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In a widely circulated manuscript from the 1990s, I.G. Macdonald introduced certain higher-rank analogs of the classical hypergeometric functions $_pF_q$, which are expressed as explicit series in Jack and Macdonald polynomials in one and two sets of variables. For special choices of parameters, these series reduce to the hypergeometric functions of matrix argument introduced earlier by C. Herz and A.T. James, which have numerous applications in number theory, multivariate statistics, signal processing, and random matrix theory.

The classical hypergeometric functions are solutions to the hypergeometric differential equation. Macdonald raised the problem of providing an analogous characterization for higher-rank functions by means of differential equations. Over the years, this problem was solved for a small number of cases where $p$ and $q$ are at most $3$. However, as the operators become increasingly complicated, the general problem remained open for 40 years. In this talk, we will present a complete solution. This is joint work with Hong Chen.

Motivated by the representation theory of finite-dimensional algebras, we recently investigated the notions of left modularity and extremality in (completely) semidistributive lattices. For lattices of torsion classes, we obtain a simultaneous characterization of left modularity and extremality in terms of the behavior of certain indecomposable modules, called bricks. Our results extend the classical theory beyond the realm of finite lattices, while remaining within the framework of (completely) semidistributive lattices. Time permitting, I will also discuss extensions of these results to arbitrary infinite lattices that are completely semidistributive and weakly atomic. This talk is based on recent joint work with Sota Asai, Osamu Iyama, and Charles Paquette.

During the pre-seminar, after reviewing some basic notions from the representation theory of finite-dimensional algebras through the language of quiver representations, I will recall the classical notion of directedness and compare it with our new generalization, called brick-directedness. I will then discuss some basic properties of torsion classes and compare the notions of splitting and brick-splitting torsion pairs. No prior background in the representation theory of algebras will be assumed.

Positroids were introduced by Postnikov in 2006 as a special class of matroids with nice combinatorial properties. Since 2008, starting with Suho Ho, several attempts have been made to describe the poset of quotients for this class of matroids in a combinatorial way. However, these descriptions are incomplete and always come from the same perspective. That is why we will explore new combinatorial objects and the context in which they arise (magic, polytopes, and antisymmetric algebras) to see if it is possible to describe this poset.

The satisfiability threshold of a random constraint satisfaction problem (CSP) is the density of constraints at which a random CSP instance transitions from being satisfiable to unsatisfiable with high probability. Much of the research on well known CSPs, including the $k$-SAT problem, $k$-XORSAT problem, hypergraph colouring, and systems of linear equations, has focused on determining satisfiability thresholds.

In this talk we consider systems of linear equations over finite commutative rings as CSPs, and build on the work of Ayre, Coja-Oghlan, Gao, and Müller, who determined the satisfiability threshold for random linear equations over a finite field. We determine when the satisfiability threshold is linear in the number of variables, and show that any linear threshold over a principal ideal ring coincides with the (unique) linear threshold over fields. We also determine the satisfiability threshold for some examples of non-principal ideal rings.

This is joint work with Jane Gao.

Around 1900 Young and Frobenius (independently, and through very different techniques) obtained a formula for the dimensions of the irreducible representations of the symmetric group. Some 53 years later, Frame, Robinson and Thrall noticed that the Young-Frobenius formula simplified into the now famous hook length formula. Nowadays there are many proofs, but the hook length formula remains something of a mystery, as if some deeper understanding lies just out of reach. One aspect of this mystery is that none of the proofs seem to indicate how one might come up with the formula in the first place, other than just guessing.

I will attempt to answer that question. It is an improbable tale that meanders through scenes of Young symmetrizers, Schur-Weyl duality, Weyl algebras, elementary combinatorics, and Plücker relations. All because Google's AI gave me a very obviously wrong answer when I was trying to find out the square of a Young symmetrizer.

The sign coherence of $c$-vectors is one of the fundamental theorems of cluster algebras with principal coefficients. Gekhtman and Nakanishi posed the Asymptotic Sign Coherence Conjecture for cluster algebras with arbitrary coefficients, which says sign coherence should eventually hold in any sufficiently generic infinite mutation sequence. We prove that for cluster algebras from quivers of arbitrary rank, their conjecture holds with probability 1 for a random mutation sequence. Our results also establish the conjecture in full generality for many families of quivers. This is joint work with Amanda Burcroff.

When you encounter an algebraic variety in the wild, you might ask: What do its components look like? Which components are smooth? How do the components intersect? For Springer fibres, answers to these questions are only known in some very special cases. This is particularly surprising because other aspects Springer fibres have been studied for the past 50 years and they appear throughout combinatorics and adjacent areas. For just one example: Hall–Littlewood polynomials can be obtained from the cohomology of Springer fibres by taking graded Frobenius characteristic.

One classical theorem says that the components of Springer fibers are indexed by standard Young tableaux. In this talk, we will discuss the benefits of instead using webs to index the components in two cases: the "two row” case, and our recent contributions in the “two column” case. We will see that in these cases, webs both characterize and describe the smooth components of Springer fibres, and give a geometric interpretation of rotation of webs.

The totally asymmetric simple exclusion process (TASEP) is a finite Markov chain of particles hopping between adjacent sites on a one-dimensional lattice. The multispecies TASEP is a generalization in which particles have different types. These processes have interesting connections to algebraic combinatorics: the stationary distribution of the TASEP on a circle is connected to Macdonald polynomials at $t=0$, whereas the stationary distribution of the open-boundary TASEP is connected to Koorwinder polynomials at $t=0$.

Multiline queues were introduced by Ferrari and Martin (2007) to compute the stationary distribution of the multispecies TASEP on a circle. It has been a long-standing open problem to find a combinatorial formula for the stationary distribution of the multispecies TASEP with open boundaries. Recently, we studied the combinatorics of Ferrari–Martin multiline queues using type A crystals. In this talk, we use crystals of type C to construct an analog of multiline queues and give a combinatorial formula for the stationary distribution of the multispecies open-boundary TASEP for a certain specialization of the boundary parameters. This is joint work with Olya Mandelshtam.

Standard tableaux are certain grids of numbers that lead a double life in algebraic combinatorics, with distinct roles in geometry and in representation theory. Extending the geometry to K-theory led to a corresponding extension of the combinatorics to a theory of increasing tableaux. I will discuss a longstanding plot by such tableaux to prevent me from explicating their combinatorial dynamics. Despite their reticence, we seem to be uncovering that these tableaux also have a mysterious second life in representation theory.

An orbital scheme D of type $M^2 = 0$ is the closure of a conjugacy class of some set of $n \times n$ upper triangular matrices which are nilpotent of order 2. The geometric components of the orbit scheme are called orbital varieties of type $M^2 = 0$, and recently their invariants have been connected to statistical mechanics. In the setting of $M^2 = 0$, there are combinatorial methods for studying these invariants via the action of the Borel group of upper triangular invertible matrices. In this talk, we introduce a new pipe dream framework for computing and understanding these invariants. This is joint work with Megumi Harada, Illya Kierkosz, Allen Knutson, Emma Naguit, Brett Nasserden, Naveena Rangunathan, and Adam van Tuyl.

The theory of log concave polynomials has recently been developed to study objects and problems in combinatorics and other subfields in mathematics. Particular classes of log concave polynomials called Lorentzian polynomials and denormalized and dually Lorentzian polynomials have been used to prove log concavity statements for various combinatorial sequences. This includes the strongest form of Mason's log concavity conjecture on the independent sets of matroids and the log concavity of sequences of Kostka numbers.

In this talk, we develop an analogous class of power series called denormalized Lorentzian (DL) Laurent series. This class is the natural generalization of DL polynomials to homogeneous power series with the benefit of capturing a number of combinatorial generating series including the Kostant partition function for integer flows of directed graphs. We then analyze specific DL Laurent series to obtain new bounds for integral flows on general directed acyclic graphs and new bounds for the dimensions of weight spaces of parabolic $\mathfrak{sl}_{n+1}(\mathbb{C})$ Verma modules.

Tensor integrals are the generating functions of triangulations of pseudo-manifolds. Such triangulations are constructed by gluing simplices along facets. These generating functions satisfy an infinite system of recursive equations called the Dyson-Schwinger equations, derived by reclusively gluing together triangulations. Such integrals also satisfy positivity constraints. By combining the Dyson-Schwinger equations and positivity constraints in a process called bootstrapping we are able to deduce known results for the generating functions of certain classes of triangulations as well as find new explicit formulae. This talk is based on joint work with Carlos I. Perez-Sanchez and Brayden Smith.

For all positive integers $\ell$ and $r$, we determine the maximum number of elements of a simple rank-$r$ positroid without the rank-$2$ uniform matroid $U_{2,\ell+2}$ as a minor, and characterize the matroids with the maximum number of elements. This result continues a long line of research into upper bounds on the number of elements of matroids from various classes that forbid $U_{2,\ell+2}$ as a minor, including works of Kung, of Geelen–Nelson, and of Geelen–Nelson–Walsh. This is the first paper to study positroids in this context, and it suggests methods to study similar problems for other classes of matroids, such as gammoids or base-orderable matroids. This project is based on joint work with Zach Walsh.

Serrano (2010) introduced the shifted plactic monoid, governing Haiman's (1989) mixed insertion algorithm, as a type B analogue of the classical plactic monoid that connects jeu de taquin of Young tableaux with the Robinson–Schensted–Knuth insertion algorithm. Serrano proposed a corresponding definition of skew shifted plactic Schur functions. Cho (2013) disproved Serrano's conjecture regarding this definition, by showing that the functions do not live in the desired ring and hence cannot provide an algebraic interpretation of tableau rectification or of the corresponding structure coefficients. Cho asked for a new definition with particular properties. We introduce such a definition and prove that it behaves as desired. We also introduce the first jeu de taquin theory that computes mixed insertion. This is joint work with Oliver Pechenik.

The order dimension of a partially ordered set (poset), which is often difficult to compute, is a measure of its complexity. Dilworth proved that the dimension of a distributive lattice is the width of its subposet on its join-irreducible elements. We generalize this result by showing that the dimension of a semidistributive extremal lattice is the chromatic number of the complement of its Galois graph (see Section 3.5 of arXiv:2511.18540). We apply this result to prove that the dimension of the lattice of torsion classes of a gentle tree with $n$ vertices is equal to $n$. No advanced background is required to follow the talk.

We use a new combinatorial construction and a sign-reversing involution to simplify an alternating sum that arises naturally in intersection theory on moduli spaces of curves. In particular, it is well known that a product of $\psi$ classes on the moduli space $\bar{M}_{0,n}$, the most commonly studied compactification of the moduli space $M_{0,n}$ of choices of $n$ distinct marked points on $\mathbb{P}^1$, is equal to a multinomial coefficient and has many natural combinatorial interpretations.

There are similar $\psi$ class products on other compactifications of $M_{0,n}$, including the "multicolored" spaces, in which the answer is a positive integer and yet only signed summation formulas were known. We simplify the alternating sum formula in the multicolored case to give a positive combinatorial rule, and discuss some applications of the formula. This is joint work with Vance Blankers and Jake Levinson.

In 1995 Kuperberg introduced a collection of trivalent web bases encoding tensor invariants of $U_q(\mathfrak{sl}_3)$. Extending these bases to general $\mathfrak{sl}_r$ has remained an open problem. We present a solution for the case $r=4$ by introducing hourglass plabic graphs, a new generalization of Postnikov's plabic graphs. This is joint work with Christian Gaetz, Oliver Pechenik, Jessica Striker, and Joshua Swanson.

In the space of type A quiver representations, putting rank conditions on the maps cuts out subvarieties called "open quiver loci." These subvarieties are closed under the group action that changes bases in the vector spaces, so their closures define classes in equivariant cohomology, called "quiver polynomials." Knutson, Miller, and Shimozono found a pipe dream formula to compute these polynomials in 2006. To study the geometry of the open quiver loci themselves, we might instead compute "equivariant Chern-Schwartz-MacPherson classes," which interpolate between cohomology classes and Euler characteristic. I will introduce objects called "chained generic pipe dreams" that allow us to compute these CSM classes combinatorially, and along the way give streamlined formulas for quiver polynomials.

Causal Set Theory (CST) is a theory of quantum gravity where spacetime is taken to be a locally finite poset, called a causal set. A central problem in CST is to determine physically relevant properties from the purely combinatorial information of the causal set. In 2014 Glaser and Surya demonstrated that the distribution of interval sizes of a causal set sprinkled into a region of Minkowski space contains information about the dimension of the underlying spacetime. In 2026 Surya showed that this distribution can be used to define a “closeness” function on causal sets that distinguishes by dimension and global topology. In this talk we present work motivated by these results which investigates the more general notion of convex sets, instead of intervals, of a poset. First, we will introduce a generating polynomial for convex sets in a finite poset and explore some of its properties. We will then show that this polynomial is a complete invariant for the family of series-parallel posets. Lastly, we discuss early results on the utility of this polynomial in CST. The pre-seminar will introduce relevant background on CST.

Cosmological correlation functions probe the quantum origins of structure in the universe and are a prototype for many calculations in physics. I will share recent work on the complexes, fans and polytopes associated to these functions based on arXiv:2602.21194 and ongoing work. Two structures lie at the heart of this story: (1) incidence relations between chains and anti-chains, (2) a notion of "nested sets of nested sets". Both structures can be studied for an arbitrary lattice, but building sets of the boolean lattice are my main motivation.

Webs are graphical objects that give a tangible, combinatorial way to compute and classify tensor invariants. Recently, Gaetz, Pechenik, Pfannerer, Striker, and Swanson (arXiv:2306.12501) found a rotation-invariant web basis for $\mathrm{SL}_4$, as well as its quantum deformation $U_q(\mathfrak{sl}_4)$, and a bijection between move equivalence classes of $\mathrm{SL}_4$-webs and fluctuating tableaux such that web rotation corresponds to tableau promotion. They also found a bijection between the set of plane partitions in an $a\times b\times c$ box and a benzene move equivalence class of $\mathrm{SL}_4$-webs by determining the corresponding oscillating tableau. In this talk, I will similarly find the oscillating tableaux corresponding to plane partitions in certain symmetry classes by characterizing them via certain lattice words. A dynamical action on tableaux, called promotion, corresponds to rotation of $\mathrm{SL}_4$-webs. I will show how promotion of certain subtableau align with rotation of their respective webs. I will also show that this correspondence maps through a projection to either $\mathrm{SL}_2$ or $\mathrm{SL}_3$ webs. Moreover, that this projection is exactly a partial evaluation of webs. This talk will be given through the lens of the combinatorics of webs and tableau. Some of this work is joint with Jessica Striker.

Exceptional collections are a powerful tool for understanding the derived category of coherent sheaves on algebraic varieties, with applications in commutative algebra, birational geometry, and mirror symmetry. While the existence of exceptional collections is known for classical varieties such as Grassmannians and flag varieties, constructing explicit collections for toric varieties presents challenges in combinatorial algebraic geometry. In this talk I will describe a computational approach to constructing full strong exceptional collections consisting of complexes of line bundles for toric varieties.
No background in derived categories is assumed.

A cosmological correlator is an Euler integral, associated to a graph $G$, which encodes information about the state of the early universe. Evaluation of these integrals is extremely challenging, even in simple cases. However, it turns out the integrand can be identified with the so-called canonical form of the cosmological polytope, revealing a rich combinatorial structure and allowing the application of techniques from commutative algebra. I'll sketch my contribution to this story and advertise the fledgling field of positive geometry, which seeks to generalize the notion of canonical forms to geometric objects more exotic than polytopes.

The set of dimers (aka perfect matchings) of a connected bipartite plane graph $G$ is a distributive lattice, as shown by Propp. The order relation on this lattice comes from the "height" of a dimer, which is a vector of nonnegative integers. In this talk, I'll focus on the dimer face polynomial of $G$, which is the height generating function of all dimers of $G$. This polynomial has close connections to knot invariants on the one hand, and cluster algebras on the other. I'll discuss joint work with Mészáros, Musiker and Vidinas in which we explore these connections. No knowledge of knot theory or cluster algebras will be assumed.

Let $u$ be a word over the positive integers. Motivated in part by a question of representation theory, we study the centralizer set $C(u)$, which consists of words $w$ for which $wu$ and $uw$ are Knuth-equivalent. In particular, we characterize $C(u)$ when $u$ has few letters or is a certain type of decreasing sequence, and we address related enumerative questions.

The goal of this talk is to introduce Calabi-Yau toric geometry from a purely combinatorial perspective, through the rich structures carried by an embedded bipartite graph on a torus called a dimer model.

In the pre-talk, we will demonstrate how matchings, tilings, and quivers naturally encodes deep geometric and physical content. No background in algebraic geometry will be assumed; instead, we’ll build the story from the ground up, with an emphasis on visual intuition and discrete structures.

Next, I will discuss how dimer models simultaneously encode the data of a toric Calabi-Yau singularity and its mirror dual, unifying perspectives from the B-model and A-model of homological mirror symmetry through noncommutative algebras. We will proceed with some recent developments and open problems, including connections to the symplectic geometry of Landau-Ginzburg models and Van den Bergh’s noncommutative resolution conjecture for toric Gorenstein affine singularities. To conclude the talk, we will discuss an example of a higher dimensional generalization and its connection to my recent research works.

This talk is also partially based on two undergraduate research projects in the present semester, where we study variations of dimer models of the same lattice polygon from different choices in the fast inverse algorithm, and implications on the two sides of mirror symmetry:
- (DRP) Dimer Variation and Geometry of Landau-Ginzburg Models, with Kenneth Xiao and Elizabeth Cai (A-model)
- (NSERC USRA) Dimer Variation and NCCR Mutations, with Filip Mildrag and Elana Kalashnikov (B-model)

A planar stuffed map is an embedding of a graph into the 2-sphere, considered up to orientation-preserving homeomorphisms, such that the complement of the graph is a collection of disjoint topologically connected components that are each homeomorphic to the 2-sphere with multiple boundaries. This is a generalization of planar maps whose complement of the graph is a collection of disjoint topologically connected components that are each homeomorphic to a disc. In this talk I will outline my work in constructing a bijection between bipartite planar stuffed maps and collections of integer-labelled trees connected by hyperedges such that they form a hypertree. This bijection directly generalizes the Bouttier-Di Franceso-Guitter bijection between bipartite planar maps and mobiles. Additionally, we show that the generating functions of these trees of mobiles satisfy both an algebraic equation, generalizing the case of ordinary planar maps, and a new functional equation. As an example, we explicitly enumerate a class of stuffed quadrangulations.

The distance matrix of a tree appears in a range of contexts, from phylogenetics to physical chemistry. I will describe a new result about the spectrum of this matrix, one that gives affirmative answers to two questions about matroid positivity properties. These are both strengthenings of Mason's conjectures about the log-concavity of sequence of numbers of independent sets of a matroid, proposed by Igor Pak and by Giansiracusa, Rincón, Schleis, Ulirsch, respectively.

This is joint work with Federico Ardila, Sergio Cristancho, Chris Eur, June Huh, and Botong Wang.

Jacobi’s classical result expresses the generating function for closed walks at a vertex of a graph as the ratio of two characteristic polynomials. We find a hypergraph analogue of this relationship, showing that the same rational function for a hypergraph also counts combinatorial substructures within it, called infragraphs. We use Viennot’s Heaps of Pieces framework to establish this result for the adjacency tensor of a hypergraph. As an immediate consequence, we obtain an alternative proof for the monotonicity of the principal eigenvalue of a hypergraph. This is based on joint work with Joshua Cooper and Utku Okur (arXiv: 2411.03567).

The Martin invariant of an even degree regular graph is an integer, defined by a recurrence at a vertex. In this talk, we define this invariant and review some of its properties, and then focus on the following question: which graphs minimise the Martin invariant?

