. We note that Danilov, Karzanov and Koshevoy extended one of our main results in a preprint from July 2020. (slides) (video)

Alastair King

I will explain several consequences of thinking about perfect matchings as modules for a dimer algebra. The context will be, as far as possible, for the dimer model on a disc associated to a general Postnikov diagram. The main goal is to relate the Marsh-Scott partition function to the cluster character formula. (slides) (video)

Tri Lai

"The modifications in size and orientation of the holes in a punctured region would lead to unpredictable changes in the tiling generating function. However, in some particular cases, the tiling generating function is changed by only a simple multiplicative factor. More interestingly, in these cases, the generating functions maybe not simple products. In other words, there are many `sibling' regions, whose tiling generating functions are not given by closed-form formulas, but the ratio of their tiling generating functions is an elegant product formula. We call this the `tiling shuffling phenomenon.' In this talk, I will present various examples of the phenomenon. (slides) (video)

Thomas Lam

A bipartite graph embedded into a disk gives rise to a point in a Grassmannian Gr(k,n), obtained by counting dimer configurations with various boundary conditions. I will discuss the interplay between the coordinate ring of the Grassmannian and the combinatorics of dimers.

One construction of the coordinate ring of the Grassmannian uses a planar graphical model, known as "webs". I will explain a duality between the combinatorics of dimers and of webs, based on joint work with Chris Fraser and Ian Le. (slides) (video 1) (video 2)

Greg Muller

A "quasiperiodic space" is a vector space of sequences which are periodic up to a constant factor. The moduli of such vector spaces are 1-dimensional extensions of Grassmannians, and there are analogous positroid stratifications of the former. I will demonstrate that these "quasiperiodic positroid varieties" have a Y-type cluster structure that is mirror dual to the X-type cluster structure on (the Plucker cone over) the corresponding positroid variety. This structure is defined by extending a version of Postnikov's boundary measurement map to the quasiperiodic case.

Time permitting, I will discuss an alternative construction of this boundary measurement map, which uses the twist to construct a linear recurrence whose solutions are the space in question. This provides a generalization of MGOST's connection between linear recurrences, friezes, and the Gale transform. A motivating goal of this project is to understand the tropical points of these quasiperiodic positroid varieties, as they parametrize the canonical basis of theta functions on (the Plucker cone over) the corresponding positroid variety." (slides) (video)

Alex Postnikov

We will give an introduction to the positive Grassmannian and discuss its links with dimers, plabic graphs, positroids, membranes, and related combinatorial structures. (slides) (video 1) (video 2)

Matthew Pressland

A consistent dimer model on the disk determines a cluster algebra structure on the coordinate ring of a positroid variety in the Grassmannian. I will explain how the dimer model can also be used to give a categorification of this cluster algebra, as a result of certain Calabi-Yau symmetries in the dimer algebra. (slides) (video)

Jim Propp

In this suite of mini-talks I'll advertise three unrelated topics that deserve more attention. Why do formulas for the number of perfect matchings in certain families of graphs display symmetries suggestive of an Ehrhart-type reciprocity? If most random domino tilings of an Aztec diamond manifest a circular arctic region, why is it so hard to give a deterministic recipe for one that does without “cheating”? When one introduces dynamics on the set of lozenge tilings of a regular hexagon of side n, why do so many orbits have size divisible by 3n-1? (slides) (video)

Marianna Russkikh

We establish a correspondence between dimer models on a bipartite graph and circle patterns with the combinatorics of that graph. This correspondence is the key for studying Miquel dynamics, a discrete integrable system on circle patterns. We describe how to construct a 't-embedding' (or a circle pattern) of a dimer planar graph using its Kasteleyn weights, and develop a relevant theory of discrete holomorphic functions on t-embeddings; this theory unifies harmonic functions on T-graphs and s-holomorphic functions coming from the Ising model.

Based on: joint works with D. Chelkak, R. Kenyon, W. Lam, B. Laslier and S. Ramassamy. (slides) (video)

Sibylle Schroll

We give a categorification of the coordinate rings of Grassmannians of infinite rank in terms of graded maximal Cohen-Macauley modules over a hypersurface singularity. This gives an infinite rank analogue of the Grassmannian cluster categories introduced by Jensen, King, and Su. We show that there is a structure preserving bijection between the generically free rank one modules in a Grassmannian category of infinite rank and the Plücker coordinates in a Grassmannian cluster algebra of infinite rank. For this we will introduce a new combinatorial tool, which we call staircase paths, which determine the dimensions of the extension spaces of two generically free rank one modules. This is joint work with Jenny August, Man-Wai Cheung, Eleonore Faber and Sira Gratz. (slides) (video)

Jeanne Scott

In joint work with François David, we consider three discrete operators — the Beltrami-Laplace operator, the David-Eynard Kähler operator, and the conformal Laplacian— each defined over a general Delaunay triangulation. When the triangulation is isoradial all three operators coincide with the critical Laplacian but otherwise they are distinct. Using R. Kenyon's formula for the Green's function of the critical Laplacian, we formally compute a perturbative expansion of the log-determinant of each operator over a Delaunay triangulation obtained by smoothly deforming the embedding of a given isoradial triangulation. For the Beltrami-Laplace and Kähler operators, the second order term in this expansion has an asymptotic form which admits a continuum limit whose value is independent of the initial isoradial triangulation. (Note: The same is true in the case of the conformal Laplacian however some regulation on the density of pairs of concyclic triangles in the initial isoradial triangulation must be imposed.) This continuum limit can be interpreted in light of the Operator Product Expansion in conformal field theory; in particular a "central charge" can be identified. (slides) (video)

Harold Williams

In this talk we explain an interpretation of the Kasteleyn operator of a doubly-periodic bipartite graph from the perspective of homological mirror symmetry. Specifically, given a consistent bipartite graph G in T^2 with a complex-valued edge weighting E we show the following two constructions are the same. The first is to form the Kasteleyn operator of (G,E) and pass to its spectral transform, a coherent sheaf supported on a spectral curve in (C*)^2. The second is to take a certain Lagrangian surface L in T^* T^2 canonically associated to G, equip it with a brane structure prescribed by E, and pass to its homologically mirror coherent sheaf. This lives on a toric compactification of (C*)^2 determined by the Legendrian link which lifts the zig-zag paths of G (and to which the noncompact Lagrangian L is asymptotic). As a corollary, we obtain a complementary geometric perspective on certain features of algebraic integrable systems associated to lattice polygons, studied for example by Goncharov-Kenyon and Fock-Marshakov. This is joint work with David Treumann and Eric Zaslow. (slides) (video)

Benjamin Young

In this talk we describe a combinatorics problem that arises in algebraic geometry - namely, that the topological vertices in Donaldson-Thomas theory and Pandharipande-Thomas theory are equal. We prove a combinatorial version of this conjecture, by translating to the single- and double- dimer models, respectively, and then applying the double-dimer model condensation techniques developed by Jenne. (video)

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Workshop supported by NSF DMS-1854225.