Amrita Acharyya
Title: Set partition statistics on standard, ordered and Type B set partitions
Esther Mae Banaian
Title: Algebras from orbifolds.
Abstract: We discuss two algebras associated to triangulated unpunctured orbifolds with all orbifold points of order three - a gentle algebra and a generalized cluster algebra, in the sense of Chekhov and Shapiro. To each algebra, we associate a map which can be seen as taking arcs on the orbifold to Laurent polynomials. The first map was defined by Caldero and Chapoton; the second is the snake graph map, defined for surfaces by Musiker-Schiffler-Williams and for orbifolds by B.-Kelley. We show that the outputs of these two maps agree. This poster is based on joint work with Yadira Valdivieso.
Jonah Berggren
Title: Consistent Dimer Models on Surfaces with Boundary
Abstract: A dimer model is a quiver with faces embedded in a surface. Dimer models and their consistency conditions have been studied extensively in the case of the disk and torus. We define and investigate notions of consistency for dimer models on general surfaces with boundary which restrict to well-studied consistency conditions in the disk and torus case. We give a definition of weak consistency in terms of the associated dimer algebra and show that it is equivalent to the absence of bad configurations on the strand diagram. We define strong consistency to require weak consistency as well as nondegeneracy, meaning that every arrow of the dimer model lies in some perfect matching. We show that the completed and noncompleted dimer algebra of a strongly consistent dimer model is bimodule internally 3-Calabi-Yau with respect to its boundary idempotent. As a consequence, we obtain an additive categorification of the cluster algebra given by the internal subquiver of the dimer model through the stable Gorenstein-projective module category of the completed boundary algebra.
Seok Hyun
Title: Lozenge tilings of hexagons with intrusions
Abstract: MacMahon’s classical theorem on the number of boxed plane partitions has been generalized in several different directions. One way to generalize the theorem is to view. boxed plane partitions as lozenge tilings of a hexagonal region, then to generalize it by making some holes in the region and counting its tilings. Recently, the number of lozenge tilings of hexagonal regions with several consecutive unit triangles removed from a certain axis was studied. After reviewing some previous results in this research direction, we will present a new result that generalizes the aforementioned result. This is a joint work with Tri Lai.
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.
Ga Yee Park
Title: Minimal skew semistandard Young tableaux and the Hillman--Grassl correspondence
Abstract: Standard tableaux of skew shape are fundamental objects in enumerative and algebraic combinatorics and no product formula for the number is known. Naruse presented a formula as a positive sum over excited diagrams of products of hook-lengths. Shortly after, Morales, Pak, and Panova gave a $q$-analogue of Naruse’s formula for semi-standard tableaux of skew shapes in terms of restricted excited arrays. They also showed, partly algebraically, that the Hillman-Grassl map restricted to skew shapes is the bijection between skew SSYTs and excited arrays. We study the problem of making this argument completely bijective. For a skew shape, we define a new set of semi-standard Young tableaux, called the \emph{minimal SSYT}, that are equinumerous with excited diagrams via a new description of the Hillman--Grassl bijection and have a version of excited moves. The minimal skew SSYT are the natural objects to compare with the terms of the Okounkov-Olshanski formula for counting SYT of skew shape. We prove that the number of summands in the Okounkov-Olshanski formula is larger than the excited diagrams in NHLF.
Mikhail Skopenkov
Title: A generalization of Cardy’s and Schramm’s formulae
Abstract: We study critical site percolation on the triangular lattice. We find the difference of the probabilities of having a percolation interface to the right and to the left of two given points in the scaling limit. This generalizes both Cardy’s and Schramm’s formulae. The generalization involves a new interesting discrete analytic observable and an unexpected conformal mapping.
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.
Trung Vu
Title: Arctic Phenomenon of the $T$-system with Slanted Initial Data
Abstract: We study the $T$-system of type $A_\infty$, also known as the octahedron recurrence/equation, viewed as a $2+1$-dimensional discrete evolution equation. Generalizing previous result for flat initial data by Di Francesco, we consider initial data along parallel “slanted” planes perpendicular to an arbitrary admissible direction $(r,s,t)\in \Z_+^3$. The solution of the T-system is interpreted as the partition function of a dimer model on some suitable “pinecone” graph introduced by Bousquet-Mélou, Propp and West. The T-system formulation and some exact solutions in uniform or periodic cases allow us to explore the thermodynamic limit of the corresponding dimer models and to derive exact arctic curves separating the various phases of the system.
Zhengye Zhou
Title: Orthogonal Dualities and Asymptotics of Dynamic Stochastic Higher Spin Vertex Models.
Abstract: We introduce a new, algebraic method to construct duality functions for integrable dynamic models. This method will be implemented on dynamic stochastic higher spin vertex models, where we prove the duality functions are the $ _3 \varphi_2$ functions. A degeneration of these duality functions are orthogonal polynomial dualities of dynamic ASEP. The method involves using the universal twister of $\mathcal{U}_q(\mathfrak{sl}_2)$, viewed as a quasi--triangular, quasi--$^*$--Hopf algebra. As an application of the duality, we prove that the asymptotic fluctuations of the dynamic stochastic six vertex model with step initial conditions are governed by the Tracy--Widom distribution.