In 1990, Conway and Lagarias introduced tilability criteria for tilability for finite regions of the hexagon and square grids. In the same year, Thurston expanded upon their work, introducing the height function criterion. We will discuss some new results in tilability: A new tilability criteria via the $\mathrm{SL}_2(\mathbb{C})$ double dimer model, and enumeration of tilings of special regions called Benzels, introduced in 2020 by Propp, using a technique called compression. If time allows, we will also discuss ongoing work that expands Thurston's height function to stone-and-bone tilings of the hexagon grid.

Two finite words are said to be abelian equivalent if one is a permutation of the letters of the other. For an infinite word, one can investigate the associated complexity function, called abelian complexity, which is a classical object of study in combinatorics on words. In particular, many works study the abelian complexity of automatic sequences, where a longstanding conjecture states that the abelian complexity of an automatic sequence is a regular sequence. We have studied when the abelian complexity can be computed efficiently, in particular using the theorem prover Walnut. To this end, we study words that are fixed points of Pisot-type substitution and prove that these words satisfy the conjecture.

If time permits, I will present $k$-abelian complexities, which are intermediate complexities between the abelian complexity and the factor complexity. I will also explain how our results can be extended to these complexities and how we can obtain a two-dimensional linear representation of some examples.

This talk is based on joint work with J-M Couvreur, M. Delacourt, N. Ollinger, J. Shallit, and M. Stipulanti (arXiv: 2504.13584).

Permuted-basement Macdonald polynomials $E_\alpha^\sigma(x_1, \dots, x_n; q, t)$ are nonsymmetric generalizations of symmetric Macdonald polynomials indexed by a composition $\alpha$ and a permutation $\sigma$. They can be described combinatorially as generating functions over augmented fillings of shape $\alpha$ and basement $\sigma$.

We construct deterministic and probabilistic bijections on fillings that prove identities relating $E_\alpha^\sigma$, $E_\alpha^{\sigma s_i}$, $E_{s_i \alpha}^\sigma$, and $E_{s_i \alpha}^{\sigma s_i}$. These identities arise from two operations on the shape and basement: swapping adjacent parts of the shape, which expands $E_\alpha^\sigma$ into $E_{s_i \alpha}^\sigma$ and $E_{s_i \alpha}^{\sigma s_i}$; and swapping adjacent basement entries, which gives $E_\alpha^\sigma = E_\alpha^{\sigma s_i}$ when $\alpha_i = \alpha_{i+1}$.

This is joint work with Olya Mandelshtam.

The set of orthonormal bases for $k$-planes in $\mathbb{R}^n$ is cut out by the equations $X X^T = I$ where $X$ is a $k \times n$ matrix of variables and $I$ is the $k \times k$ identity matrix. This space, known as the Stiefel manifold $\mathrm{St}(k,n)$, generalizes the orthogonal group and can be realized as the homogeneous space $O(n)/O(n-k)$. Its algebraic closure gives a complex affine variety, and thus, it has a degree.

I will discuss our derivation of these degrees. Extending 2017 work on the degrees of special orthogonal groups, joint work with Fulvio Gesmundo gives a combinatorial formula in terms of non-intersecting lattice paths. This result relies on representation theory, commutative algebra, Ehrhart theory, polyhedral geometry, and enumerative combinatorics.

I will conclude with some open problems inspired by these objects.

In this talk, we establish connections between hypersurface arrangements and likelihood geometry. The central object is the likelihood correspondence which captures the dependence between data and critical points of the likelihood function of a statistical model parametrized by the polynomials defining the arrangement. In particle physics, this same object is known as the scattering correspondence. The connection to hypersurface arrangements leads to a new description of the prime ideal of the likelihood correspondence, which is often computationally advantageous. This description is based on the Rees algebra of the likelihood module of the arrangement, a module closely related to the module of logarithmic derivations. We present results for generic and graphic arrangements.

For a hypergraph $\mathbb{H}$ on $[n]$, the hypergraphic poset $P_\mathbb{H}$ is the transitive closure of the oriented $1$-skeleton of the hypergraphic polytope $\Delta_\mathbb{H}$, which is the Minkowski sum of the standard simplices $\Delta_H$ for each hyperedge $H \in \mathbb{H}$. In 2019, C. Benedetti, N. Bergeron, and J. Machacek established a remarkable correspondence between the transitive closure of the oriented $1$-skeleton of $\Delta_\mathbb{H}$ and the flip graph on acyclic orientations of $\mathbb{H}$. Viewing an orientation of $\mathbb{H}$ as a map $A$ from $\mathbb{H}$ to $[n]$, we define the sources of the acyclic orientations as the values $A(H)$ for each hyperedge $H \in \mathbb{H}$. In a recent paper, N. Bergeron and V. Pilaud provided a characterization of $P_\mathbb{H}$ based on the sources of acyclic orientations for interval hypergraphs. Specifically, two distinct acyclic orientations $A$ and $B$ of $\mathbb{H}$ are comparable in $P_\mathbb{H}$ if and only if their sources satisfy $A(H) \le B(H)$ for all hyperedges $H \in \mathbb{H}$. The goal of this work is to extend this source characterization of $P_\mathbb{H}$ to arbitrary hypergraphs on $[n]$.

Given a finite poset, Greene introduced a rational function obtained by summing certain rational functions over the linear extensions of the poset. This function has interesting interpretations, and for certain families of posets, it simplifies surprisingly. In particular, Greene evaluated this rational function for strongly planar posets in his work on the Murnaghan–Nakayama formula. Develin, Macauley, and Reiner introduced toric posets, which combinatorially are equivalence classes of posets (or rather acyclic quivers) under the operation of flipping maximum elements into minimum elements and vice versa. In this work, we introduce a toric analogue of Greene's rational function for toric posets, and study its properties. In addition, we use toric posets to show that the Kleiss–Kuijf relations, which appear in scattering amplitudes, are equivalent to a specific instance of Greene's evaluation of his rational function for strongly planar posets. Also in this work, we give an algorithm for finding the set of toric total extensions of a toric poset.

Stanley recently introduced the shifted parking function symmetric function as a shifted analogue of the parking function symmetric function and posed the question of what the corresponding combinatorial objects are. This talk will answer that question and explain how the answer connects to projective representations of the symmetric group. Based on joint work with Zach Hamaker.

In 2016, Ardila and Boocher were investigating the variety obtained by taking the closure of a linear space within $\mathbb{A}^n$ in its compactification $(\mathbb{P}^1)^n$; later work named this the "matroid Schubert variety". Its Gröbner degenerations led them to define, and study the commutative algebra of, the _external activity complex_ of a matroid. If the matroid is on $n$ elements, this is a complex on $2n$ vertices whose facets encode the external activity of bases.

In recent work with Andy Berget on Speyer's $g$-invariant, we required a generalisation of the definition of external activity where the input was a pair of matroids on the same ground set. We generalise many of the results of Ardila--Boocher to this setting. Time permitting, I'll also present the tropical intersection theory machinery we use to understand the external activity complex of a pair.

For those who attended my talk at this year's CAAC on this paper, the content of the present talk is meant to be complementary.

The Kromatic symmetric function was introduced by Crew, Pechenik, and Spirkl (2023) as a K-analogue of Stanley's chromatic symmetric function. While the chromatic symmetric function encodes proper colorings of a graph (where each vertex gets a color and adjacent vertices get different colors), the Kromatic symmetric function encodes proper set colorings (where each vertex gets a nonempty set of colors and adjacent vertices get non-overlapping color sets). The expansion of the chromatic symmetric function in the basis of power sum symmetric functions has several nice interpretations, including one in terms of source components of acyclic orientations, due to Bernardi and Nadeau (2020). We lift that expansion formula to give expansion formulas for the Kromatic symmetric function using a few different K-analogues of the power sum basis. Our expansions are based on Lyndon heaps, introduced by Lalonde (1995), which are representatives for certain equivalence classes of acyclic orientations on clan graphs (graphs formed from the original graph by removing vertices and adding extra copies of vertices).

The Abelian/non-Abelian correspondence gives rise to a natural basis for the cohomology of flag varieties, which - except for Grassmannians - is distinct from the Schubert basis. I will describe this basis and its multiplication rules, and explain how to relate it to the Schubert basis for two-step flag varieties. I will then explain how this leads to new tableaux Littlewood--Richardson rules for many products of Schubert classes. This is joint work (separately) with Wei Gu and Linda Chen.

The graded Möbius algebra $B(M)$ of a matroid $M$ contains much combinatorial information about the flats of $M$. Its algebraic properties were instrumental in the proof of the Dowling—Wilson top-heavy conjecture. We will introduce a skew-commutative analogue $\mathrm{OB}(M)$ associated to every oriented matroid $M$, and discuss its algebraic structure. This is part of ongoing work joint with Yu Li.

The harmonics of quasisymmetric polynomials in superspace are the orthogonal complement of the ideal generated by quasisymmetric polynomials without constant term. In this talk, I will discuss the harmonics and present the first basis of this space, which is indexed by a specific family of nested forests.

Here is a purely combinatorial problem that arose in causal set theory. Let $\{P_1, \dots, P_k\}$ be distinct unlabelled posets all with $n$ elements. Suppose there is a poset $Q$ such that $\{P_1, \dots, P_k\}$ is exactly the set of downsets of $Q$ of size $n$ up to isomorphism. Given $n$ and $k$ can we give a tight upper bound on the minimum size of such a $Q$? As with newspaper headlines, the answer to the question is no, at least for the moment, but I'll explain what we do know. Joint work with Jette Gutzeit, Kimia Shaban, and Stav Zalel.

Quantum field theorists study Feynman periods, which are obtained by integrating expressions related to the spanning tree polynomials of graphs known as Feynman diagrams. But if you are like me and know nothing about physics, the good news is that doing quantum field theory often leads one to play with combinatorial objects. In particular, if one wishes to efficiently compute Feynman periods, they would likely be faced with unanswered questions about set partitions, determinantal identities, spaces of polynomials, and the all-minors matrix-tree theorem, many of which are quite accessible. I will present these questions and their relevant background in the context of my work on spaces of quadratic spanning forest identities, supervised by Dr. Karen Yeats.

It is a classical theorem that the diagonal of any bivariate rational power series is algebraic; that is, it satisfies a polynomial equation. We will discuss an algorithm that efficiently computes this polynomial.

Given a rational generating function $F(z_1, \dots, z_d)$ we are interested in the asymptotic behaviour of its coefficient sequence in a specified direction $(r_1, \dots, r_d)$. Although this problem is difficult in general, when $d=2$ and when some conditions are satisfied, there is a known algorithm that resolves it. We will explain the basics of analytic combinatorics in several variables and show how this algorithm operates.

It can often be tricky to find a combinatorial specification of a counting sequence for a given set of objects. A recently developed framework called "Combinatorial Exploration" aims to automate the process of finding combinatorial specifications. It has successfully been used to find specifications for several new permutation classes and looks promising for several other objects. In this talk, I will briefly explain how Combinatorial Exploration works, and how I am using it to automate finding specifications for families chord diagrams.

The field of analytic combinatorics in several variables (ACSV) is dedicated to the creation of effective techniques to study the large-scale behaviour of combinatorial objects. This talk provides results for two areas of ACSV: limit theorems and asymptotics of algebraic generating functions. First, I will describe an automated approach to proving local central limit theorems and its applications to a variety of examples. Included in these examples will be a family of permutations with restricted cycles, integer compositions with tracked summands and $n$-colour compositions with tracked summands. The second half of the talk will survey techniques for analyzing multivariate algebraic generating functions, going into detail specifically for the process of embedding an algebraic generating function into a sub-series of a rational function of more variables. For both parts of the talk, SageMath code which automates the methods discussed will be demonstrated for various examples.

A plane partition, whose 3D Young diagram is made of unit cubes, can be approximated by a "coarser” plane partition, made of cubes of side length 2. Two such approximations can be obtained by "rounding up” or "rounding down” to the nearest cube. We relate this coarsening (or downsampling) operation to Young's squish map, introduced in earlier work. We exhibit a related measure-preserving map between the 2-periodic single dimer model on the honeycomb graph, and a particular instance of Kenyon's $\mathrm{SL}_2$ double dimer model on a coarser honeycomb graph. This allows us to apply existing computations from the 2-periodic single dimer partition function to a portion of the parameter space of the the harder double dimer model. We also specialize our map and exhibit new criterion for the signed-tilability of a closed region on the honeycomb graph.

Causal set theory is an approach to quantum gravity where the underlying spacetime is a locally finite poset. This opens up many interesting combinatorial questions on posets that are either useful to the physics or that are asked by the physics but wouldn't necessarily be asked from a purer perspective. This talk is about one of the latter questions. Glaser gave a formula for the causal set theory analogue of the d'Alembertian in general dimension (growing out of previous work of Sorkin, Benincasa and Dowker, and Dowker and Glaser). The formula contains integer coefficients. Who can resist trying to find something that they count -- not me! -- so I will tell you about such a something.

We introduce equivariant quasisymmetry, a version of quasisymmetry for polynomials in two sets of variables. Using this definition we define double fundamental polynomials and double forest polynomials, a quasisymmetric generalizations of the theory of double Schur and double Schubert polynomials, where the subset of noncrossing permutations play the role of $S_n$.This combinatorics is governed by the quasisymmetric flag variety, a new geometric construction which plays the role for equivariant quasisymmetry what the usual flag variety plays in the classical story.

In this talk, I will focus on the combinatorial aspect first, and with the remaining time discuss the geometrical implications.

In this talk we will introduce a new combinatorial approach to understanding quasisymmetric polynomials via a quasisymmetric analogue of "divided differences". The resulting combinatorial theory is paired with a geometric theory that closely follows the classical development of Schubert calculus and Schubert varieties. Next week Nantel Bergeron will continue with the torus-equivariant story.

Causal set theory is an approach to quantum gravity that proposes that spacetime is fundamentally discrete and the causal relations among the discrete elements play a prominent role in the physics. Progress has been made in recognizing and understanding how some continuumlike features can emerge from causal sets at macroscopic scales, i.e., when the number of elements is large. An important result in this context is that a causal set is well approximated by a continuum spacetime if there is a number-volume correspondence between the causal set and spacetime.

This occurs when the number of elements within an arbitrary spacetime region is proportional to its volume. Such a correspondence is known to be best achieved when the number of causal set elements is randomly distributed according to the Poisson distribution. I will discuss the Poisson distribution and the statistical fluctuations it induces in the causal set-continuum correspondence, highlighting why it is important and interesting. I will also discuss new tools and techniques that facilitate such analyses.

A fundamental problem in spherical distance geometry aims to recover an $n$-tuple of points on a 2-sphere in $\mathbb{R}^3$, viewed up to oriented isometry, from $O(n)$ input measurements. This talk will discuss an algebraic solution to this problem using only the four arithmetic operations. We will show how a new type of frieze pattern can be employed to arrange the measurement data. These friezes exhibit glide symmetry and a version of the Laurent phenomenon.

Two classical lattices are the Tamari lattice on bracketings of a word and the weak order on permutations. The Hasse diagram of each of these lattices is the oriented 1-skeleton of a polytope, the associahedron and the permutohedron respectively. We examine a poset on bracketings of rooted trees whose Hasse diagram is the oriented 1-skeleton of a polytope called the operahedron. We show this poset is a lattice which answers question of Laplante-Anfossi. These lattices provide an extremely natural generalization of both the Tamari lattice and the weak order.

Whether it is decidable to determine when sequences satisfying linear recurrences with constant coefficients have all positive terms is a notorious problem in enumerative combinatorics that has essentially been open for around 90 years. Nevertheless, a "meta-principle" states that all such sequences arising from combinatorial counting problems belong to a special class where positivity (and more general asymptotic behaviour) is decidable. Here we discuss new software for determining positivity for sequences satisfying linear recurrences with *polynomial* coefficients. Originally motivated by a novel approach to proving genus one solution uniqueness for the Canham model for biomembrane shapes, our algorithm combines rigorous numeric analytic continuation of functions satisfying linear ODEs with singularity analysis techniques from analytic combinatorics. The main talk will be presented using a live Sage Jupyter notebook, and audience members who have access to Sage with a recent version of the ore_algebra package installed (available at https://github.com/mkauers/ore_algebra) will be able to follow along and play with the package during the talk.

There are three rings-with-bases whose multiplicative structure constants are computed by the same rule: the cohomology ring of the Grassmannian $\mathrm{Gr}(k,n)$, the representation ring of $\mathrm{GL}_k$ (made by stabilizing the previous in $n$), and one made by summing all representation rings of the symmetric groups (made by stabilizing the previous in $k$). The most famous rules, typically involving counting Young tableaux, are for the most stable version, but the unstable version admits the most generalizations, to K-theory, equivariant cohomology, quantum cohomology, and to other homogeneous varieties. I'll explain how to compute the multiplication in many of these cases by counting "puzzles".

This work is joint with Terry Tao and Paul Zinn-Justin.

I will talk about recent joint work with Chiara Meroni and Max Wiesmann, where we showed that specific exponential bivariate integrals serve as generating functions of labeled edge-bicolored graphs. Based on this, we prove an asymptotic formula for the number of regular edge-bicolored graphs with arbitrary weights assigned to different vertex structures.

The asymptotic behavior is governed by the critical points of a polynomial. An interesting application of this purely combinatorial work to mathematical physics is the Ising model on a random graph. I will explain how its phase transitions arise from our formula.

The quasisymmetric Macdonald polynomials $G_\gamma(X; q, t)$ are a quasisymmetric refinement of the symmetric Macdonald polynomials that specialize to the quasisymmetric Schur functions $QS_\alpha(X)$. We study the $t = 0$ specialization $G_\gamma(X; q, 0)$, which can be described as a sum over weighted multiline queues. We show that $G_\gamma(X; q, 0)$ expands positively in the quasisymmetric Schur basis and give a charge formula for the quasisymmetric Kostka-Foulkes polynomials $K_{\gamma,\alpha}(q)$ in the expansion $G_\gamma(X; q, 0) = \sum K_{\gamma,\alpha}(q) QS_\alpha(X)$. The proof relies heavily on crystal operators, and if you do not know what that means, come find out! This is joint work with Olya Mandelshtam.

Over one hundred years ago, Radon proved that any set of $d+2$ points in $\mathbb{R}^d$ can be partitioned into two sets whose convex hulls intersect. I will talk about Radon partitions when the points are selected randomly. In particular, if the points are independent normal random vectors, let $p_k$ be the probability that the Radon partition has size $(k, d+2-k)$. Answering a conjecture of Kalai and White, we show that the sequence $(p_k)$ is ultra log-concave and that, in fact, a balanced partition is the most likely. Joint work with Swee Hong Chan, Gil Kalai, Bhargav Narayanan, and Moshe White.

A well-known identity involving Eulerian numbers and combinations of a single variable $x$, sometimes called the Worpitzky identity, can be interpreted as a relationship between permutations and non-decreasing functions from $[n]$ to $[x]$. In this talk, we present a new identity that arises when extending this perspective to all functions from $[n]$ to $[x]$. We will provide a combinatorial proof of this identity by using trees, Dyck paths, and the theory of continued fractions.

Perturbative quantum field theory typically leads to asymptotic expansions with 0 radius of convergence. A formal series which is the asymptotic expansion of a function can sometimes be made into an actual function via Borel summation; when this works we say the series is Borel summable. Resurgence is a collection of techniques which can sometimes extract meaningful answers when Borel summation fails. In particular, this is governed by the alien calculus, where the alien derivatives measure the singularities in the Borel plane.

In this talk I'll tell you about recent work with David Sauzin where we find a recursive system of Dyson-Schwinger equations, expand it combinatorially as a sum over rooted trees equipped with a binary tubing, and find that the alien derivatives can be computed directly in terms of tree surgery operations on the tubings.

