The UC Berkeley Combinatorics Seminar

DATE	SPEAKER	TITLE (click to show abstract)
January 29th	Annie Raymond, UMass Amherst	The generalized Pitman-Stanley polytope The eponymous Pitman-Stanley polytope introduced in 1999 is related to plane partitions of skew shape with entries 0 and 1. It has been well studied because of its connections to probability, parking functions, generalized permutahedra, and flow polytopes. We consider a generalization of this polytope related to plane partitions with entries 0, 1, ... , m. We show that this polytope can also be realized as a flow polytope of a grid graph. We give multiple characterizations of its vertices in terms of plane partitions of skew shape and integer flows. We also study formulas for the volume as well as for the number of lattice points and vertices of this polytope. This is joint work with William Dugan, Maura Hegarty and Alejandro Morales.
February 5th	Eugene Gorsky, UC Davis	Skewing formulas for Delta Conjecture Delta Conjecture of Haglund, Remmel and Wilson is the identity describing the action of Macdonald operators on elementary symmetric functions. The conjecture was proved independently by Blasiak-Haiman-Morse-Pun-Seelinger, and D'Adderio-Mellit. In this talk, I will explain yet another proof of Delta Conjecture by applying a Schur skewing operator to both sides of another identity known as Rational Shuffle Theorem. I will also describe geometric models for Delta conjecture and Rational Shuffle Theorem using affine Springer fibers. This is a joint work with Sean Griffin and Maria Gillespie.
February 12th	Stephanie van Willigenburg, UBC	The $e$-positivity of chromatic symmetric functions We will meet the chromatic symmetric function, dating from 1995, which is a generalization of the chromatic polynomial. We will also hear about a famed problem regarding it, called the Stanley-Stembridge (3+1)-free problem. This has been the focus of much research lately including resolving another problem of Stanley of whether the (3+1)-free problem can be widened. The resulting paper on the latter problem was recently awarded the 2023 MAA David P. Robbins Prize, and we will hear this story too. This talk requires no prior knowledge and will be suitable for a broad audience. Pre-talk Title: Noncommutative chromatic symmetric functions revisited Pre-talk Abstract: The chromatic polynomial of a graph dates from 1912, when Birkhoff created it while trying to solve the 4-colour problem. In 1995 Stanley generalized it to the chromatic symmetric function, which stored more information about the graph. This was then generalized again to noncommuting variables, by Gebhard-Sagan in 2001. In this talk we will meet these newest chromatic functions, the space they live in, and discover new results for them. No background is needed for this talk and it is suitable for students.
February 19th	Daniel Kráľ, Masaryk University -> Leipzig University	Extremal problems with quasirandom constraints A combinatorial structure is said to be quasirandom if it resembles a random structure in a certain robust sense. Classical work of Rödl, Thomason, Chung, Graham and Wilson from the 1980s led to the notion of quasirandom graphs, which is nowadays considered to be well-understood. In this talk, we first review classical and recent results on quasirandom combinatorial structures, and we then focus on problems from extremal combinatorics with additional quasirandom constraints. The study of such extremal problems was initiated by Erdős and Sós in the early 1980s, however, substantial progress appeared only recently with use of the hypergraph regularity method, which was independently developed by Kohayakawa, Nagle, Rödl, Schacht and Skokan, and Gowers. We will present some of recent results, e.g. a solution of a 40-year-old problem of Erdős and Sós concerning the uniform Turán densities of K_4^3-, introduce methods developed that have been developed to tackle extremal problems with quasirandom constraints, and discuss some of many open problems concerning extremal problems with quasirandom constraints. The talk will include results obtained jointly with various collaborators, particularly, with Matija Bucić, Jacob W. Cooper, Frederik Garbe, Daniel Iľkovič, Filip Kučerák, Ander Lamaison, Samuel Mohr, David Munhá Correia and Gábor Tardos.
February 26th	Talia Blum, Stanford	Bosonic bicolored solvable lattice models The study of solvable lattice models originated in statistical mechanics, and has since formed rich connections with areas of math including combinatorics, probability, and representation theory. Lattice models are called solvable when they can be studied using the Yang-Baxter equation. The partition function of a system, which captures global information about the lattice model, is at the heart of many of these connections with other areas. To compute the partition function, one method is to identify boundary conditions that give systems with a unique state, from which other systems can be computed by Demazure recursion relations coming from the Yang-Baxter equation. Bosonic and colored variants of solvable lattice models have been studied in recent years by Aggarwal, Borodin, Brubaker, Buciumas, Bump, Gustafsson, Naprienko, Wheeler, and others. We will define a class of these models which are bosonic and include two types of colors, generalizing the now widely-studied colored models. These bicolored bosonic models satisfy the Yang-Baxter equation, which gives a four-term recurrence relation on the partition function. We will give conditions on the number of states of the model based on boundary conditions in terms of the Bruhat order, and discuss connections with Gelfand-Tsetlin patterns.
March 5th	Warut Thawinrak, UC Davis	Ehrhart polynomials of generalized permutohedra from A to B I’ll show how to derive a formula for the Ehrhart polynomials for the type-$B$ generalized permutohedra, which offer a more concise alternative to the recent formula obtained by Eur, Fink, Larson, and Spink from their study of delta-matroids. My approach utilized some of the techniques and tools introduced around 20 years ago by Postnikov from his study of generalized type-$A$ permutohedra, a family of polytopes that interconnects with many mathematical concepts such as matroids, graphs, and Weyl groups. If time permits, I’ll discuss about some progress on computing their h-polynomials.
