Christian Gaetz
Title: An SL(4) web basis from hourglass plabic graphs
Abstract: The SL(3) web basis is a special basis of certain spaces of tensor invariants developed in the late 90’s by Khovanov and Kuperberg as a tool for computing quantum link invariants. Since then this basis has found connections and applications to cluster algebras, canonical bases, dimer models, and tableau combinatorics. A main open problem has remained: how to find a basis replicating the desirable properties of this basis for SL(4) and beyond? I will describe joint work with Oliver Pechenik, Stephan Pfannerer, Jessica Striker, and Josh Swanson in which we construct such a basis for SL(4). Modified versions of plabic graphs and the six-vertex model and new tableau combinatorics will appear along the way.
Nick Early
Title: Nonplanar Scattering Amplitudes: Oriented Matroids, Positive del Pezzo Moduli and Beyond
Abstract: In this talk, we go for a short romp through recent developments at the intersection of theoretical particle physics with real, complex, and tropical algebraic geometry -- all held together with deep combinatorics from the theory of matroids and their subdivisions. The physical heart is the Cachazo-He-Yuan scattering equations formalism, generalized in work of Cachazo-Early-Guevara-Mizera from the Riemann sphere to higher dimensional projective spaces. We walk through recent joint work with F. Cachazo and Y. Zhang, where we bootstrapped the assignment on one side, to any realizable oriented uniform ranks matroid on n points for n<9, and similarly for rank 4, a differential form and an amplitude, and on the other, to give a purely combinatorial formula for that amplitude, a subfan of the tropical Grassmannian, called the chirotopal tropical Grassmannian. We explore the incorporation (with A. Geiger, M. Panizzut, B. Sturmfels and C. Yun) of the coconic condition into the moduli spaces of n=6,7 points in the projective plane, essentially dividing the positive (tropical) Grassmannian Trop+G(3,6) in half, connecting to del Pezzo surfaces of degree 9-n. We provide a positive parameterization for n=6 a la the one of Speyer-Williams for the positive (tropical) Grassmannian.
Yi-Lin Lee
Title: Off-diagonally symmetric domino tilings of the Aztec diamond
Abstract: We introduce a new symmetry class of domino tilings of the Aztec diamond, called the off-diagonal symmetry class, which is motivated by the off-diagonally symmetric alternating sign matrices introduced by Kuperberg in 2002. We use the method of non-intersecting lattice paths and a modification of Stembridge’s Pfaffian formula for families of non-intersecting lattice paths to enumerate our new symmetry class. The number of off-diagonally symmetric domino tilings of the Aztec diamond can be expressed as a Pfaffian of a matrix whose entries satisfy a nice and simple recurrence relation.
Matthew Nicoletti
Title: Perfect t-embeddings of uniform Aztec diamond graphs
Abstract: In this talk we study a sequence of ’perfect t-embeddings’ of the uniformly weighted Aztec diamond. We believe that these t-embeddings always exist and that they are good candidates to recover the complex structure of big bipartite planar graphs carrying a dimer model. The graphs of the associated origami maps converge to a Lorentz-minimal surface S. We show that in the setup of the uniformly weighted Aztec diamond, perfect t- embeddings can be used to prove convergence of gradients of height fluctuations to those of the Gaussian free field in the intrinsic metric of the surface S. This provides the first example where the general theory of perfect t-embeddings is used to prove convergence to the Gaussian free field.
Nicholas Ovenhouse
Title: Higher Dimer Covers and Continued Fractions
Abstract: There is a construction by Canakci and Schiffler which associates to each finite continued fraction a certain planar graph (called "snake graph"), such that the number of perfect matchings (or "dimer covers") of this graph is equal to the numerator of the rational number represented by the continued fraction. There is also a well-known relationship between continued fractions and certain matrix products in PSL(2,Z). Recently, in joint work with Schiffler, Musiker, and Zhang, we were able to enumerate multi-dimer covers on snake graphs using similar matrix products. Also, we used this to define a new type of higher continued fractions, which allows for a generalized version of the Canakci-Shiffler theorem. I will discuss these enumerative formulas, and some elementary properties of the higher continued fractions.
Sri Tata
Title: 2D Dirac Fermions from Dimers
Abstract: The dimer model and the 2D Dirac fermion are two important examples of critical systems: the former being a stat mech model and the latter being simple quantum field theory. We show that these theories are broadly speaking the same. We show that the partition functions of both in the presence of a background gauge field gives the same quantity in the continuum limit, known mathematically as the isomonodromic tau function. The computation on the dimer side involves lattice-level identities of discrete holomorphic functions and leads to an interesting expansion of the tau function in terms of a series of holomorphic integrals.
Kayla Wright
Title: Double and Triple Dimers and Grassmannian Cluster Algebras
Abstract: In this talk, we will discuss work with Moriah Elkin and Gregg Musiker about a combinatorial model for Grassmannian cluster algebras in the case of Gr(3,n). In Grassmannian cluster algebras, Plücker coordinates give us a subset of the cluster variables and we have a lovely combinatorial understanding of these cluster variables through dimers on plabic graphs. However, some cluster variables are more mysterious and are not merely just Plückers. These non-Plücker cluster variables are given by certain combinations of Plücker coordinates e.g. in . In our work, we aim to give a combinatorial meaning to these expressions by modeling these non-Plücker cluster variables as certain superimpositions of dimers. In particular, in certain Grassmannians, we are able to model these more unwieldy cluster variables using double and triple dimers. We use double and triple dimers, governed by web combinatorics, to give a dimer partition function for the twist of these non-Plücker cluster variables.