Finding the capacity of a graph is an old and important problem in theoretical computer science, combinatorics, and information theory. By the max-flow min-cut theorem, the capacity of a graph can be found in polynomial time via linear programming. In this talk, we consider the related counting problem: what is the number of flows of capacity at most 1 on a given directed graph with non-negative edge capacities?

Because this problem is #P-hard, our goal is to efficiently find a deterministic approximation. We will discuss a framework for polynomial capacity optimization which yields deterministic approximation algorithms for many #P-hard counting problems, including the flow counting problem. Time permitting, we will also discuss how this framework gives an algorithm for approximately computing the permanent of a non-negative matrix.

Positivity of sequence terms is a fundamental property in combinatorics and has applications across physics, algebra, and geometry. However, determining the positivity of sequences generated by linear recurrence relations is challenging. In 2022, Melczer and Mezzarobba utilized the tools of Analytic Combinatorics in Several Variables (ACSV) to determine the positivity of terms in asymptotic expansions. Building on this, we present an algorithmic implementation in Sage to establish the positivity of sequences using ACSV, with a particular focus on D-finite sequences.

Many algebraic and combinatorial objects are graded by the integers and have a cyclic action which cyclically permutes the grades. A natural generating function which captures this structure is the cyclotomic generating function (CGF). We will survey recent work on CGFs, including connections to the cyclic sieving phenomenon, unimodality, and asymptotic normality.

Joint work with Sara Billey.

Kazhdan-Lusztig polynomials are of foundational importance in geometric representation theory. Yet the Combinatorial Invariance Conjecture, due to Lusztig and to Dyer, suggests that they only depend on the combinatorics of Bruhat order. I'll describe joint work with Grant Barkley in which we adapt the hypercube decompositions introduced by Blundell-Buesing-Davies-Veličković-Williamson to prove this conjecture for Kazhdan-Lusztig R-polynomials in the case of elementary intervals in the symmetric group. This significantly generalizes the main previously known case of the conjecture, that of lower intervals.

A pair of skew shapes are said to be skew equivalent if they admit the same number of semistandard Young tableaux of each weight, or in other words if the skew Schur functions they define are equal. A conjecture of McNamara and van Willigenburg gives necessary and sufficient combinatorial conditions for shapes to be skew equivalent, but neither direction was known to hold in general. We prove sufficiency. The techniques used are Hopf-algebraic in spirit and extend ideas used by Yeats to prove a simple case.

Positroid varieties are subvarieties in the Grassmannian defined by cyclic rank conditions and which are related to Schubert varieties. We will provide a criterion for whether positroid varieties are smooth at certain distinguished points, and we will show that this information is sufficient to determine smoothness for the entire positroid variety. This will involve looking at combinatorial diagrams called "affine pipe dreams." We can also form a partial order on positroid varieties given by deletion and contraction, such that there is closure for smooth positroid varieties, and we will characterize the minimal singular elements in this order. Finally, we will discuss a couple of connections between the techniques of this work and planar $N=4$ SYM: the BCFW bridge decomposition and inverse soft factors.

Lusztig's fake degree is the generating polynomial for the major index of standard Young tableaux of a given shape. Results of Springer and James & Kerber imply that, mysteriously, its evaluation at a $d$-th primitive root of unity yields the number of border strip tableaux with all strips of size $d$, up to sign. This is essentially the special case of the Murnaghan-Nakayama rule for rectangular partitions as cycle type. We refine this result to standard Young tableaux and border strip tableaux with a given number of descents. To do so, we introduce a new descent statistic for border strip tableaux, extending the classical definition for standard Young tableaux.

Schubert coefficients are nonnegative integers that arise in Algebraic Geometry and play a central role in Algebraic Combinatorics. It is a major open problem whether they have a combinatorial interpretation, i.e, they are in #P. In this talk we discuss the closely related problem of the vanishing of Schubert coefficients. We prove that this vanishing problem is rather low in the polynomial hierarchy and discuss implications of this result.

This is joint work with Igor Pak.

Consider coloring the positive integers either red or blue one at a time in order. Van der Waerden's classical theorem states that no matter how you color the integers, you will eventually have $k$ equally spaced integers all colored the same for any $k$. But, how can we minimize the number of times $k$ equally spaced integers are colored the same? Even for $k = 3$, this question is unsolved. We will discuss progress towards proving an existing conjecture by leveraging a connection to coloring the continuous interval $[0,1]$. Our strategy relies on identifying classes of colorings with permutations and then using mixed integer linear programming. Joint work with Jonathan Kariv and Noah Williams.

Hessenberg varieties were introduced by De Mari, Procesi, and Shayman in the early 1990s and lie at the intersection of geometry, representation theory, and combinatorics. In 2012, Insko and Yong studied a class of Hessenberg varieties using patch ideals, a technique dating back to at least the 1970s from the study of Schubert varieties. In this talk, we will derive patch ideals and use them to study two classes of Hessenberg varieties. We will see the combinatorics that govern the behaviour of these patch ideals and translate these results to the geometric setting. Based in part on work with Sergio Da Silva, Megumi Harada, and Jenna Rajchgot.

In the pre-seminar we will review Birkhoff's structure theory for finite distributive lattices and its consequences for the geometry of some algebraic varieties associated with lattices by Hibi. In the seminar itself we will look more closely at the geometry of these varieties, motivating the definition of an interesting class of partial orders and raising several open problems.

In this talk, I will discuss connections between configuration spaces, an important class of topological space, and combinatorial algebras arising from the theory of reflection groups. In particular, I will present work relating the cohomology rings of some classical configuration spaces—such as the space of $n$ ordered points in Euclidean space—with Solomon’s descent algebra and the peak algebra. The talk will be centered around two questions.

First, how are these objects related?

Second, how can studying one inform the other? This is partially joint work with Marcelo Aguiar and Vic Reiner.

Let $(X, \{~,~\})$ be an (affine) Poisson variety. A log-canonical cluster structure on $X$ is a cluster structure on the coordinate ring of $X$ such that $\{\phi, \psi\} = \mathrm{const} \cdot \phi \psi$ for any two cluster variables $\phi$ and $\psi$ in the same cluster. It is known that log-canonical cluster structures can be used to construct integrable systems on Poisson varieties.

In this talk, we consider the Poisson variety $\mathfrak{g}^*$, the dual space of a Lie algebra $\mathfrak{g}$ equipped with the Lie-Poisson bracket. We focus on two cases: (1) $\mathfrak{g}=\mathfrak{b}$, the Borel subalgebra of a complex semi-simple Lie algebra; and (2) $\mathfrak{g}=\mathfrak{gl}_n(\mathbb{R})$. In both cases, we show that log-canonical cluster structures on $\mathfrak{g}^*$ give rise to interesting (new) integrable systems.

Given a permutation, we define its interval poset to be the set of all intervals ordered by inclusion. In this framework, a 'tube' is a convex connected subset, while a 'tubing' denotes a collection of tubes, that are pairwise either nested or disjoint. The interval poset associahedron is a polytope, whose faces correspond to proper tubes and whose vertices correspond to maximal tubings of the interval poset of a given permutation.

If we start with a simple permutation, the resulting interval poset associahedron will be isomorphic to the permutahedron. And if we consider inverse permutations, it turns out, that they yield identical associahedra.

If there is time, I will discuss another order on permutations, the Bruhat order, and compare it to the permutahedron.

Analytic combinatorics in several variables is a field of study focused on the derivation of limit behaviours of multivariate sequences. In this talk, I will describe a local central limit theorem and its applications to a variety of examples. Included in these examples will be a family of permutations with restricted cycles, integer compositions with tracked summands and $n$-colour compositions with tracked summands. The proof of this new, automated, local central limit theorem will also be briefly discussed, using techniques from the theory of analytic combinatorics in several variables. I will also provide background for proving central limit theorems probabilistically.

I will discuss some combinatorics of non-crossing chord diagrams that arose in the context of proving a conjecture from my coauthor's thesis which said that the global Schwinger formula of Cachazo and Early could be decomposed into a sum over cones indexed by non-crossing chord diagrams and that the amplitudes can be read off the chord diagrams by a triangulation construction.

In "Higher Operations in Perturbation Theory", Gaiotto, Kulp, and Wu discussed Feynman integrals that controls certain deformations in quantum field theory. These integrals themselves are differential forms, and the authors conjectured that one class of them squares to zero. This phenomenon can be interpreted as absence of quantum corrections in topological quantum field theories with more than one topological direction, or as an analogue of Kontsevich's formality theorem. In my talk, I will present a purely combinatorial proof of the conjecture for arbitrary graphs. It is based on graph matrices and graph polynomials, and a careful analysis of the involved signs and multiplicities. No knowledge or intution of the underlying physics is required.

In the preseminar, I will review the necessary definitions and properties of graph polynomials, and how they are typically applied in Feynman integrals. If time permits, I might also comment on the physical background.

The chromatic symmetric function is a symmetric function generalization of the chromatic polynomial that encodes the ways to color a graph such that no two adjacent vertices get the same color. We will discuss two different analogues of the chromatic symmetric function: a K-theoretic analogue called the Kromatic symmetric function, and a categorification called the chromatic symmetric homology. We show that certain properties of a graph can be recovered given its Kromatic symmetric function, and we give some formulas for special cases of the chromatic symmetric homology.

The asymmetric simple exclusion process (ASEP) is a parametrised Markov process on a state space of particle permutations and comes from statistical physics. It is a generalisation of the simplest system that displays more than two distinct "phases", analogous to solids, liquids, and gasses in matter. The problem at hand looks at the ASEP on a loop with a fixed number of particles of a given type. Its was proved by James Martin that the stationary distribution can be computed by taking the sum of weights over a certain combinatorial class called the multi-line queues (MLQs), which he introduced. He also introduced a modification of them, the alternative multi-line queues (AMLQs), that have a slightly simpler weight scheme and conjectured that they also may be used to compute the stationary distribution of the ASEP on a loop. However, despite being widely recognised as likely being true, the result was only known for the case when $N$, the number of distinct particle types, is $2$. This talk will sketch a proof of his conjecture for arbitrary $N$.

A poset $P = (X, \le)$ is a set $X$ equipped with a partial order $\le$.

The dimension of $P$ is the minimum number of linear orderings on $X$ required so that their intersection is $P$. We investigate two problems regarding the dimension of posets. The first problem is a conjecture by Bollobás and Brightwell from 1997 that the poset with a unicyclic cover graph has dimension at most $3$. The second problem was proposed by Erdős in 1991 about the dimension of the random partial order obtained by taking the transitive closure of the random graph $G(n,p)$, and the random bipartite graph $B(n,n,p)$. The random graph order is a type of classical sequential growth model that physicists use to model relations of spacetime events in the Minkowski space, and thus finds its applications in the causal set approach to quantum physics.

The well-known Cayley Hamilton theorem tells us that the unital algebra generated by a single $n \times n$ matrix has dimension at most $n$. In 1961 Gerstenhaber proved that the unital algebra generated by a pair of commuting $n \times n$ matrices has dimension at most $n$. The analogous statement for triples of pairwise commuting matrices remains an open problem, named the Gerstenhaber Problem. In this talk, we introduce the problem and reformulate it in commutative algebraic language, then we narrow our focus to a special class of matrices that arise from plane partitions.

The talk will discuss controlling the Kerov-Kirillov-Reshetikhin (KKR) bijection between semistandard tableaux and rigged configurations with a particular emphasis on the standard case. We introduce theorems and techniques to control the shape of the first rigged partition. We also introduce an operation on a standard tableau which induces a very small, very controlled change in the riggings of the corresponding rigged configuration. Despite how specific this operation seems, it can be used to manipulate all the riggings on the first rigged partition of a rigged configuration. It can also be used to give an alternate method to Kuniba et al. in order to "unwrap" the natural nesting structure on rigged configurations. The connection to the multi colour box ball system is discussed.

We establish, for every family of orthogonal polynomials in the Askey scheme and the $q$-Askey scheme, a combinatorial model for mixed moments and coefficients in terms of paths on the lecture hall lattice. This generalizes to all families of orthogonal polynomials in the Askey scheme previous results of Corteel and Kim for the little $q$-Jacobi polynomials. This is joint work with Sylvie Corteel, Bhargavi Jonnadula, and Jon Keating.

As Littlewood-Richardson rules compute linear representation theory of symmetric groups and cohomology of ordinary Grassmannians, shifted Littlewood-Richardson rules compute analogous projective representation theory of symmetric groups and cohomology of orthogonal Grassmannians. The first shifted Littlewood-Richardson rule is due to Stembridge (1989), building on a natural generalization by Sagan and Worley (1979/1984) of the jeu de taquin algorithm to shifted Young tableaux. We give a new shifted Littlewood-Richardson rule that requires consideration of fewer tableaux than Stembridge's rule and appears to involve an easier check on each. Our rule derives from applying old ideas of Lascoux and Schützenberger (1981) to the study of Haiman's mixed insertion (1989) and Serrano's shifted plactic monoid (2010). (Joint work with Oliver Pechenik).

In this talk, I will first provide a brief overview of causal set theory, an approach to quantum gravity. This theory proposes that spacetime is fundamentally characterized by a partially ordered set (poset), in which the partial order represents causal relations and the number of elements signifies the volume of a spacetime manifold region. I will then discuss how efforts to find a discrete counterpart of the d'Alembertian operator on a poset led to the formulation of the causal set action $S_{\mathrm{BDG}}$. This action is defined as a linear combination of the counts of various order intervals. Further analysis has shown that while KR posets are predominant in the number of posets of size $n$, the quantum dynamics imposed by $S_{\mathrm{BDG}}$ suppresses them for large $n$. Finally, I will propose a method to derive the combinatorial analogue of Einstein's field equations on posets.

Demazure atoms and characters are polynomials that each form a $\mathbb{Z}$-basis for polynomials in $n$ variables. The product of a Schur polynomial with a Demazure atom (resp. character) expands into a linear combination of Demazure atoms (resp. characters) with positive integer structure coefficients. There are known combinatorial rules that compute these coefficients using "skyline tableaux" given by Haglund, Luoto, Mason and Willigenburg. I have found alternative rules using the theory of integrable vertex models, inspired by a technique introduced by Zinn-Justin.

I use "coloured" vertex models for atoms and characters obtained from Borodin and Wheeler's models for non-symmetric Macdonald polynomials (setting $q=t=0$). The structure coefficients are then obtained as the number of fillings of a "diamond" vertex model that is compatible with both Schur (uncoloured) and Demazure (coloured) vertex models. The proof is completely combinatorial and very pretty.

I'm going to tell you more of the details behind my Martin polynomial work last year with Erik Panzer which led to the proof of the $c_2$ completion conjecture. I will actually describe the key bijection that proves it all, say some things about the permanent invariant, and cover other details that there hadn't been time for in the colloquium level presentation of the result.

Every linear representation of a matroid determines a matroid Schubert variety whose geometry encodes combinatorics of the matroid. When the representation is over the real numbers, we show that the topology of these varieties is completely determined by the combinatorics of zonotopes. As an application, we compute the fundamental groups. When the real matroid Schubert variety comes from a Coxeter arrangement, we show that the equivariant fundamental group is a “virtual” analogue of the corresponding Weyl group.

Daugherty and Orellana introduced a new diagram algebra called Quasipartition algebras in 2014. In this talk we will construct quasi-partition algebras and half quasi-partition algebras using an idempotent. Then using set-valued tableaux we give a description of a complete set of simple modules.Planar (half) quasi-partition algebras are sub-algebras of (half) quasi-partition algebras. The set-valued tableaux cannot be used in this setting, but for certain values of $x$, this sub-algebra is isomorphic to the Motzkin-Riordan algebra in which it is easier to find the representations. Finally, we use some idempotents of the Motzkin-Riordan algebra to compute the character table.

The enumeration of maps originates from a series of works by Tutte in the 1960’s. This work later went on to find uses in physics in the enumeration of Feynman diagrammatic expansions of matrix integrals.

In this talk I will discuss how maps with colored edges glued from 2-cells with one or two boundaries arise in recent work in the construction of path integrals over finite dimensional noncommutative spaces. Explicit formulae for the enumeration of such planar maps can be found by solving generalizations of Tutte’s equations. The generating functions of higher genus maps can also be computed from planar map generating functions using a process called Topological Recursion. This talk is based on joint work with Hamed Hessam and Masoud Khalkhali.

A numerical semigroup is a subset of the natural numbers that is closed under addition, contains 0, and has finite complement. Each numerical semigroup $S$ with fixed smallest positive element $m$ corresponds to an integer point in a polyhedral cone $C_m \subset \mathbb{R}^{m-1}$ called the Kunz cone. Moreover, numerical semigroups corresponding to points on the same face $F$ of $C_m$ are known to share many properties, such as the number of minimal generators. But not all faces of the Kunz cone contain integer points corresponding to numerical semigroups. In this talk, we will classify all the faces that do contain such points. Additionally, we will present sharp bounds on the number of minimal generators of $S$ in terms of the dimension of the face of $C_m$ containing the point corresponding to $S$.

This is joint work with Levi Borevitz, Tara Gomes, Jiajie Ma, Christopher O'Neill, Daniel Pocklington, Rosa Stolk, Jessica Wang, and Shuhang Xue.

In this talk, I will describe some recently discovered connections between one-dimensional interacting particle models (the ASEP and the TAZRP) and Macdonald polynomials and show the combinatorial objects that make these connections explicit. I will give a new compact tableau formula for the symmetric Macdonald polynomials $P_{\lambda}(X;q,t)$ in terms of a queue inversion statistic on certain sorted non-attacking tableaux. The nonsymmetric components of our formula specialize to the probabilities of the asymmetric simple exclusion process (ASEP) on a circle; moreover, the queue inversion statistic is naturally related to the dynamics of the ASEP. The new formula arises from the plethystic correspondence between the classical and modified Macdonald polynomials, which is closely related to fusion in the setting of integrable systems which connects the ASEP to the TAZRP.

Analytic combinatorics in several variables uses complex analytic results to find coefficients in the series expansions of meromorphic functions. Typically this is used to find asymptotics of sequences by examining their associated generating functions. In the 2000's Bousquet-Melou (and others) used the kernel method, introduced in the late 60s and 70s, to find generating function expressions (in terms of certain multivariate rational functions) for certain kinds of walks in restricted regions. In particular Melczer and Mishna found asymptotics for these restricted walks when the step sets are symmetric in every axis. Further work by Melczer and Wilson found asymptotics under the weaker assumption that the step set is symmetric in all but one axis, except in the special case that the vector sum of all the steps is equal to zero. In the pre-seminar I will discuss the kernel method. In the main talk I will discuss how to solve the case where the vector sum is equal to zero. If time permits I will talk a little about the interesting combinatorial behaviour in this case and the treatment of the integrals that appear.

In this talk, I will present the results of my masters thesis. I examine three applications of analytic methods to problems in combinatorics. By coincidence, each problem involves a combinatorial structure named for a plant--AVL trees, cactus graphs, and sunflowers--which we refer to collectively as combinatorial plants.

In our first result, we use a novel decomposition to create a succinct encoding for tree classes satisfying certain properties. This has applications to the study of data structures in computer science. To analyze our encoding, we derive asymptotics for the information-theoretic lower bound on the number of bits needed to store these trees. Our analysis applies to AVL trees (a commonly studied self-balancing binary search tree in computer science) as a special case. Joint work with Stephen Melczer, J. Ian Munro, and Ava Pun.

Next, we study the hat guessing game on cactus graphs and cycles. In this game, a player is placed on each vertex $v$ of a graph $G$ and assigned a colored hat from $h(v)$ possible colors. Each player makes a deterministic guess on their hat color based on the colors assigned to the players on neighboring vertices, and the players win if at least one player correctly guesses his assigned color. Joint work with I.M.J McInnis, Mehrdad Sohrabi and Shriya Kaistha.