March 12th	Peter Winkler, Dartmouth and SLMath	Optimizing on the Fly How should you make decisions in an uncertain world, in which you can change your mind later? Suppose there are several tokens taking random walks, and you one of them to reach a target state ASAP. You can choose any token to take a move, and if you don't like where it goes, switch to another one. Amazingly, there's an efficiently-calculable strategy for optimal play. Joint work with Ioana Dumitriu and Prasad Tetali, based on great stuff from John Gittins and Richard Weber.
March 19th	Sara Billey, University of Washington	Enumerating Quilts of Alternating Sign Matrices and Generalized Rank Functions We present new objects called quilts of alternating sign matrices with respect to two given posets. Quilts generalize several commonly used concepts in mathematics. For example, the rank function on submatrices of a matrix gives rise to a quilt with respect to two Boolean lattices. When the two posets are chains, a quilt is equivalent to an alternating sign matrix and its corresponding corner sum matrix. Such rank functions are used in the definition of Schubert varieties in both the Grassmannian and the complete flag manifold. Quilts also generalize the monotone Boolean functions counted by the Dedekind numbers, which is known to be a #P-complete problem. Quilts form a distributive lattice with many beautiful properties and contain many classical and well known sublattices, such as the lattice of matroids of a given rank and ground set. While enumerating quilts is hard in general, we prove two major enumerative results, when one of the posets is an antichain and when one of them is a chain. We also give some bounds for the number of quilts when one poset is the Boolean lattice. Several open problems will be given for future development. This talk is based on joint work with Matjaz Konvalinka in arxiv:2412.03236.
March 26th	No seminar - spring break
April 2nd	Matthew Nicoletti, UC Berkeley	Perfect t-embeddings and Lozenge Tilings We construct and study the asymptotic properties of "perfect t-embeddings" of uniformly weighted hexagon graphs. Hexagon graphs are subgraphs of the honeycomb lattice, and the corresponding dimer model is equivalent to the model of uniformly random lozenge tilings of the hexagon. We provide exact formulas describing the perfect t-embeddings of these graphs, and we use these to prove the convergence of naturally associated discrete surfaces (coming from the "origami maps") to a maximal surface in Minkowski space carrying the conformal structure of the limiting Gaussian free field (GFF). The emergence of such a maximal surface is predicted to hold for a large class of dimer models by Chelkak, Laslier, and Russkikh. In addition, we check all conditions of a theorem of Chelkak, Laslier, and Russkikh which uses perfect t-embeddings to prove convergence of height fluctuations to the GFF, and thus we complete give a new proof, via t-embeddings, of convergence to the GFF. This is based on joint work with Marianna Russkikh and Tomas Berggren.
April 9th	Greta Panova, USC	Algebra meets probability: permutons from pipe dreams via integrable probability Pipe dreams are tiling models originally introduced to study objects related to the Schubert calculus and K-theory of the Grassmannian. They can also be viewed as ensembles of random lattice walks with various interaction constraints. In our quest to understand what the maximal and typical algebraic objects look like, we revealed some interesting permutons. The proofs use the theory of the Totally Asymmetric Simple Exclusion Process (TASEP). Deeper connections with free fermion 6 vertex models and domino tilings of the Aztec diamond and Alternating Sign Matrices allow us to describe the extreme cases of the original algebraic problem. This is based on joint work with A. H. Morales, L. Petrov, D. Yeliussizov.
April 16th	Kayla Wright, Oregon
April 23rd	Stephan Pfannerer, Waterloo
Friday April 25th, 3:10pm-4:00pm, Evans 891 (no pre-talk)	Joshua Swanson, USC	Tanisaki witness relations In recent years, anti-commuting analogues of the classical coinvariant algebra of the symmetric group have received significant attention. The main problem is determining the multigraded Frobenius series. Despite great progress, this problem remains open even for one set of commuting and one set of anti-commuting variables. We present a novel approach to the problem which constructs a filtration with Tanisaki quotients as composition factors. These quotients arise in the theory of Springer fibers. The key difficulty in this approach is finding sufficiently many relations between certain harmonic differential forms, which we call "Tanisaki witness relations". We give two highly non-trivial families of such relations representing opposite extremes. They exhibit interesting structure, such as graded-non-negativity. Our methods are entirely combinatorial, and further progress seems likely to hinge on discovering more refined algebraic or geometric insight.
April 30th	Jessica Striker, NDSU	Pipe dream, pattern, and polytope perspectives on alternating sign matrices and plane partitions Alternating sign matrices are certain {0,1,-1}-matrices known to be equinumerous with plane partitions in the totally symmetric self-complementary symmetry class (TSSCPP), but no meaningful bijection is known. In joint work with Daoji Huang, we give such a bijection in the reduced, 1432-avoiding case, using the bijection of Gao and Huang between reduced bumpless pipe dreams and reduced pipe dreams. In joint work with Mathilde Bouvel and Rebecca Smith, we discuss the related notion of key-avoidance in alternating sign matrices. In joint work with Vincent Holmlund, we transform TSSCPPs to {0,1,-1}-matrices we call magog matrices, and investigate their enumerative and geometric properties.

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