Finally, we study the sunflower problem. A sunflower with $r$ petals is a collection of $r$ sets over a ground set $X$ such that every element in $X$ is in no set, every set, or exactly one set. We study the case where the pairwise intersections of the set family are restricted, proving new bounds.

Schubert polynomials are distinguished representatives of Schubert cells in the cohomology of the flag variety. Pipedreams (PD) and bumpless pipedreams (BPD) are two combinatorial models of Schubert polynomials. There are many classical perspectives to view PDs: Fomin and Stanley represented each PD as an element in the NilCoexter algebra; Lenart and Sottile converted each PD into a labeled chain in the Bruhat order. In this talk, we unravel the BPD analogues of both viewpoints.

One application of our results is a simple bijection between PDs and BPDs via Lenart's growth diagram.

A sunflower with $r$ petals is a collection of $r$ sets over a ground set $X$ such that every element in $X$ is in no set, every set, or exactly one set. Erdős and Rado showed that a family of sets of size $n$ contains a sunflower if there are more than $n!(r-1)^n$ sets in the family. Alweiss et al. and subsequently Rao and Bell et al. improved this bound to $(O(r \log n))^n$.

In this talk, I will discuss the sunflower problem with an additional restriction, a bound on the size of pairwise intersections in the set family. In particular, I will show an improved bound for set families when the size of the pairwise intersections of any two sets is in a set $L$. This talk is based on arXiv:2307.01374.

Deodhar decomposition of the Grassmannian is finest decomposition (that we know of) for which the components are homeomorphic to affine spaces. So, it's natural to be interested in their topology. In the talk we will try to describe a combinatorial rule that can possibly describe the closure of Deodhar decomposition. This work is joint with Olya Mandelshtam and Kevin Purbhoo.

This talk is about a different part of the quantum field theory story than I usually talk about. Wilson loop diagrams can be used to index amplitudes in a theory known as $N=4$ SYM. Suitably nice Wilson loop diagrams are also associated to positroids. For both mathematical and physical reasons it would be nice to have a diagrammatic understanding of the boundaries of the positroid cells of all co-dimensions. While we do not yet have a full understanding, we can build many boundaries with certain diagrammatic moves.

Joint work with Susama Agarwala and Colleen Delaney.

The type A coinvariant algebra, obtained by quotienting the polynomial ring by the ideal of positive degree symmetric polynomials, is a rich and active object of study. A distinguished basis for this quotient is given by Schubert polynomials. There is a dual to this story involving degree polynomials studied in depth by Postnikov-Stanley who shed light on their combinatorics.

I will describe the analogous picture in the context of the quotient of the polynomial ring modulo the ideal of positive degree quasisymmetric polynomials. The Schubert polynomials will be replaced by forest polynomials, while the degree polynomials will be replaced certain volume polynomials. This is joint work with Philippe Nadeau (CNRS and Univ. Lyon).

The connection between enumerating spanning trees of a graph and the theory of (linear) electrical networks goes all the way back to Kirchhoff's 1847 paper. It is physically sensible that if one increases the conductance of one wire in an electrical network, then the overall conductance of the network can not decrease. This corresponds to the less obvious fact that any two distinct edges are non-positively correlated, when one chooses a random spanning tree. Covariance is the 2-point "Ursell function'', and expectation is the 1-point Ursell function. For any subset of edges there is an associated Ursell function, and these are related to occupation probabilities by Möbius inversion. I will discuss some situations in which the signs of these Ursell functions can be predicted, yielding higher-order correlation inequalities for random spanning trees.

We study Ehrhart theory of lattice path matroid polytopes motivated by a conjecture by De Loera, Haws and Köppe. More specifically, we aim to understand the $h^*$-vector of this family of matroid polytopes. To do so, we subdivide them into smaller matroid polytopes such that each piece is a snake, which are matroids such that their matroid polytope coincides with the order polytope of a fence poset. Together with the triangulation of order polytopes given by Stanley, we give a combinatorial interpretation for the $h^*$-vector of lattice path matroid polytopes. For the special case of Schubert matroids of rank $2$ we go one step further and prove that these $h^*$-vectors count permutations with one descent filtered by initial gaps, and prove a recurrence relation for their volumes.

This talk is based in joint work with Carolina Benedetti and Kolja Knauer.

The extended Schur basis and the shin basis generalize the Schur functions to the dual algebras of the quasisymmetric functions and the noncommutative symmetric functions. We define a creation operator and a Jacobi-Trudi rule for certain shin functions and show that a similar matrix determinant expression does not exist for every shin function. We also define and study skew extended Schur functions which connect to skew Schur functions and to the multiplicative structure of the shin functions. Then, we introduce two new pairs of dual bases that result from applying certain involutions to the extended Schur and shin functions. These bases are defined combinatorially by variations on shin-tableaux much like the row-strict extended Schur functions. We will also discuss colored generalizations of the shin and extended Schur functions.

The chromatic symmetric function (CSF) of Dyck paths of Stanley and its Shareshian--Wachs $q$-analogue ($q$-CSF) have important connections to Hessenberg varieties, diagonal harmonics and LLT polynomials. In the, so called, abelian case they are related to placements of non-attacking rooks by results of Stanley-Stembridge (1993) and Guay-Paquet (2013).

In the first part of the talk, I will discuss a linear relation of the $q$-CSF for abelian paths in terms of the Garsia--Remmel $q$-rook and $q$-hit numbers originally due to Guay-Paquet and its relation to the $e$-positivity conjecture of Stanley--Stembridge and Shareshian--Wachs. This is joint work with Colmenarejo and Panova. In the second part of the talk, I will discuss the Newton polytope of CSFs of Dyck paths, whether it is saturated, and a conjectured Lorenztian property for these CSFs that is true for the abelian case. This is joint work with Matherne and Selover.

We’ll look at different ways of deforming the multiplicative structure of “classical” algebras to obtain new algebras and explain how this algebraic structure can often be understood combinatorially. We’ll then look at a special class of algebras one can produce this way called filtered deformations and we’ll discuss a conjecture of Etingof which asserts that in positive characteristic that filtered deformations of commutative rings should be in some natural sense very close to being commutative themselves. Not much background will be assumed.

The classic coinvariant space is a graded representation of the symmetric group with deep connections to permutation statistics and Hall-Littlewood polynomials. Its generalization, the diagonal harmonics, has a rich connection to Macdonald polynomials and the $q,t$-Catalan numbers. In this talk, we consider the variant of the classical coinvariant story in the superspace. We briefly survey its connections and recent developments. We introduce polarization operators and discuss a new basis for its alternating component. We also discuss a folklore on cocharge and propose a basis for the super harmonics.

Manjul Bhargava and the speaker introduced a functorial Galois closure operation for finite-rank ring extensions, generalizing constructions of Grothendieck and Katz-Mazur. In this talk, we use Galois closures to construct new components of Hilbert schemes of points, which are fundamental objects in algebraic geometry whose component structure is largely mysterious. We answer a 35 year old open problem posed by Iarrobino by constructing an infinite family of low dimensional components. This talk is based on joint work with Andrew Staal. No prior knowledge of Hilbert schemes will be assumed.

In a story some of you have heard from me before, we get subHopf algebras of the Connes-Kreimer Hopf algebra of rooted trees from certain simple tree classes which correspond to solutions to combinatorial analogues of Dyson-Schwinger equations in quantum field theory. Another important subHopf algebra of the Connes-Kreimer Hopf algebra is the Connes-Moscovici Hopf algebra which can be viewed as coming from rooted trees grown by adding leaves.

On the other hand, causal set theory is an approach to quantum gravity where in place of spacetime we have a locally finite poset with the poset relation interpreted as the causal relation between spacetime points. The classical sequential growth (CSG) model builds finite posets one element at a time with certain weights and is used in causal set theory.

I will give a common framework for all these classes/models, and discuss a new result with Stav Zalel on when certain models related to the CSG model give subHopf algebras of the poset Hopf algebra.

The presentation will be combinatorial and will not assume knowledge of causal set theory or quantum field theory.

Joint work with Stav Zalel.

I'll give two talks on the representations of the symmetric group over small finite fields, in particular, their block structure, with an emphasis on the perspective from categorical actions of Lie algebras. No previous background in modular representation theory will be assumed.

The classical determinantal ideals play an important role in commutative algebra, algebraic geometry, representation theory and combinatorics. They can be generalized to bipartite determinantal ideals which are the defining ideals of Nakajima's affine graded quiver variety. In this talk, we will introduce a combinatorial model called concurrent vertex maps to describe the Stanley-Reisner complex of the initial ideal of any bipartite determinantal ideal, and study properties and applications of this model.

Multiline queues were introduced by Ferrari and Martin to model the stationary states of the TASEP, a 1D non-equilibrium particle model. Later, Corteel, Mandelshtam, and Williams gave a formula for the Macdonald polynomial using a $(X,q,t)$-weighted version of multiline queues. This talk aims to develop the combinatorics of such objects in the $t=0$ case. We introduce an insertion algorithm on multiline queues that leads to new proofs of the charge expansion of $q$-Whittaker polynomials in the Schur basis, the Littlewood--Richardson rule and the Cauchy identity for binary matrices. We generalize these ideas to multiline diagrams, a plethystic analog of multiline queues, yielding a cocharge formula for the modified Hall--Littlewood polynomials and another proof of the second Cauchy identity for integer matrices. This is joint work with Olya Mandelshtam.

In 1957, Motzkin conjectured that the maximum number of faces possible for a polytope on $n$ vertices in $d$-space is achieved by the convex hull of $n$ points on the moment curve in $d$-space. This conjecture, called the Upper Bound Conjecture, was proved by McMullen in 1970 and generalized by Stanley in 1975. On the road to Stanley's proof, a correspondence between squarefree monomial ideals and simplicial complexes was born. That correspondence is called the Stanley--Reisner correspondence. It has come to occupy a central place in combinatorial algebraic geometry.

A large share of this talk will be devoted to exposition on the Stanley--Reisner correspondence, Hilbert functions, and Cohen--Macaulayness and how they served the combinatorial aim of proving the Upper Bound Conjecture. We will then define at most as many of the notions from Gorenstein linkage as we have to to state an open problem in that field. We will reexpress an important special case of this problem in terms of questions about triangulations of polytopes, and we will state some partial results from ongoing joint work with Jenna Rajchgot (McMaster) and Matt Satriano (Waterloo).

The ring of multivariate polynomials carries a natural action of the symmetric group. Quotienting by the ideal generated by the polynomials which are invariant under this action yields the "coinvariant algebra," an object with many beautiful algebraic and combinatorial properties. We will survey these properties and then discuss recent generalizations where the multivariate polynomials may contain anti-commuting ("superspace") variables. This talk is based on joint work with Brendon Rhoades.

The nabla operator is certainly one of the most exploited ones in the study of: Macdonald symmetric functions and their occurrences in several areas of investigations. After recalling some of these, as well as the historical role of many related operators, we will describe a new “super” version that unifies this whole area of investigation. The talk is illustrated by explicit examples, in an effort to make it accessible to a “general” audience. The new aspects are joint work with Jim Haglund, Alessandro Iraci, and Marino Romero.

In this talk we use the history of cluster combinatorics to motivate the study of certain subcategories of modules called wide subcategories, coming from quiver representations. In the context of cluster combinatorics, the classical noncrossing partition lattice is isomorphic to a poset of wide subcategories for a type A linearly oriented quiver. We will play on this connection, and review some still open problems related to the W-noncrossing partition lattice. Our main result is an EL-labeling of the poset of wide subcategories which can be encoded in terms of the Kreweras complement (joint with Eric Hanson).

Dyson-Schwinger equations are integro-differential equations satisfied by correlation functions in quantum field theory, which play the role of the "equations of motion" of the theory. They have a recursive, tree-like structure which enables these equations and their solutions to be studied combinatorially. Marie and Yeats showed that in a special case, the solution could be expanded as a sum over connected chord diagrams; this was generalized to many more cases by Hihn and Yeats. Using Hopf algebra techniques we give new combinatorial expansions for a much larger class of Dyson-Schwinger equations and systems as sums over rooted trees equipped with a kind of recursive decomposition we call a "binary tubing". This talk is based on joint work with Paul-Hermann Balduf, Amelia Cantwell, Kurusch Ebrahimi-Fard, Lukas Nabergall, and Karen Yeats.

There is a natural map from the symmetric polynomial ring in $r_1$ variables to the quantum cohomology ring of a type A flag variety $\mathrm{Fl}(n, r_1, \dots, r_k)$, given by evaluating Schur polynomials in the Chern roots of the first tautological bundle. I’ll explain how for a large class of Schur polynomials, the result is a Schubert class that can be obtained by dividing the partition into a quantum-hook and smaller partitions. Surprisingly, this is the key result proving a mirror theorem for type A flag varieties. A function $W$ is a mirror of a Fano variety $X$ if enumerative information of $X$ can be determined from $W$: for example, the Jacobi ring of $W$ should be the quantum cohomology ring of $X$. Mirrors for Fano toric varieties are well-understood; and more recently Plücker coordinate mirrors have been proposed for a variety of homogeneous spaces. We use quantum hooks to prove that the Plücker coordinate mirror of the flag variety computes quantum cohomology relations. This is joint work with Linda Chen.

Analytic combinatorics in several variables (ACSV) analyzes the asymptotic growth of series coefficients of multivariate rational functions in an exponent direction r by analyzing the singular set V of a multivariate rational function. The poly-torus of integration T that arises from the multivariate Cauchy Integral Formula (it would be a circle in one complex variable) is deformed away from the origin into cycles around critical points of a “height function" h on V. The deformation can sometimes flow to infinity at finite height in the presence of a critical point at infinity (CPAI): a sequence of points on V approaching a point at infinity, and such that the log-normals to V converge projectively to the direction of r. The CPAI is called heighted if the height function also converges to a finite value. In this talk we discuss under what conditions we know that all CPAI are heighted, and in which directions CPAI can occur, by compactifying in a toric variety. In smooth cases under generically satisfied conditions, CPAI must always be heighted. Non-generic cases are also studied under other conditions.

Petersen and Tenner defined the depth statistic for Coxeter group elements which, in the symmetric group, can be described in terms of a cost-minimization problem over the factorizations of a permutation into transpositions. We generalize that cost function to the other classical (finite and affine) Weyl groups, letting the cost of an individual reflection $t$ be the distance between the integers transposed by $t$ in the combinatorial representation of the group (à la Eriksson and Eriksson). Arbitrary group elements then have a well-defined cost, obtained by minimizing the sum of the transposition costs among all factorizations of the element. We show that the cost of arbitrary elements can be computed directly from the elements themselves using a simple, intrinsic formula. This work is joint with Bridget Tenner.

Most decompositions of the Grassmannian are described as subsets of the Grassmannian by setting certain Plücker coordinates to zero, demanding certain other Plücker coordinates to be non-zero, and leaving the remaining Plücker coordinates unspecified. In the case of the Deodhar decomposition, these coordinates are determined by the location of stones in the corresponding Go-diagram. We shall be interested in answering the question as to when one Deodhar component lies in closure of another by looking at their corresponding Go-Diagrams. We will define restricted paths on a graph determined by the Go-diagram and show how they can be used to solve the above problem. This work is joint with Kevin Purbhoo and Olya Mandelshtam.

This talk is about finding a quasisymmetric variety (QSV): a subset of permutations which (i) is a basis for the Temperley--Lieb algebra $\mathrm{TL}_n(2)$, and (ii) has a vanishing ideal (as points in $n$-space) that behaves similarly to the ideal generated by quasisymmetric polynomials. While this problem is primarily motivated by classical (co-)invariant theory and generalizations thereof, the course of our investigation uncovered a number of remarkable combinatorial properties related to our QSV, and I will survey these as well. Of particular interest is a new equivalence relation on permutations defined using their excedance sets. This relation has many nice properties: each equivalence class is naturally indexed by a noncrossing partition and also forms an interval in the (strong) Bruhat order. This allows us to define a quotient Bruhat order and gives a simple method for constructing many new bases of $\mathrm{TL}_n(2)$, generalizing known results of Williams--Gobet and Zinno. Surprisingly, the combinatorics of this equivalence relation turn out to be key in solving the QSV problem: collecting the maximal element of each excedance class produces a QSV, and the ensuing noncrossing partition combinatorics are essential to prove this fact. Based on joint work with Nantel Bergeron; arXiv:2302.10814.

For each arrangement of (pseudo)lines on the projective plane, it is possible to construct a differential form that captures its combinatorial structure. The forms have simple poles whenever triangles shrink to a point in the arrangement, and share the same residue when two arrangements are connected via a "triangle flip". In this talk I will explain the construction and give evidence for the conjecture that integrating such differential forms, with the appropriate measure, computes generalized scalar scattering amplitudes. These amplitudes are defined as sums over arrangements of metric trees or generalized Feynman diagrams. While generic arrangements of metric trees span the Dressian, it is also conjectured that the "physical" ones define cones in the tropical Grassmannian.

Lascoux polynomials are K-theoretic analogues of the key polynomials. They both have combinatorial formulas involving tableaux: reverse set-valued tableaux (RSVT) rule for Lascoux polynomials and reverse semistandard Young tableaux (RSSYT) rule for key polynomials. Besides, key polynomials have a simple algorithmic model in terms of Kohnert diagrams, which are in bijection with RSSYT. Ross and Yong introduced K-Kohnert diagrams, which are analogues of Kohnert diagrams. Ross and Yong conjectured a K-Kohnert diagram rule for Lascoux polynomials. We establish this conjecture by constructing a weight-preserving bijection between RSVT and K-Kohnert diagrams.

The plethysm operation $f[g]$ of two symmetric functions is of foundational interest in algebraic combinatorics. On the representation theoretic side, it corresponds to the composition of symmetric group class functions under the Frobenius characteristic. On the symmetric function side, plethysms occur frequently as one of the most natural ways to describe operators and mappings in the space of symmetric functions.

In this talk, I will give an overview of the fundamentals of plethysm, and give a new combinatorial interpretation for any plethysm by interpreting it in terms of a signed sum over proper colorings of acyclically oriented graphs. I will demonstrate that this graph-theoretic interpretation unifies previous results and gives rise to new plethystic identities.

This is based on joint work with Sophie Spirkl.

To put it simply, Pauli's exclusion principle is the reason why we can't walk through walls without getting hurt. Pauli won the Nobel Prize in Physics in 1945 for the formulation of this principle. A few years later, this principle received a geometrical formulation that is still overlooked today. This formulation uses the eigenvalues of certain matrices (which represent a system of elementary particles, for example electrons). These eigenvalues form a symmetric geometric object obtained by cutting a hypercube: it is a hypersimplex.

To represent systems of particles with a non-zero temperature, it is necessary to generalize the hypersimplex to obtain what is called "lineup polytopes". These polytopes are defined using classical notions of combinatorics and discrete geometry. Moreover, they produce new exclusion principles which refine Pauli's principle that shall be put to the test by experimentalists. During this talk, we will see the history behind the introduction of these polytopes and give a presentation of some properties.

This is joint work with physicists Julia Liebert, Christian Schilling and mathematicians Eva Philippe, Federico Castillo and Arnau Padrol.

The first half of this talk will be expository and devoted to a discussion of (quasi)symmetric functions and tableaux.

We define new families of quasisymmetric functions, in particular the new basis of row-strict dual immaculate functions, with an associated cyclic, indecomposable $0$-Hecke algebra module. Our row-strict immaculate functions are related to the dual immaculate functions of Berg-Bergeron-Saliola-Serrano-Zabrocki (2014-15) by the involution $\psi$ on the ring $\mathrm{QSym}$ of quasisymmetric functions. We uncover the remarkable properties of the immaculate Hecke poset induced by the $0$-Hecke action on standard immaculate tableaux, revealing other submodules and quotient modules, often cyclic and indecomposable.

As in the dual immaculate case, the row-strict dual immaculate function is the generating function of a suitable set of tableaux, defined by a specific descent set. We complete the combinatorial and representation-theoretic picture by constructing $0$-Hecke modules for the remaining variations on descent sets. We show that the generating functions of all the possible variations of tableaux are characteristics of these $0$-Hecke modules, captured in the immaculate Hecke poset.

This talk is based on joint work with Elizabeth Niese, Stephanie van Willigenburg, Julianne Vega and Shiyun Wang.

In this talk, I will give a brief introduction on Catalanimal, a tool that helps us to prove the shuffle theorem under any line, the extended delta conjecture and the Loehr- Warrington conjecture. I will then focus on its variant "Macanimal" which gives us an explicit raising operator formula for the modified Macdonald polynomials. Our method just as easily yields a formula for an infinite series of $\mathrm{GL}_l$ characters which truncates to the modified Macdonald polynomials.

This is a joint work with Jonah Blasiak, Mark Haiman, Jennifer Morse and George Seelinger.

The hat guessing number $HG(G)$ of a graph $G$ on $n$ vertices is defined in terms of the following game: $n$ players are placed on the $n$ vertices of $G$, each wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.

In this talk, I will begin with an illustrative example and then show the lower bound on $HG(G(n,1/2))$, where $G(n,1/2)$ denotes the random graph on $n$ vertices where each edge is included uniformly and independently with probability $1/2$. I will also discuss the linear hat guessing number.

This is based on joint work with Noga Alon.

The collection of parking functions under a natural $S_n$-action (which has Catalan-many orbits) has been a central object in Algebraic Combinatorics since the work of Haiman more than 30 years ago. One of the lines of research spawned around it was towards defining and studying analogous objects for real and complex reflection groups $W$; the main candidates are known as the $W$-non-nesting and $W$-non-crossing parking functions.

The $W$-non-nesting parking functions are relatively well understood; they form the so called algebraic $W$-parking-space which has a concrete interpretation as a quotient ring. The $W$-non-crossing ones on the other hand have defied unified explanations while simultaneously proving themselves central in the study of Coxeter and Artin groups (their geometric group theory, representation theory, and combinatorics). One of the main open problems in the field since the early 2000's has been to give a type-independent proof of the $W$-isomorphism between the algebraic and the non-crossing $W$-parking spaces. In this talk, I will present such a proof, solving the more general Fuss version of the problem, that proceeds by comparing a collection of recursions that are shown to be satisfied by both objects. This relies on a variety of recent techniques we introduced, in particular the enumeration of certain flats of full support via Crapo's beta invariant, the $W$-Laplacian matrices for reflection arrangements and, in collaboration with Matthieu Josuat-Verges, the refined $f$- to $h$-transformation between the cluster complex and the non-crossing lattice of $W$.

In this presentation we will see how to apply the theory of complex analysis to study multivariate generating series by looking at several examples. Specifically, given a rational bivariate generating function $G(x, y)/H(x, y)$ with coefficients $f_{i, j}$ the objective is algorithmically determine asymptotic formulas to approximate $f_{rn, sn}$ as $n$ goes to infinity, for fixed positive integers $r$ and $s$. In this presentation we demonstrate two approaches for determining the asymptotic formulae, each of which involve determining so-called minimal critical points of the denominator $H(x, y)$ in the direction $(r, s)$. The first approach uses numerical methods to solve systems of polynomial equations which depend on the given bivariate generating function to determine minimal points of the denominator, while the second involves analyzing a map $h$ from the zero-set of $H$ to the real numbers, known as a height map. Software developed for both of these purposes will be demonstrated.

I will present results on the asymptotic growth rate of the Euler characteristic of Kontsevich's commutative graph complex. By a work of Chan-Galatius-Payne, these results imply the same asymptotic growth rate for the top-weight Euler characteristic of $\mathcal{M}_g$, the moduli space of curves, and establish the existence of large amounts of unexplained cohomology in this space. This asymptotic growth rate follows from new generating functions for the edge-alternating sum of graphs without odd automorphisms. I will give an overview on this interaction between topology and combinatorics and illustrate the combinatorial and analytical tools that were needed to obtain these generating functions.

Numerous combinatorial enumeration problems reduce to the study of functional equations which can be solved by a uniform method introduced by Bousquet-Mélou and Jehanne in 2006. In my talk, I will first briefly explain this result and its proof. Then I will present a new generalization of it to the case of systems of functional equations with one catalytic variable. The method is constructive and yields an algorithm for computing the minimal polynomials of interest.

The talk is based on joint work with Hadrien Notarantonio.

Analytic combinatorics in several variables is a field of study focused on the derivation of limit behaviours of multivariate sequences. In this talk, I will provide a variety of examples of applying an automated version of a local central limit theorem. Included in these examples will be a family of permutations with restricted cycles, integer compositions with tracked summands and n-colour compositions with tracked summands. The proof of this automated local central limit theorem will also be briefly discussed, using techniques from the theory of analytic combinatorics in several variables.

Coxeter groups, such as the symmetric group of permutations are groups generated by reflections. In this talk, I will discuss the poset structure defined as the weak order of Coxeter groups and introduce the Sperner property. While it is known the weak order of the symmetric group of permutations is strongly Sperner, this talk will focus on extending this result for all finite Coxeter groups. Solving the $1$-Sperner case using a bipartite matching algorithm to find the maximum size antichain will be explained. I will define the construction of root systems by Coxeter groups of type $A_n$, $B_n$, and $D_n$ and discuss its applications towards the order of the Weyl groups listed.

We describe a Yang-Baxter integrable colored vertex model, from which we construct a class of partition functions that equal the LLT polynomials of Lascoux, Leclerc, and Thibon. Using the vertex model formalism, we can prove many properties of these polynomials. We also use the vertex model to study $k$-tilings ($k$-tuples of domino tilings) of the Aztec diamond of rank $m$, where we assign a weight to each $k$-tiling depending on the number of vertical dominos and the number of "interactions" between the tilings. We compute the generating polynomials of the $k$-tilings, and prove some combinatorial results about $k$-tilings in certain limits of the interaction strength.

The Hopf algebra of symmetric functions is spanned by several important bases, including by power sum symmetric functions, which encode the class values of the characters of the symmetric group under the Frobenius characteristic map. We introduce in this talk the basis of combinatorial power sums, naturally refining the power sum symmetric functions, for the larger Hopf algebra of quasisymmetric functions. Our construction is motivated by the theory of (weighted) $P$-partitions, the combinatorics of which will allow us to describe formulas for products, coproducts and classical quasisymmetric involutions, as well as give combinatorial rules for the expansion into the monomial and fundamental bases of quasisymmetric functions. Joint work with Farid Aliniaeifard and Stephanie van Willigenburg.

The cyclic sieving phenomenon (CSP) connects a cyclic group action on a family of combinatorial objects with some q-analog of that set. We discuss some recent results and open problems for standard and semistandard tableaux, as well as some other families of combinatorial objects.

Several open problems with various levels of difficulty will be presented.

Functions of permutations are studied in a wide variety of fields including probability, statistics and theoretical computer science. I will introduce a method for studying such functions using representation theory and symmetric functions. As a consequence, one can extract detailed information about asymptotic behavior of many permutation statistics with respect to non-uniform measures that are invariant under conjugation. The key new tool is a combinatorial formula called the path Murnaghan-Nakayama rule that gives the Schur expansion of a novel basis of the ring of symmetric functions. This is joint work with Brendon Rhoades.

Bruhat interval polytopes are a well-studied class of generalized permutohedra which arise as moment map images of various toric varieties and totally positive spaces in the flag variety. I will describe work in progress in which I study the $1$-skeleta of these polytopes, viewed as posets interpolating between weak order and Bruhat order. In many cases these posets are lattices and the polytopes, despite not being simple, have interesting $h$-vectors.

Krattenthaler and Thibon discovered a beautiful formula for the determinant of the matrix indexed by permutations whose entries are $q^{\operatorname{maj}(u v^{-1})}$, where $\operatorname{maj}$ is the major index. Previous proofs of this identity have applied the theory of nonsymmetric functions or the representation theory of the Tits algebra to determine the eigenvalues of the matrix. I will present a new, more elementary proof of the determinantal formula. Then I will explain how we used this method to prove several conjectures by Krattenthaler for variations of the major index over signed permutations and colored permutations. This is based on joint work with Donald Robertson and Clifford Smyth.

A box-ball system (BBS) is a discrete dynamical system whose dynamics come from the balls jumping according to certain rules. A permutation on $n$ objects gives a BBS state by assigning its one-line notation to $n$ consecutive boxes. After a finite number of steps, a box-ball system will reach a steady state. From any steady state, we can construct a tableau called the soliton decomposition of the box-ball system. The shape of the soliton decomposition is called the BBS partition. An exciting discovery (made in 2019 by Lewis, Lyu, Pylyavskyy, and Sen) is that the BBS partition and its conjugate record permutation statistics similar to the classical Greene’s theorem statistics.

The well-known Robinson—Schensted algorithm is a bijection from permutations $w$ to pairs of standard tableaux $P(w), Q(w)$ of the same shape. We will discuss a few new results which relates BBS to these $P$ and $Q$ tableaux:

(1) The soliton decomposition of a permutation $w$ is a standard tableau if and only if it is equal to $P(w)$.

(2) The $Q$ tableau of a permutation completely determines the dynamics of the corresponding box-ball system.

(3) We present a bijection between Motzkin paths and a class of involutions whose soliton decompositions are standard.

This talk is based on joint work with B. Drucker, E. Garcia, A. Rumbolt, R. L. Silver (arxiv.org/abs/2112.03780); M. Cofie, O. Fugikawa, M. Stewart, D. Zeng (SUMRY 2021); and S. Hong, M. Li, R. Okonogi-Neth, M. Sapronov, D. Stevanovich, and H. Weingord (SUMRY 2022).

Come with me on a magical journey into the mysterious world of inverse Wronskians.

Thresholds for increasing properties of random structures are a central concern in probabilistic combinatorics and related areas. In 2006, Kahn and Kalai conjectured that for any nontrivial increasing property on a finite set, its threshold is never far from its "expectation-threshold," which is a natural (and often easy to calculate) lower bound on the threshold. In this talk, I will present recent progress on this topic. Based on joint work with Huy Tuan Pham.

The original problem of classical invariant theory was to describe the invariants of $\mathrm{SL}_m$ acting on a polynomial ring in an $m \times n$ matrix of variables. One way to solve this problem is to consider the polynomial ring as a $\mathrm{GL}_m \times \mathrm{GL}_n$ representation, and decompose this representation into its irreducible components.

Berenstein and Kazhdan's theory of geometric crystals gives rise to two families of rational actions on the space of $m \times n$ complex matrices, which we view as "crystallized versions" of the usual $\mathrm{GL}_m$ and $\mathrm{GL}_n$ actions. We describe the invariants of these two families of actions. Our main tool is Noumi and Yamada's geometric lifting (or de-tropicalization) of the RSK correspondence, which is analogous to the above-mentioned irreducible decomposition in the classical setting. This is joint work with Ben Brubaker, Pasha Pylyavskyy, and Travis Scrimshaw.

I will not assume prior knowledge of crystals or geometric crystals; the basic building blocks of these theories will be introduced through examples.

We explain how perfect matchings and set-partitions (as well as their weighted versions) manifest in Law of Large Numbers from Random Matrix Theory. One of the examples discussed is a deformation of the operation of "free convolution" from the theory of Free Probability. At the end, we discuss open questions in probability motivated from the combinatorial point of view. No advanced knowledge of probability is needed to understand the theorems. The new results from this presentation are joint work with Florent Benaych-Georges and Vadim Gorin.

Springer fibers are a family of varieties that have remarkable connections to combinatorics and representation theory. Springer used them to geometrically construct all of the irreducible representations of the symmetric group (Specht modules). Moreover, they give a geometric meaning to Hall-Littlewood symmetric functions. In this talk, I will introduce a generalization of Springer fibers called $\Delta$-Springer varieties, a special case of which gives a new geometric meaning to the expression in the Delta Conjecture at $t=0$. We’ll then use these varieties to geometrically construct induced versions of the Specht modules. This is joint work with Jake Levinson and Alexander Woo.

This talk will introduce semidistrim lattices, which generalize semidistributive lattices and trim lattices; these two families, in turn, generalize distributive lattices. We will discuss structural, topological, and dynamical properties of semidistrim lattices. In particular, we will see how one can define a certain bijective operator on a semidistrim lattice called rowmotion; this definition unifies the definition that Barnard gave for semidistributive lattices and the definition that Thomas and Williams gave for trim lattices. Somewhat surprisingly, rowmotion for semidistrim lattices is intimately connected with a noninvertible operator called pop-stack sorting, which can be defined for any lattice. This talk is based on joint work with Nathan Williams.

This talk is intended for an algebraic combinatorial community and no prior knowledge is required. All the difficult words (Hurwitz numbers, KP hierarchy, HCIZ and BGW integrals, Jack symmetric functions, the $b$-conjecture) will be explained and gently introduced.

The monotone Hurwitz numbers can be understood combinatorially as cardinalities of monotone transitive walks in the Cayley graph of the symmetric group. They share many beautiful properties with ordinary Hurwitz numbers, but one of their most interesting properties is that their generating function coincides with the topological expansion of the celebrated Harish-Chandra—Itzykson—Zuber integral (in the case of double numbers) and the Brezin—Gross—Witten integral (in the case of single numbers). Using standard tools from algebraic combinatorics, one can express this generating function in terms of Schur symmetric functions and prove that it is a solution of the infinite system of partial differential equations called the KP hierarchy.

Inspired by a mysterious conjecture of Goulden and Jackson which connects generating function of Jack symmetric functions with enumeration of combinatorial maps, we define (following joint work with Chapuy) the generating function of $b$-deformed monotone Hurwitz numbers by replacing Schur symmetric functions by their one-parameter deformation — Jack symmetric functions. We show that it has an explicit combinatorial interpretation, which gives a topological expansion of the $\beta$-HCIZ integral. Finally, we show that for $b=1$ this generating function has a very interesting structure — it is a solution of the infinite system of Partial Differential Equations called the BKP hierarchy. We prove it by finding an explicit expansion in Schur symmetric functions, which surprisingly involves dimensions of the irreducible representations of the orthogonal group. As an application, we deduce an explicit Pfaffian formula for the Brezin—Gross—Witten integral over the orthogonal group. This is joint work with Valentin Bonzom and Guillaume Chapuy.

The set of all possible spectra of 1-reduced density operators for systems of $N$ particles on a $d$-dimensional Hilbert space is a polytope called hypersimplex and this is related to Pauli's exclusion principle. If the spectrum of the original density operators is fixed, the set of spectra (ordered decreasingly) of 1-reduced density operators is also a polytope. A theoretical description of this polytope using inequalities was provided by Klyachko in the early 2000's.

Adapting and enhancing tools from discrete geometry and combinatorics (symmetric polytopes, sweep polytopes, and the Gale order), we obtained such necessary inequalities explicitly, that are also valid for arbitrarily large $N$ and $d$.

This approach leads to a new class of polytopes called lineup polytopes.

This is joint work with physicists Jean Philippe Labbe, Julia Liebert, Eva Philippe, Arnau Padrol, and Christian Schilling.

The Stirling numbers of the first and second kind are classical objects in enumerative combinatorics which count the number of permutations or set partitions with a given number of blocks or cycles, respectively. Carlitz and Gould introduced $q$-analogues of the Stirling numbers of the first and second kinds, which have been further studied by many authors including Gessel, Garsia, Remmel, Wilson, and others, particularly in relation to certain statistics on ordered set partitions. Separately, type B analogues of the Stirling numbers of the first and second kind arise from the study of the intersection lattice of the type B hyperplane arrangement. We combine the two directions and introduce new type B $q$-analogues of the Stirling numbers of the first and second kinds. We will discuss connections between these new $q$-analogues and generating functions identities, inversion and major index-style statistics on type B set partitions, and aspects of super coinvariant algebras which provided the original motivation for the definition. This is joint work with Bruce Sagan.

I discovered an intriguing linear algebra relationship which I call a multijection. I used this construction to solve an open problem about higher-dimensional sandpile groups, but I think that it has more to say. In this talk, I will focus on sharing the most general version of the multijection that I know of, which involves a family of beautiful periodic tilings. This talk uses mostly linear algebra, so it should be accessible to a general math audience.

Sorting probability for a partially ordered set $P$ is defined as the $\min |\Pr[x < y] - \Pr[y < x]|$ going over all pairs of elements $x, y \in P$, where $\Pr[x < y]$ is the probability that in a uniformly random linear extension (extension to total order) $x$ appears before $y$. The celebrated 1/3--2/3 conjecture states that for every poset the sorting probability is at most 1/3, i.e. there are two elements $x$ and $y$, such that $1/3 \le \Pr[x < y] \le 2/3$. The asymptotic extension of this conjecture states that the sorting probability goes to 0 as the width (maximal antichain) of the poset grows to infinity. We will prove the last conjecture for Young diagrams, where the linear extensions are Standard Young Tableaux.

Beyond SYTs, these conjectures bring out a variety of poset inequalities, which have connections to both algebra as in group actions and probability as in random walks. Based on joint works with Swee Hong Chan and Igor Pak.

Schubert polynomials are a very interesting family of polynomials in algebraic geometry due to their relation with the cohomology of the flag variety. Moreover, they are also very interesting from a combinatorial point of view because they can be considered generalizations of Schur functions. In this talk, we will talk about how to multiply a Schubert polynomial by a Schur function indexed by a hook and how we can extend this multiplication to the quantum world. This is a current work with C. Benedetti, N. Bergeron, F. Saliola, and F. Sottile.

In 1995, Stanley introduced the chromatic symmetric functions. The study of chromatic symmetric functions of graphs inspired two main research directions. The first research direction is to prove the Stanley-Stembridge conjecture: if a poset is (3+1)-free, then the chromatic symmetric function of its incomparability graph is $e$-positive, i.e., a nonnegative linear combination of elementary symmetric functions. The second research direction is to determine whether two non-isomorphic trees can have the same chromatic symmetric function. In this talk, we present several modular relations between chromatic symmetric functions and apply them to show that the Stanley-Stembridge conjecture is true for several new families of graphs. Moreover, using the modular relations, we give an algorithm to write the chromatic symmetric functions of trees in terms of the chromatic symmetric functions of paths. (Joint work with Victor Wang and Stephanie van Willigenburg).

Elements in the group algebra of a symmetric group $S_n$ are known to have an interpretation in terms of card shuffling. I will discuss a new family of such elements, recently constructed by Nadia Lafrenière. Given a positive integer $n$, we define $n$ elements $t_1, t_2, \dots, t_n$ in the group algebra of $S_n$ by $t_i =$ the sum of the cycles $(i)$, $(i, i+1)$, $(i, i+1, i+2)$, $\dots$, $(i, i+1, \dots, n)$, where the cycle $(i)$ is the identity permutation. The first of them, $t_1$, is known as the top-to-random shuffle and has been studied by Diaconis, Fill, Pitman (among others).

The $n$ elements $t_1, t_2, \dots, t_n$ do not commute. However, we show that they can be simultaneously triangularized in an appropriate basis of the group algebra (the "descent-destroying basis"). As a consequence, any rational linear combination of these $n$ elements has rational eigenvalues. The maximum number of possible distinct eigenvalues turns out to be the Fibonacci number $f_{n+1}$, and underlying this fact is a filtration of the group algebra connected to "lacunar subsets" (i.e., subsets containing no consecutive integers). This talk will include an overview of other families (both well-known and exotic) of elements of these group algebras. I will also briefly discuss the probabilistic meaning of these elements as well as some tempting conjectures.

This is joint work with Nadia Lafrenière.

A striking result in graph theory is that the property of a graph being quasirandom (i.e. resembling a random graph) is characterized by the number of edges and the number of 4-cycles being close to the expected number in a random graph. Král’ and Pikhurko (2013) proved an analogous result for permutations; i.e. that quasirandom permutations are characterized by the densities of all permutations of length 4. We improve on this result by showing that there is a single algebraic expression consisting of a sum of densities of 8 permutations of length 4 whose value forces quasirandomness. Moreover, we characterize all permutation expressions of this type which force quasirandomness. These results have direct implications on the problem of independence testing in non-parametric statistics. Joint work with Timothy F. N. Chan, Daniel Král’, Yanitsa Pehova, Maryam Sharifzadeh and Jan Volec.

The Newell-Littlewood numbers are defined in terms of the Littlewood-Richardson coefficients. Both arise as tensor product multiplicities for a classical Lie group. A. Klyachko connected eigenvalues of sums of Hermitian matrices to the saturated LR-cone and established defining linear inequalities. We prove analogues for the saturated NL-cone. This is based on work with Gidon Orelowitz, Nicolas Ressayre and Alexander Yong; see arxiv.org/abs/2005.09012, arxiv.org/abs/2009.09904, and https://arxiv.org/abs/2107.03152.

The $q$-Whittaker symmetric function associated to an integer partition is a $q$-analogue of the Schur symmetric function. We give a new formula for the $q$-Whittaker function in terms of partial flags compatible with a nilpotent endomorphism over the finite field of size $1/q$. We show that considering pairs of partial flags and taking Jordan forms leads to a probabilistic bijection between nonnegative-integer matrices and pairs of semistandard tableaux of the same shape, which we call the $q$-Burge correspondence. In the $q \to 0$ limit, we recover a known description of the classical Burge correspondence (also called column RSK). This is joint work with Hugh Thomas.

Jenna Rajchgot observed that the Castelnuovo-Mumford regularity of matrix Schubert varieties is computed by the degrees of the corresponding Grothendieck polynomials. We give a formula for these degrees. Indeed, we compute the leading terms of the top degree pieces of Grothendieck polynomials and give a complete description of when two Grothendieck polynomials have the same top degree piece (up to scalars). Our formulas rely on some new facts about major index of permutations. (Joint work with David Speyer and Anna Weigandt.)

A class of combinatorial structures is hereditary if membership in the class is closed under taking substructures. Hereditary classes have been extensively studied for a variety of objects, notably graphs and permutations. A central problem is to determine the number of objects of size $n$ in a given hereditary class. We discuss this problem for chord diagrams, perfect matchings of $[2n]$. After discussing past work enumerating hereditary classes defined by forbidding subdiagrams of size $2$ and $3$, we consider forbidding certain graphically-inspired infinite sets of subdiagrams. Rich enumerative relationships seem to emerge from these classes after imposing one of several connectedness notions. In particular, these classes connect to combinatorial maps, Catalan lattices, and uniquely-sorted permutations, conjecturally allowing for their enumeration.

Recently Okada defined algebraically ninth variation skew $Q$-functions, in parallel to Macdonald's ninth variation skew Schur functions. Here we introduce a skew shifted tableaux definition of these ninth variation skew $Q$-functions, and prove by means of a non-intersecting lattice path model a Pfaffian outside decomposition result in the form of a ninth variation version of Hamel's Pfaffian outside decomposition identity. As corollaries to this we derive Pfaffian identities generalizing those of Josefiak-Pragacz, Nimmo, and most recently Okada. This is joint work with Ron King.

The limit shape phenomenon is a "law of large numbers" for random surfaces: the random surface looks macroscopically like the "average surface". The first result of this kind was the celebrated arctic circle theorem for domino tilings of the aztec diamond. The limit shape has macroscopic regions with different qualitative behavior, and the arctic curve is the boundary separating these regions. The work of Kenyon, Okounkov, Sheffield and others has shown that periodic lattices with non-trivial Newton polygons lead to rich arctic curves with many frozen and gaseous regions. Groves are another model, closely related to spanning trees, that exhibits an arctic circle theorem, due to Petersen and Speyer. We compute arctic curves for groves with non-trivial Newton polygons using analytic combinatorics results of Baryshnikov, Pemantle and Wilson, and provide a geometric description of asymptotic edge probabilities.

The random-to-below shuffle of a deck of cards consists of removing any card randomly (with uniform probability), and inserting it anywhere below (with uniform probability). When looking at the eigenvalues of its transition matrix, they all seem to be rational and positive. This is surprising for a non-symmetric matrix, and suggests some combinatorial interpretation. We give a recursive explanation that involves standard Young tableaux and makes connection with the well studied top-to-random shuffle.

Since the invention of the compound microscope in the early seventeenth century, scientists have marvelled over red blood cells and their surprising shape. An influential model of Canham predicts the shapes of blood cells and similar biomembranes come from a variational problem minimizing the "bending energy" of these surfaces. Because observed (healthy) cells have the same shape in humans, it is natural to ask whether the model admits a unique solution. Here, we prove solution uniqueness for the genus one Canham problem. The proof builds on a result of Yu and Chen that reduces solution uniqueness to proving non-negativity of a sequence defined by an explicit linear recurrence relation with polynomial coefficients. We combine rigorous numeric analytic continuation of D-finite functions with classic bounds from singularity analysis to derive an effective index where the asymptotic behaviour of the sequence, which is positive, dominates the sequence behaviour. Positivity of the finite number of remaining terms can then be checked computationally.

We give an overview of the theory of growth functions for associative algebras and explain their significance when trying to understand algebras from a combinatorial point of view. We then give a classification for which functions can occur as the growth function of a finitely generated associative algebra up to asymptotic equivalence. This is joint work with Efim Zelmanov.

Recently, a formula for the symmetric Macdonald polynomials $P_{\lambda}(X;q,t)$ was given in terms of objects called multiline queues, which also compute probabilities of a statistical mechanics model called the multispecies asymmetric simple exclusion process (ASEP) on a ring. It is natural to ask whether the modified Macdonald polynomials $\widetilde{H}_{\lambda}(X;q,t)$ can be obtained using a combinatorial gadget for some other statistical mechanics model. We answer this question in the affirmative. In this talk, we will give a new formula for $\widetilde{H}_{\lambda}(X;q,t)$ in terms of fillings of tableaux called polyqueue tableaux. We define a multispecies totally asymmetric zero range process (TAZRP) on a ring with parameter $t$, whose (unnormalized) stationary probabilities are computed by polyqueue tableaux, and whose partition function is equal to $\widetilde{H}_{\lambda}(X;1,t)$. This talk is based on joint work with Arvind Ayyer and James Martin.

Dynamical Algebraic Combinatorics studies objects important in algebraic combinatorics through the lens of dynamical actions. In this talk, we give a flavor of this field by investigating ever more general domains in which the actions of promotion on tableaux (or tableaux-like objects) and rowmotion on order ideals (or generalizations of order ideals) correspond. This is based on joint works with J. Bernstein, K. Dilks, O. Pechenik, C. Vorland, and N. Williams.

In these difficult times, what we need to feel better is some colorful and elegant bijections. This talk introduces the work we did with Andrew Elvey-Price (Tours, France) and Irène Marcovici (Nancy, France). Together we answered an open question from Mortimer and Prellberg, asking for a bijection between a family of walks inside a bounded triangular domain (think about a large equilateral triangle subdivided in several smaller equilateral triangles) and the famous Motzkin paths, but which have bounded height. The used techniques for the proof are quite elementary, and seem to be robust. Indeed, in addition to solving Mortimer and Prellberg's conjecture, our approach enabled us to find a new surprising bijection between 3D-walks constrained inside a pyramid and some 2D-walks in a squared grid.

The $q,t$-Catalan numbers $\mathrm{Cat}_n(q,t)$ are polynomials in $q$ and $t$ that reduce to the ordinary Catalan numbers when $q=t=1$. These polynomials have important connections to representation theory, algebraic geometry, and symmetric functions. Work of Garsia, Haglund, and Haiman has given us combinatorial formulas for $\mathrm{Cat}_n(q,t)$ as sums of Dyck vectors weighted by area and dinv. This talk narrates our ongoing quest for a bijective proof of the notorious symmetry property $\mathrm{Cat}_n(q,t) = \mathrm{Cat}_n(t,q)$. We describe some structural decompositions of integer partitions into infinite chains that can be paired to prove the symmetry of certain coefficients in $\mathrm{Cat}_n(q,t)$. The chains are built from initial objects by applying an operator $\mathrm{NU}$ that increases dinv by $1$ and reduces area by $1$. A remarkable feature of these chains is that they are independent of $n$ and explain symmetry for all $n$ simultaneously. Our chain construction leads to a combinatorial proof that for all $k < 12$ and all $n$, the terms in $\mathrm{Cat}_n(q,t)$ of total degree $n(n-1)/2 - k$ satisfy the required symmetry property.

Several algebraic and geometric structures are most naturally encoded via graphs. These include restrictions, such as trees, and decorations, such as planar graphs, ribbon graphs, bi-partite graphs (aka. hypergraphs), directed versions, etc. Particularly nice properties satisfy some kind of hereditary condition. This affords a dual perspective. Either as (nested) subsets and decomposition, or as composition, gluing locally. Both views relate to category theory, algebra, and combinatorics in terms of finite sets, cospans etc. We will give examples of these phenomena and provide a general background.

I will talk about Wilson loop diagrams, explain a bit about what they are, and some of the combinatorial questions that come out of them, with a focus on when they are equivalent. This is joint work with Susama Agarwala and Zee Fryer.

In 1978, Neggers conjectured that a certain transform of the order polynomial of a partially ordered set (poset) has only real roots.

In the late 1980s, Stanley gave this to me as a thesis project, generalized to labelled posets. For my thesis I proved the conclusion for series-parallel labelled posets and a bit more. Brändén, and later Stembridge, found counterexamples to the conjecture in general.

But the conjecture remains open for some notable subclasses of the class of all posets. Ferrers posets -- unknown -- clearly important.

Posets for which the Hasse (covering) graph is a tree might be possible. The case in which the Hasse graph is a path is still open, but seems almost within reach given some new ideas.

Schubert polynomials form a basis of all polynomials and appear in the study of cohomology rings of flag manifolds. The vanishing problem for Schubert polynomials asks if a coefficient of a Schubert polynomial is zero. We give a tableau criterion to solve this problem, from which we deduce the first polynomial time algorithm. These results are obtained from new characterizations of the Schubitope, a generalization of the permutahedron defined for any subset of the n x n grid. In contrast, we show that computing these coefficients explicitly is #P-complete. This is joint work with Anshul Adve and Alexander Yong.

Intrinsic volumes of a convex body provide scalar data (volume, surface area, Euler characteristic, etc. ) about the geometry of a convex body independent of the ambient space. Minkowski tensors are the tensor-valued generalization of intrinsic volumes. They provide more complex geometric information about a convex body, such as its shape, orientation, and more.

Minkowski volume tensors are closely linked to the moments of the uniform distribution on a convex body, and a rational generating function for these moments allows us to extract the tensors symbolically. In this talk, we explain this connection and show that it can be extended to the setting of Minkowski "surface tensors". We demonstrate how this generating function approach allows us to give an explicit formula for these surface tensors in the case of simplicial polytopes.

No prior knowledge of Minkowski tensors will be assumed.

This is a joint work with Büşra Sert and Niklas Livchitz

The Harish-Chandra/Itzykson-Zuber (HCIZ) and Brezin-Gross-Witten (BGW) integrals are a pair matrix integrals which play a prominent role in quantum field theory. Remarkably, these ubiquitous special functions are also significant from the perspective of algebraic combinatorics: they are generating functions for certain classes of Hurwitz numbers. I will explain the connection between the HCIZ/BGW integrals and Hurwitz theory, and explain how it leads to a proof of some old conjectures concerning the asymptotic behavior of these integrals. This is based in part on joint work with Ian Goulden and Mathieu Guay-Paquet.

A Hopf algebra of double posets was introduced by Claudia Malvenuto and Christophe Reutenauer in 2011, motivated by the study of pictures of tableaux as defined by Zelevinsky. Starting from the correspondence between top-cones in the braid arrangement and partial orders, we investigate several properties of the Hopf algebra of double posets as the image of a Hopf monoid (via the Fock functor). In particular, we obtain a non-cancellative formula for the antipode. A description of the primitive space is also discussed.

We consider symmetric function analogues of the chromatic and Tutte polynomials on graphs whose vertices have positive integer weights. We show that in this setting these functions admit edge deletion-contraction relations akin to those of the corresponding polynomials, and we use these relations to give enumerative and/or inductive proofs of properties of these functions. In particular we note that the Tutte symmetric function in this form is related to a family of vertex-weighted graph functions, from which we derive a recipe theorem and a spanning-tree expansion. This is joint work with Sophie Spirkl.

In this talk we consider a poset structure on projective sign vectors. We show that the order complex of this poset is partitionable and give an interpretation of the $h$-vector using type B descents of the type D Coxeter group. Based on joint work with Nantel Bergeron and John Machacek, and is a continuation of the work by John Machacek. arXiv preprint: 2008.03794

The study of factorizations in the symmetric group is related to combinatorial objects like graphs embedded on surfaces and non-crossing partitions. We consider analogues for complex reflections groups of certain factorization problems of permutations first studied by Jackson, Schaeffer, Vassilieva and Bernardi. Instead of counting factorizations of a long cycle given the number of cycles of each factor, we count factorizations of Coxeter elements by fixed space dimension of each factor. We show combinatorially that, as with permutations, the generating function counting these factorizations has nice coefficients after an appropriate change of basis. This is joint work with Joel Lewis.

This talk will introduce spherical elements in a finite Coxeter system. These spherical elements are a generalization of Coxeter elements, that conjecturally, for Weyl groups, index Schubert varieties in the flag variety $G/B$ that are spherical for the action of a Levi subgroup. We will see that this conjecture extends and unifies previous sphericality results for Schubert varieties in G/B due to P. Karuppuchamy, J. Stembridge, P. Magyar–J. Weyman-A. Zelevinsky. In type A, the combinatorics of Demazure modules and their key polynomials, multiplicity freeness, and split-symmetry in algebraic combinatorics are employed to prove this conjecture for several classes of Schubert varieties. This talk is based on joint work with Alexander Yong.

The Robinson--Schensted (RS) correspondence is a bijection between permutations and pairs of standard Young tableaux which plays a central role in the theory of Schur polynomials. In this talk, I will present a $(q,t)$-dependent probabilistic deformation of Robinson--Schensted which is related to the Cauchy identity for Macdonald polynomials. By specialising $q$ and $t$, one recovers the row and column insertion algorithm as well as $q$- and $t$-deformations of RS; these have been introduced in recent years and are related to $q$-Whittaker and Hall-Littlewood polynomials respectively. I will also explain connections to a $(q,t)$-generalization of the Greene--Nijenhuis--Wilf random hook walk and the $q$-Plancherel measure. This is joint work with Gabriel Frieden.

This talk concerns the cohomology rings of complex Grassmannians. In 2003, Reiner and Tudose conjectured the form of the Hilbert series for certain subalgebras of these cohomology rings. We build on their work in two ways. First, we conjecture two natural bases for these subalgebras that would imply their conjecture using notions from the theory of $k$-Schur functions. Second we formulate an analogous conjecture for Lagrangian Grassmannians.

Joint work with Michael Feigen, Victor Reiner, and Ajmain Yamin.

Partial difference sets are highly structured group subsets that occur in various guises throughout design theory, finite geometry, coding theory, and graph theory. They admit only two possible nontrivial character sums and so are often studied using character theory. The central question is to determine which groups contain a partial difference set with two specified nontrivial character sums. We consider an apparently more difficult question: which groups contain a large disjoint collection of such partial difference sets? This leads us to identify a certain subgroup as containing important structural information about the packing. With this insight, we are able to formulate a recursive construction of packings in abelian groups of increasing exponent. This allows us to unify and extend numerous previous results about partial difference sets using a common framework.

This is joint work with Shuxing Li, a 2019-2021 PIMS Postdoctoral Fellow.

About fifteen years ago F. Brenti and V. Welker showed that the face enumerating polynomial of the barycentric subdivision of any Cohen-Macaulay simplicial complex has only real roots. It is natural to ask whether similar results hold when barycentric subdivision is replaced by more general types of triangulations, or when simplicial complexes are replaced by more general cell complexes. This talk will report on recent progress on these questions. For various special types of triangulations, there are strong connections to traditional combinatorial themes, such as the enumeration of permutations, words, signed permutations and ordered set partitions.

Recent investigations in Ehrhart theory suggested some conjectures involving interlacing relations among polynomials with only real roots, and Veronese sections of them. Revisiting some old theorems, we find as corollaries some new lemmas which have been overlooked for a long time. One of these lemmas directly implies a strong form of the motivating conjecture. Similar applications of the other lemmas are anticipated. This is ongoing joint work with Christos Athanasiadis (U. Athens).

A twisted Hopf algebra is a Hopf algebra in the category of linear species. The Fock functors allow to recover "classical" Hopf algebras from twisted ones. Numerous constructions and results can be lifted to the level of twisted bialgebras, such that cofreeness, shuffle and quasi-shuffles products, etc. Using two tensor products in the category of species, we define the notion of cointeraction of twisted bialgebras. Examples on finite topologies and graphs allow to reconstruct Ehrhart polynomials and chromatic polynomials in a canonical way from cointeraction of twisted bialgebras.

Given a graph and a set of colors, a coloring of the graph is a function that associates each vertex in the graph with a color. In 1995, Stanley generalized this definition to symmetric functions by looking at the number of times each color is used and extending the set of colors to $\mathbb{Z}^+$. In 2012, Shareshian and Wachs introduced a refinement of the chromatic functions for ordered graphs as $q$-analogues.

In the particular case of Dyck paths, Stanley and Stembridge described the connection between chromatic symmetric functions of abelian Dyck paths and square hit numbers, and Guay-Paquet described their relation to rectangular hit numbers.

Recently, Abreu-Nigru generalized the former connection for the Shareshian-Wachs $q$-analogue, and in unpublished work, Guay-Paquet generalized the latter. Both of these generalizations use the Garsia-Remmel $q$-hit numbers.

In this talk, I want to give an overview of the framework and present another proof of Guay-Paquet's identity using $q$-rook theory and use it to give a new proof of the Abreu-Nigru identity. This is recent work with Alejandro H. Morales and Greta Panova.

Calculations in perturbative QED (and also in QCD) use a reduction from Feynman integrals to 'master integrals'. In general, the reduction to master integrals is performed by excessive use of computer power. Some of the reduction identities, however, are very combinatorial (others not) in the sense that they have a simple graph theoretical description. I will (ab-)use the seminar to ask the following question: Is it possible to understand the (partial) reduction by these 'combinatorial' identities in a mathematically more satisfactory way? Note that this will not be an expert talk on QED. Nor will this talk present any deep results. It should rather be considered as a problem session.

Springer fibers and Nakajima quiver varieties are amongst the most important objects in geometric representation theory. While Springer fibers can be used to geometrically construct and classify irreducible representations of Weyl groups, Nakajima quiver varieties play a key role in the geometric representation theory of Kac--Moody Lie algebras. I will begin by first recalling some background on the objects of interest mentioned above. I will then connect Springer fibers and quiver varieties by realizing the irreducible components of two-row Springer fibers inside a suitable Nakajima quiver variety and describing the resulting subvariety in terms of explicit quiver representations. Next, consider certain fixed-point subvarieties of these quiver varieties, which were studied by Henderson--Licata and Li with the goal of developing the geometric representation theory for certain coideal subalgebras. By applying this machinery, I will give an explicit algebraic description of the irreducible components of all two-row Springer fibers for classical types, thereby generalizing results of Fung and Stroppel--Webster in type A. This is joint with C.-J. Lai and A. Wilbert.

The chromatic polynomial was generalized to the chromatic symmetric function by Stanley in his seminal 1995 paper. This function is currently experiencing a flourishing renaissance, in particular the study of the positivity of chromatic symmetric functions when expanded into the basis of elementary symmetric functions, that is, $e$-positivity.

In this talk we approach the question of $e$-positivity from various angles. Most pertinently we resolve the 1995 statement of Stanley that no known graph exists that is not contractible to the claw, and whose chromatic symmetric function is not $e$-positive.

This is joint work with Soojin Cho, Samantha Dahlberg, Angele Foley and Adrian She, and no prior knowledge is assumed.

Motivated by the universal property of the Connes-Kreimer Hopf algebra of rooted trees and Hopf subalgebras arising from so-called combinatorial Dyson-Schwinger equations, we introduce a class of two-variable recursive functional equations involving Hochschild 1-cocycle operators. An instance of this equation has been studied in the context of quantum field theory and found to be solved by an expansion over connected chord diagrams. We extend and generalize this line of work to show that these equations are solved by weighted generating functions for certain classes of connected weighted chord diagrams. We then look towards explaining why chord diagrams appear in the solutions by proving that the 1-cocycle property is equivalent to a differential equation related to Stein's recurrence for the number of connected chord diagrams and discuss how this work relates to other combinatorial objects, including weighted ordered trees and Stirling permutations.

An integer partition is called graphical if it can be realized as the size-ordered degree sequence of a simple graph (with no loops or multiple edges). In his 1736 paper on the Königsberg bridge problem, arguably the origin of graph theory, Euler gave a necessary condition for a partition to be graphical: its sum must be even. In the nineteenth century, counting the number of graphs with a fixed degree sequence was popularized by Cayley to describe the chemical bonds which could be formed between atoms. Here we prove that the probability that a uniformly chosen partition of size $n$ is graphical decreases to zero faster than a fixed power of $n$, answering a question of Pittel. Our probabilistic proof also implies an upper bound for the probability that two randomly chosen partitions are comparable in the dominance order.

This is joint work with Marcus Michelen (UI-C) and Somabha Mukherjee (Penn).

For an undirected graph, its sandpile group is an interesting isomorphism invariant-- it is a finite abelian group that describes the integer cokernel of the graph's Laplacian matrix. This talk will discuss joint work with G. Benkart and C. Klivans examining an analogous invariant for a complex representation of a finite group, built from what one might call its "McKay matrix". We will then discuss work with D. Grinberg and J. Huang which generalizes this to modules over finite-dimensional Hopf algebras.

A functional equation for 2-connected chord diagrams is derived, then is used to calculate asymptotic information for the number of 2-connected chord diagrams by means of alien derivatives applied to factorially divergent power series. The calculation extends the older result by D. J. Kleitman on counting irreducible diagrams. Namely, Kleitman’s result calculates the first coefficient of the infinite asymptotic expansion derived here and is therefore a linear approximation of the result presented here. In calculating the asymptotics this way we are following the approach M. Borinsky used for solving the asymptotic counting problem of general connected chord diagrams. The numbers of 2-connected chord diagrams and the sequence of coefficients of their asymptotic expansion amazingly also appeared, without being recognized, in physics contexts in the work of Broadhurst on 4-loop Dyson-Schwinger-Johnson anatomy, and among the renormalized quantities of quenched QED calculated by M. Borinsky. The underlying chord-diagrammatic structure of quenched QED and Yukawa theory is unveiled here.

I will introduce factorial Schur functions as they relate to my Master's thesis. Factorial Shur functions are a generalization of Schur functions with a second family of "shift" parameters. In 2009, Zinn-Justin reproved the answer to a tiling problem (the puzzle rule) with a toy fermionic model, using techniques from physics to extract the result. He showed the same tiles can be arranged to represent factorial Schur functions. The same facts can then be used to prove a Littlewood-Richardson rule for factorial Schur functions with the same first set of variables. I will go over the ideas of this proof and get into results that have built on top of this theory. For example, a Littlewood-Richardson rule for Grothendieck polynomials can be shown in a similar way.

The asymmetric simple exclusion process (ASEP) is a one-dimensional model of hopping particles that has been extensively studied in statistical mechanics, probability, and combinatorics. It also has remarkable connections with orthogonal symmetric polynomials in many variables such as Macdonald and Koornwinder polynomials. In this talk, I will discuss new formulas for Macdonald polynomials (joint work with Corteel and Williams) that arise from the study of the ASEP on a ring, and introduce a new notion of quasisymmetric Macdonald polynomials (joint with Corteel, Haglund, Mason, and Williams) that specialize to the quasisymmetric Schur polynomials defined by Haglund, Luoto, Mason, and van Willigenburg.

Consider a plane partition $P$ in an $a \times b \times c$ box. The rowmotion operator sends $P$ to the plane partition generated by the minimal elements of its complement. We show rowmotion resonates with frequency $a+b+c-1$, in the sense that each orbit size shares a prime divisor with $a+b+c-1$. This confirms a 1995 conjecture of Peter Cameron and Dmitri Fon-Der-Flaass. (Based on joint works with Kevin Dilks & Jessica Striker and with Becky Patrias.)

Feynman graphs in quantum electrodynamics are essentially chord diagrams with photon edges taking the role of chords attached to lines or cycles given by electron edges. The associated Feynman integrals involve traces of Dirac gamma matrices whose computation leads to large sums of scalar Feynman integrals (cf. the earlier talk by O. Schnetz). I will present a method to compute these traces combinatorially by counting certain coloured subgraphs in chord diagrams.

We will start by comparing Macdonald's formula of the generating function for the symmetric powers of a compact complex manifold and Grothendieck's formula of the zeta series of a projective variety over a finite field, an explicit version of Dwork's rationality result. After seeing a common generalization of the two formulas, we will see how it is related to a classical theorem in combinatorics called the Pólya enumeration theorem, which has to do with counting colorings of a graph modulo symmetries. If time permits, we will discuss another version of this enumeration theorem with distinct vertices, a geometric analogue of which is a joint work with Yifeng Huang.

A covering graph is a structure obtained from a graph by ‘replacing’ every vertex with a coclique of size $r$. The main focus of this talk is connections between (spectral) characteristic of a cover and properties such as being walk- or distance- regular.

The algebro-geometric invariant on Feynman Diagrams called the $c_2$ invariant is a useful tool for detecting properties of Feynman periods. We present this identity on graphs that originate from the scalar $\phi_4$-theory with a purely combinatorial perspective and go over some strategies for computing it. We will further narrow our focus onto the circulant family of graphs and present some explicit results.

Motivated by questions in quantum field theory, we introduce a purely combinatorial problem of counting subdivergence-free gluings of trees. We present closed-form expressions counting subdivergence-free gluings for four different families of trees, as well as an algorithm to count subdivergence-free gluings of arbitrary pairs of trees. This is joint work with Clair Dai and Karen Yeats.

This is a reprise of a survey talk I gave at the East Coast Combinatorics Conference in August 2019, and a variation of one I gave in the graphs and matroids seminar last year. Some of you have seen some of it before, but Karen wanted to see it and it's "in the can", so we'll do this to get the seminar started. I also promise you another talk on a different subject later in the term.

I will discuss some places matroids have appeared in my work in quantum field theory, including some older work on numerator structure with Dirk Kreimer and some work in progress with Iain Crump on period identities. I will also explain why I think matroids should appear even more. I will not, except in passing, talk about the story of positroids even though they also fit the title. This talk stands alone but is also my response to and reason for requesting David's Jan 16 talk.

A "pattern of length $k$" is simply a permutation of $\{1, \dots, k\}$. This pattern is said to be contained in a permutation of $\{1, \dots, N\}$ (for $N > k$) if there is a subsequence of $k$ elements of the (long) permutation that appears in the same relative order as the pattern. (E.g. the pattern $312$ is contained in the permutation $2463175$ because the latter contains the subsequence $615$.) A permutation avoids the pattern $P$ if it does not contain $P$. For a given $P$, let $\mathrm{AV}[N;P]$ be the set of permutations of $\{1, \dots, N\}$ that avoid $P$. The cardinality of $\mathrm{AV}[N;P]$ has been extensively studied by combinatorialists. This talk looks at properties of permutations drawn uniformly at random from $\mathrm{AV}[N;P]$ for large $N$. When such a permutation is plotted as a function from $\{1, \dots, N\}$ to itself, some striking structure appears. I shall describe what is known probabilistically about such structure, including clustering and large empty regions in the plot. I shall also describe an attempt (with Justin Troyka) to study these plots under periodic boundary conditions, inspired by work in statistical physics. This led us to consider affine permutations on the integers with a new boundedness condition.

In this talk, we consider families of finite sets that we call shellable and that have been characterized by Chang and Hirst and Hughes as being the families of sets that admit unique solutions to Hall's matching problem. We prove that shellable families can be characterized by using a generalized notion of hook-lengths; hook-lengths originate from the hook-length formula which is used to determine the number of standard Young tableaux on partition shapes. Then, we introduce a natural generalization of standard skew tableaux and Edelman and Greene's balanced tableau, then prove existence results about such a generalization using our characterization of shellable families. We also calculate the average number of such tableaux using a hook-length formula.

Springer fibers are subvarieties of the flag variety that play an important role in combinatorics and geometric representation theory. In this talk, I will discuss joint work with Martha Precup where we analyze the equivariant cohomology of Springer fibers in type A. We define a basis for the equivariant cohomology of a Springer fiber, generalizing a monomial basis of the ordinary cohomology defined by De Concini and Procesi and studied by Garsia and Procesi. Our construction yields a combinatorial framework with which to study the equivariant and ordinary cohomology rings of Springer fibers. As an application, we identify an explicit collection of Schubert classes whose images in the cohomology ring of a given Springer fiber form a basis.

The Hall algebra of a finitary category is an associative (and sometimes Hopf) algebra whose structure constants count the number of extensions between objects. Classical examples include categories of quiver representations over a finite field, in which case the Hall algebra contains (half) the corresponding quantum group. When this construction is applied to non-additive categories built from combinatorial objects, such as trees, graphs, matroids, etc. it produces combinatorial Hopf algebras (some previously studied, some new).

One source of combinatorial examples arises from algebraic geometry over $\mathbb{F}_1$ -- the field of one element. I will discuss joint work with Jaiung Jun which attaches to a smooth projective toric variety a Hall algebra of coherent sheaves in this setting. These can be thought of as gluing together skew shapes.

In 1993, Jim Haglund conjectured the following. If $A$ is a square matrix of real numbers which are weakly decreasing down each column, and $J$ is the all-ones matrix of the same size, then the permanent of the matrix $xJ+A$ is a polynomial with only real roots. Jim, Ken Ono and I proved this for $0$-$1$ matrices in 1995. Using the multivariate generalization of polynomials with only real roots developed by Borcea and Brändén, in 2010 Brändén, Haglund, Visontai and I proved a multivariate generalization of the whole thing. I will present the main ideas of the proof.

Independence polynomials are generating functions for the number of independent sets of each cardinality in a graph $G$. In addition to encoding useful information about the graph (such as the number of vertices, the number of edges and the independence number), the analytic and algebraic properties can say much about the shape and inter-dependence of the coefficients. In this talk we'll focus on the nature and location of the roots of such polynomials, and even cross paths with a fractal or two! This research is joint with Ben Cameron, Iain Beaton, Karl Dilcher, Richard Hoshino and Richard Nowakowski.

We begin the talk by discussing a question in Invariant Theory: given a representation $V$ of a Lie group $G$, when if the invariant ring $k[V]^G$ a polynomial ring? We give a conjectural answer which we have verified for $\mathrm{SL}_n$ and discuss some combinatorial questions motivated by the proof. This is joint work with Dan Edidin.

A pair of skew shapes are said to be (skew) equivalent if they admit the same number of semistandard tableaux of any weight; i.e. if their associated skew Schur functions are equal. A result of Billera, Thomas, and Van Willigenburg gives a complete combinatorial characterization of equivalences between ribbon shapes (those with no $2 \times 2$ square) but the general case is more complicated. I will give a brief survey of existing results in the area and then discuss some recent work of my own.

I'll talk about some joint work with Julien Courtiel where some nice enumerative combinatorics tells us something about how gauge theories are such special quantum field theories.

In 2016, Duffy et al. introduced the following process on a graph. Initially, each vertex has some integer number of ''chips'' placed there (possibly negative). Thereafter, in discrete time steps, if an edge has more chips at one end than at the other, then one chip moves along that edge from the richer to the poorer end. All edges are processed in parallel at each time step. Duffy et al. observed experimentally that the dynamics of this process was eventually periodic of period one or two. This was proven in 2017 by Long and Narayanan. I will give their proof generalized to the context of real hyperplane arrangements, explain the analogies with the heat equation, and present some conjectures about what happens when the system is held out of equilibrium by some external sources and sinks of chips.

The classic approach to scattering amplitudes sums a contribution from a (potentially very large) number of Feynman diagrams. Over the past decade, Arkani-Hamed and his collaborators have developed a new approach, in which the sum of Feynman diagrams is replaced by a single geometrical object. For $N=4$ SYM, this object is now known as the amplituhedron. More recently, Arkani-Hamed, Bai, He, and Yan, studying a simpler (biadjoint scalar $\phi^3$) quantum field theory, discovered that the object playing the role of the amplituhedron is in fact a well-known polytope: the associahedron, originally defined by Stasheff some fifty years ago in the context of algebraic topology, with a lovely combinatorial structure which I shall explain. I will present​​​​​​​ Arkani-Hamed's approach using this simple model as an example, and discuss some subsequent work in collaboration with Arkani-Hamed, He, and​​​​​​​ Salvatori, in which we were led to define an infinite-dimensional​​​​​​​ associahedron. I will not assume previous familiarity with either scattering amplitudes or associahedra.

We define two generalizations of the totally nonnegative Grassmannian and determine their topology in the case of real projective space. We find the spaces to be PL manifolds with boundary which are homotopy equivalent to another real projective space of smaller dimension. One generalization makes use of sign variation while the other uses boundary measurement. Spaces arising from boundary measurement are shown to admit Cohen-Macaulay triangulations.

I will survey some ideas related to how pair-partitions are used in non-commutative probability in order to establish simplified combinatorial versions of the well-known Central Limit Theorem. Among the probability distributions which can appear as "limit law", the emphasis of the talk will be on a 1-parameter family of laws which interpolates between the classical Gaussian law and its counterpart in free probability, the semicircle law of Wigner. On a combinatorial level, the study of this interpolating family of laws boils down to counting crossings (or, more generally, oriented crossings) in pair-partitions of the sets $\{1, 2, \dots, 2n\}$.

Given an algebraic hypersurface $H \subset \mathbb{R}^n$, we can always 2-colour the components of the complement $\mathbb{R}^n \setminus H$ such that adjacent components are of opposite colours. However, this property does not necessarily continue to hold if we replace $\mathbb{R}^n$ by a space with a non-trivial topology (e.g. a torus). We wanted to know: does this 2-colouring property hold for open Richardson varieties in the real Grassmannian? It turns out, the answer is yes. To prove this, we showed that the coordinate ring of open Richardson variety is a unique factorization domain over any field, which implies the result. Our proof uses a non-trivial theorem of Knutson-Lam-Speyer about positroid varieties. This is joint work with Jake Levinson.

We study arrangements of slightly skewed tropical hyperplanes, called blades, on the vertices of a hypersimplex $\Delta_{k,n}$. We reformulate the condition under which such an arrangement induces a matroid (in fact a positroid) subdivision as the requirement that the collection of vertices defines a weakly separated collection of $k$-element subsets, in the sense of the work of Leclerc and Zelevinsky on quasicommuting families of quantum minors.

This talk is based on 1910.11522 and 1810.0324.

A tree is invertible if and only if it has a perfect matching.

Godsil considers an invertible tree $T$ and finds that the inverse of the adjacency matrix has entries in $\{0, \pm 1\}$ and is the signed adjacency matrix of a graph which contains $T$. In this talk, we give a new proof of this theorem, which gives rise to a partial ordering relation on the class of all invertible trees on $2n$ vertices. Though properties of graphs are related to their eigenvalues, this relationship is not, in general, a quid pro quod relationship. In this case however, we are able to define an operation which changes an invertible tree $T$ to a non-isomorphic invertible tree $T'$ whose median eigenvalue is strictly greater. This extends naturally to a partial ordering of the class of invertible trees. We characterize the maximal and minimal elements of this poset and discuss some applications.

I will report on recent joint work with Karen Vogtmann on the Euler characteristic of $\mathrm{Out}(F_n)$ and the moduli space of graphs. A similar study has been performed in the seminal 1986 work of Harer and Zagier on the Euler characteristic of the mapping class group and the moduli space of curves. I will review a topological field theory proof, due to Kontsevich, of Harer and Zagier's result and illustrate how an analogous 'renormalized' topological field theory argument can be applied to $\mathrm{Out}(F_n)$.

We call a particular operation on a graph which converts one triangle into two triangles a double triangle expansion, and call all those graphs which can be obtained from repeated double triangle expansions of a fixed graph the double triangle descendants of the graph. The class of double triangle descendants of the graph $K_5$ seem to be a very important class of graphs for quantum field theory. Notably a conjecture of Brown and Schnetz says they are special since they appear to be precisely those graphs where a particular arithmetic invariant, the $c_2$ invariant, is $-1$ for all primes. Consequently it is interesting to enumerate double triangle descendants of $K_5$. I will describe some partial results in this directions.

This is joint work with Marni Mishna and Mohamed Laradji.

The Mukhin-Tarasov-Varchenko (MTV) theorem is the following statement in real algebraic geometry. If the wronskian of a set of complex polynomials has only real roots, then the vector space spanned by these polynomials is real. This may seem like innocent curiosity, but it has a variety of applications in geometry, representation theory, and combinatorics. It is also highly non-trivial. Their proof is both a tour de force, and mindbogglingly complicated.

Recently Jake Levinson and I found a new way to prove the MTV theorem. This happened because we conjectured a generalization, and then noticed that the generalization is actually easier to prove than the original theorem. I will talk about what our generalization says, and how it came about.

Given a finite lattice $L$, consider the ring of complex polynomials in indeterminates indexed by $L$, modulo the ideal generated by $(X_a X_b - X_{a \vee b} X_{a \wedge b}$ for all $a, b \in L)$. Hibi showed that this is an integral domain if and only if $L$ is distributive, in which case the corresponding projective variety is toric. We describe the orbit decomposition and singularities of this variety in terms of the poset of join-irreducibles of $L$, and introduce a class of posets motivated by other questions about its geometry.

Arborified zeta values are a generalisation to rooted trees of the usual multizeta values. I construct these objects using a universal property of the algebra of rooted forests which allows to branch morphisms. I will present this construction without spending too much time on analytical details and will instead insist on the algebraic properties of these numbers, in particular their relations with new shuffle products of trees.

In 1998, Connes and Kreimer introduced a combinatorial Hopf algebra $\mathcal{H}_{\mathrm{CK}}$ on the vector space of forests of rooted trees that precisely explains the phenomenon of renormalization in quantum field theory. This Hopf algebra has been of great interest since its inception, as it connects the disciplines of algebra, combinatorics, and physics, providing interesting questions in each. In this thesis we introduce the notion of higher-order renormalization group equations, which generalize the usual renormalization group equation of quantum field theory, and further define a corresponding notion of order on certain sequences of trees constituting elements of the completion of $\mathcal{H}_{\mathrm{CK}}$. We also give an explication of a result, due to Foissy, that characterizes which sequences of linear combinations of trees with one generator in each degree generate Hopf subalgebras of $\mathcal{H}_{\mathrm{CK}}$. We conclude with some results towards classifying these sequences by their order (when such an order is admitted), and by presenting a new family of second-order sequences of which the sequence of generators of the Connes-Moscovici subalgebra is a member.

A Knutson-Tao-Woodward puzzle is a tiling of a triangle with certain pieces that have labeled edges. The puzzle rule states that number of puzzles with a given boundary is equal to a Littlewood-Richardson coefficient. I will present a proof of this due to Zinn-Justin which relates the problem to the time evolution of a set of fermions. Transfer matrices describing discrete time steps are applied to elements in the state space of a set of fermions known as Fock space. Repeated applications of the Yang-Baxter equation can "unzip" the transfer matrices, showing they are commutative, which yields the result.

Schur functions determinantal identities (e.g. Jacobi-Trudi, Giambelli) are cornerstones of symmetric function theory. Less well-known are the Pfaffian identities for Schur $Q$-functions. In this talk we give an introduction to this parallel Pfaffian universe and review the known identities. Along the way we show that a recent result of Okada is a special case of general theorem (Hamel 1996), and that our Pfaffian identities apply to a number of Schur $Q$-function variations, including factorial.

This is joint work with Ron King.

The solution group of binary linear system is a quantum-probabilistic generalization of the solution space of the system. An incidence group is a solution group where the linear system comes from the incidence matrix of a graph. Incidence groups can be defined quite naturally in terms of the underlying graph, and appear to walk an interesting line: they are much more tractable than the class of all solution groups, but still have a non-trivial theory, which is closely connected with the theory of graph minors. In this talk, I'll review what twe know about incidence groups, and give a number of open problems. Contains joint work with Cnnor Paddock, Vincent Russo, and Turner Silverthorne.

The asymptotic study of multivariate generating functions comprises the domain of Analytic Combinatorics in Several Variables (ACSV).​​​​​​​ Although the techniques of ACSV parallel a better known​​​​​​​ univariate theory, the pathologies which arise in the analysis of multivariate functions leads to many intriguing -- perhaps, in general, undecidable -- questions. This talk focuses on two issues: asymptotic transitions between different sequences encoded by one, typically rational, multivariate generating function, and the use of Morse theory to provide strong structure results for possible asymptotic behaviour. These results can be combined with computer algebra software to provide rigorous asymptotic proofs, and present the most promising attack on the "connection problem" for so-called P-recursive sequences. Applications discussed include quantum computing, queuing theory, and automatic proofs of transcendence.

We begin with Joni and Rota's definition of the incidence coalgebra of a category or partially ordered set and then discuss some cases where a monoidal product on a category turns this coalgebra into a bialgebra. We will see that some familiar combinatorial Hopf algebras in this way, either directly or as quotient bialgebras.

Let $\lambda$ be a partition. The reverse plane partitions of shape $\lambda$ are a kind of filling of the Ferrers diagram of $\lambda$ by non-negative integers. Richard Stanley found the generating function which enumerates them according to the sum of the entries. This series suggests that reverse plane partitions should be thought of as being built out of elementary building blocks corresponding to the boxes of $\lambda$. This then leads to the question of how to divide a reverse plane partition up into its component pieces. The first way to do this was found by Hillman and Grassl, thereby giving a​​​​​​​ bijective proof of Stanley's result. I will present a simple way to accomplish the same thing which goes via quiver representations. (The simplicity is due to letting representation theory take care of the​​​​​​​ combinatorics for us.) No prior knowledge of quiver representations will be assumed. This talk is based on joint work with Al Garver and Becky Patrias, arXiv:1812.08345.

The Lewis Carroll identity expresses the determinant of a matrix in terms of subdeterminants obtained by deleting one row and column or a pair of rows and columns. Using the matrix tree theorem, we can convert this into an equivalent identity involving sums over pairs of forests. Unlike the Lewis Carroll Identity, the Forest Identity involves no minus signs. Using the Involution Principle, we can pull back Zeilberger's proof of the Lewis Carroll Identity to a bijective proof of the Forest Identity. This bijection is implemented by the Red Hot Potato algorithm, so called because the way edges get tossed back and forth between the two forests is reminiscent of the children's game of hot potato.

A standard Young tableau with entries $-M < \cdots < -1 < 1 < \cdots < M$ can be "folded" by performing certain conversions and rectifications in sequence. This operation yields a tableau with entries $1' < 1 < \cdots < M' < M$ and has a remarkable relationship to a generalization of Schensted insertion known as "mixed insertion". Using a connection between domino insertion and mixed insertion, we can show that folding yields a bijection between certain tableaux with rotational symmetry (obtained from specific shifted tableaux) and symmetric domino tableaux (that are in one-to-one correspondence with standard square tableaux). Similar bijections arise when considering other symmetries, some of which admit geometric explanations.

If we take two rooted trees with the same number of leaves and form a graph by gluing the leaves of one tree to the other, then we say the graph is subdivergence-free if no 2-edge cut have been generated. We are interested in counting the number of ways to do this and will discuss some special cases where we can obtain some enumerative results. The problem is motivated by the study of primitive Feynman diagrams with cuts, and how they can be recreated from the cut piece.

A classical problem asks us to find, for each element $A[i]$ of an array of integers, the position of the nearest smallest element. Similarly, we can ask about the dual problem: for each element of an array of integers $A[i]$, what is the position of the furthest smaller element? In our paper, we discussed both these problems from a combinatorial perspective and considered algorithms to solve them. By examining results of permutations of distinct integers and behaviour of the algorithms, we find many classical combinatorial sequences such as the Stirling numbers, the Catalan numbers, the Bell numbers, and the harmonic numbers.

In this talk I will define the so-called Feynman graphs, which are a particular class of graphs appearing in quantum field theory. Ribbon graphs (or fat-graphs or graphs on surfaces), which appear as Feynman graphs of the so-called matrix models, will then be defined. I will then introduce tensor graphs, which are a natural generalization of ribbon graphs. These tensor graphs are Feynman graphs of the so-called tensor models. In the last part of the talk I will present several results on tensor graphs, such as the identification of the class of dominant graphs in a particular expansion of these tensor models.

Our focus here is the Kronecker function whose inputs are three partitions of bounded length, and whose output is the corresponding Kronecker coefficient. The piecewise quasipolynomiality of this function has been the center of much interest. This phenomenon is a consequence of the powerful $[Q, R] = 0$ theorem of Meinrenken-Sjamaar [MS99] on the piecewise quasipolynomial behavior of multiplicity functions. This talk will describe an elementary approach to determining this property, and discuss how to effectively compute coefficients. This is work in progress with Mercedes Rosas and Sheila Sundaram.

A rooted map is a connected graph in which the half-edges have been cyclically ordered around each vertex. This talk addresses the question of the asymptotic behavior of several parameters of maps (such as number of vertices, the root degree) without any constraint on the genus of the maps. Although this perspective is quite opposed to the classical one where maps model discrete surfaces (and so where the genus is important), this has numerous applications in transverse scientific areas, like Quantum Field Theory or lambda-calculus. Thus, as a motivation, we begin by introducing the existing connexions between combinatorial maps and other families of objects. Then, we explain the (new!) techniques required to solve the underlying enumerative problem, and show why they must differ from the ones used when the genus is fixed. Finally, we stand back a bit, and ask ourselves whether the asymptotic results could have been thought ahead, given the previously mentioned combinatorial connexions. This is a joint work with Olivier Bodini, Sergey Dovgal and Hsien-Kuei Hwang.

Schubert polynomials were introduced by Lascoux and Schützenberger in 1982 to study Schubert varieties. They have been intensely studied since and remain a central object in algebraic combinatorics. Macdonald showed in 1991 that the principal specialization, i.e. setting all variables to one, gives a weighted sum over reduced words. In 2017 Stanley conjectured that the limit of this specialization exists as $n$ goes to infinity and asked the question of what kind of permutations maximize the value for fixed $n$. Merzon and Smirnov conjectured that this maximum is achieved on layered permutations. We resolve Stanley’s problem restricted to layered permutations and find the shape of such a permutation. Joint work with Igor Pak and Greta Panova. No knowledge of Schubert polynomials is required.

A $p$-tandem quarter plane walk is a walk starting at the origin and remaining in the first quadrant using the steps $(1,-1)$ and $(-n, p-n)$ for $n$ between 0 and $p$. A $p$-Łukasiewicz walk is like a Motzkin path but using the steps $(-1, 0, 1, \dots, p)$. I will describe a bijection between these classes. This bijection can be implemented in a single pass using certain transition rules and keeping track of two parameters. Unlike most of what I do there is absolutely no physics here; I hope the audience will enjoy some pure bijective combinatorics. This is joint work with Frédéric Chyzak.

The circulation algebra is a commutative graded algebra associated to a graph, introduced by Wagner in 1998 to study flows. Its graded dimension is given by a certain specialization of the graph's Tutte polynomial, and it encodes information about the combinatorics of circuits in the graph roughly equivalent to the cycle matroid. Somewhat surprisingly, algebras of essentially the same kind appear in areas as seemingly distant as numerical approximation theory and the geometry of homogeneous manifolds. I will give a survey of the basic theory of these algebras and the different contexts in which they have been studied, including my own research.

The study of integer partitions is a classic subject with applications ranging from number theory to representation theory and combinatorics. More recently, interest in the subject has been driven by a connection to Kronecker coefficients, which are at the heart of certain geometric complexity theoretic approaches to resolving P vs NP. This talk examines the number $N_n(l,m)$ of integer partitions of $n$ fitting inside an $m \times l$ rectangle, equal to the $q^n$ coefficient of the $q$-binomial $\binom{m+l}{l}_q$. We give an exact asymptotic formula when $l = Am$ and $n = Bm^2$ for constants $A$ and $B$; our approach uses a carefully chosen probability distribution on partitions to apply a local central limit theorem.

In 1856, Cayley conjectured that for fixed $l$ and $m$ the sequence $N_n(l,m)$ is unimodal in $n$ (i.e., increases until hitting a maximum at $n=lm/2$ and then decreases). This was proven by Sylvester in 1878 via the representation theory of $\mathfrak{sl}_2$, however the sequence was not known to be strictly unimodal until 2013, when Panova and Pak used the fact that consecutive differences of terms in the sequence are Kronecker coefficients. We give the first asymptotic proof of unmodality, and the first effective bounds for consecutive differences of terms.

Joint work with Greta Panova and Robin Pemantle.

Terminal chords are special chords in rooted connected chord diagrams that come up in solving Dyson-Schwinger equations in quantum field theory. Julien Courtiel and I were recently able to prove a conjecture of mine from 2014 on the asymptotics of terminal chords in connected chord diagrams. The proof proceeds by viewing connected chord diagrams as the trees in a particular family of combinatorial bridges and running the kernel method on a functional equation in three variables.

The field of analytic combinatorics studies the asymptotic behaviour of sequences through analytic properties of their generating functions. Although this approach has a long and rich history in the univariate domain, a systematic study of multivariate asymptotics using complex analysis was only initiated in the late 1990s by Pemantle and Wilson. In this talk, we give an overview of Analytic Combinatorics in Several Variables and discuss how effective computation and computer algebra are used to make these asymptotic techniques practical and explicit.

Given a permutation (or more generally an element in a finite reflection group) $w$, one can define a hyperplane arrangement $A(w)$ whose hyperplanes are the inversions of $w$. When $w$ is the longest element in the symmetric group, this is the classical braid arrangement. We study when $A(w)$ is a free arrangement in the sense of Terao. Using a geometric characterization of supersolvability, we show that $A(w)$ is free if and only if it is supersolvable for all vexillary permutations, 2143-avoiding permutations, and all permutations in $S_n$ for $n \le 7$. The characterization of freeness relies on the notion of the coessential set of $w$, which is closely related to the diagram of a permutation.

Inspired by the theory pioneered by Stanley's chromatic symmetric function and its connection to Hopf algebras, we study chromatic invariants on families of simplicial complexes. By endowing the vector space spanned by finite simplicial complexes with a Hopf structure, we construct a polynomial invariant that generalizes the chromatic polynomial of a graph and the chromatic symmetric function of Stanley. We discuss various combinatorial properties and applications to topological combinatorics.

The $g$-theorem is a powerful result that characterizes the face numbers of simplicial polytopes. The theorem was conjectured by McMullen and proved by Billera and Lee (sufficiency) and Stanley (necessity, using the Hard Lefschetz theorem from algebraic geometry). McMullen later found a proof of necessity using only elementary techniques, developing the polytope algebra and the McMullen-Hodge-Riemann relations. I will give an accessible introduction to McMullen's proof and its connections to recent developments in combinatorial Hodge theory.

From a root system, one may consider the arrangement of reflecting hyperplanes, as well as its toric and elliptic analogues. These arrangements are known to possess rich combinatorial, topological, and algebraic properties. In this talk, we consider families of arrangements associated to infinite families of Weyl groups (such as type $A$, $B$, $C$, $D$) and study the stability of the cohomology groups of their complements under the action of the corresponding Weyl groups, in the sense of Church-Ellenberg-Farb representation stability.

Resonance varieties are cohomological invariants of topological spaces; in the case of a complex hyperplane arrangement complement, they are determined by the underlying matroid. We define and investigate higher resonance varieties associated to matroids, building on the Orlik-Solomon algebra. We will discuss their geometric meaning and show connections to the combinatorics of flats and broken circuits.

Symmetric informationally complete measurements (SICs) are highly symmetric structures in the Hilbert space of quantum mechanics that correspond to equiangular lines. They have deep connections to algebraic number theory (specifically Stark conjectures and Hilbert's twelfth problem), finite geometry, and algebraic combinatorics. I will give an overview of SICs, their construction, and recent mathematical progress.

I will present a technical lemma about extending matrix algebras by an idempotent. There will be no movie references, fun, or combinatorics. Specifically, I will be presenting Jones' Basic Construction. Given a matrix algebra $A$ equipped with a trace function and a subalgebra $B$, there is a unique idempotent which projects $A$ onto $B$ in a way that plays nicely with the trace function. Jones' Basic Construction is the algebra obtained by adjoining this idempotent to $A$. Its structure is completely determined by $B$ and knowledge about the restriction from $A$ to $B$. Jones' Basic Construction has applications anywhere you might expect the words "subfactors of type $\mathrm{II}_1$ von Neumann algebras" to pop up, including: knot theory (where it is related to the Jones' polynomial), statistical mechanics (where it is related to certain transfer matrix algebras), and representation theory (where it is used to study certain towers of algebras). I will not be discussing any of these applications; I will only be giving technical results about the basic construction itself.

A parking function can be thought of as a Dyck path of length $n$ where the vertical edges are labeled with the integers $1$ through $n$, increasing in the columns. Haglund's "shuffle conjecture" from 2005 is a combinatorial formula for the symmetric function expression for $\nabla(e_n)$ with one term for each parking function. In 2008 Haglund, Morse and myself extended this conjecture to the action of $\nabla$ on a symmetric function indexed by a composition. The combinatorial formula has one term for each parking function which touches the diagonal according to the composition. In the last few years Hicks, Garsia, Xin and myself were partially able to prove the compositional shuffle conjecture by showing algebraic recurrences on coefficients exist and agree with the combinatorics. In this talk I'll show algebraic recurrences which generalize those that were used to prove those results and draws a direct connection with Macdonald symmetric functions.

We sketch the construction of a finitely generated graded abelian group $K^\bullet(X)$ associated to a graph $X$ which encodes Kirchhoff's Current Law on $X$ and all its contractions, in such a way that $\operatorname{Hom}(K^1(X), \mathbb{R})$ is the familiar (real) cycle-space of $X$. There is a split "deletion-contraction" short exact sequence which shows that $K^\bullet(X)$ is torsion-free and that its Poincaré polynomial is a specialization of the Tutte polynomial of $X$. Functoriality of $K^\bullet$ implies a functorial coalgebra structure: dualizing, we obtain a $B$-algebra structure on $\operatorname{Hom}(K^\bullet(X), B)$ for any commutative ring $B$, functorial in both arguments. This leads to some inequalities for the numbers $d_j(X) = \operatorname{rank}(K^j(X))$, and a nice presentation for $\operatorname{Hom}(K^\bullet(X), \mathbb{Q})$. This all looks suspiciously like the homology and cohomology of some complex algebraic variety $X$ which is a contravariant functor of $X$. Wish I knew what $X$ is.

Braids can be thought of in many different ways, most commonly as a geometric object or as a group. In this talk I will review a certain class of braids (positive braids) that is not a group, but a monoid. The braid monoid admits many combinatorial properties, and turns out to be a relatively easy thing to count. I will go through a few of these properties and derive a generating function with respect to length. Also come to learn about the role that the jedi force has on learning about braids.

I will talk about a class of ideals whose generators are of the form "all of the size $d$ minors of the northwest $i \times j$ portion of a matrix of variables." I'll talk about how these ideals and their generators lead to some nice combinatorics involving the antidiagonals of the determinants and how the combinatorics can be used to intersect these ideals. In other words, I'll explain how to make interesting combinatorics from algebra and then how to use the combinatorics to do algebra. I'll also say something about geometric reasons involving the flag manifold and group actions that this is an interesting class of ideals.

In this talk we look at the efficacy of encoding D-finite generating functions (those satisfying linear differential equations) as diagonals of multivariate rational functions, with a particular focus on lattice path models in restricted regions. By combining the popular "kernel method" for lattice path problems with recent results in the field of analytic combinatorics in several variables, this approach allows us to determine general formulas for the dominant asymptotics of counting sequences of certain symmetric models restricted to $d$-dimensional orthants. The exponential growth of each model is given by the number of steps, while the sub-exponential growth depends only on the dimension of the underlying lattice and the number of steps moving forward in each coordinate. These expressions are derived by analyzing the singular variety of a multivariate rational function whose diagonal counts the lattice paths in question.

Algebraic geometry poses many positivity challenges to enumerative combinatorics. Two notable such challenges are Macdonald's positivity conjecture, and structure constants in the Schubert Calculus. This talk will explain how Assaf's solution to the first, through her new method for showing symmetry and Schur-positivity of quasi-symmetric generating functions, may be applied to resolve a (by now old) problem of the positivity of some of the structure constants in the Schubert calculus of the flag manifold. This is joint with with Nantel Bergeron and Sami Assaf.

This will be an informal talk about knot invariants which will not presuppose any background in knot theory. The theory is rich in combinatorial constructions combined with specific algebras. Thus the area also serves as a rich context for studying the combinatorial and enumerative properties of these algebras. Such algebras include Hopf algebras and Lie algebras, and their combinatorialisation (for example, through Penrose's diagrammatic tensor calculus). My aim is to reach the Reshetikhin-Turaev Theorem for constructing invariants, and then to show how the Jones polynomial, the invariant of the title, may be deduced from it.

This is a continuation of a talk which began last week. It will be an informal talk about knot invariants which will not presuppose any background in knot theory. The theory is rich in combinatorial constructions combined with specific algebras. Thus the area also serves as a rich context for studying the combinatorial and enumerative properties of these algebras. Such algebras include Hopf algebras and Lie algebras, and their combinatorialisation (for example, through Penrose's diagrammatic tensor calculus).

In this talk, I will introduce an operator $B$ that acts on $1$ to create the Schur basis of $\mathrm{Sym}$. To construct this operator, we will need to consider elementary basis indexed by an integer $r$ as an operator that adds vertical strip of size $r$ on a diagram of an integer partition over all possible ways. From this we can define the dual operator of elementary operator. Go through some construction of commutation of those operator and generation functions of those operator we will find Schur basis is the coefficient of certain term of the generating function $B(z)$. This talk only requires a little background of symmetric functions.