A category central to algebraic combinatorics is that of “rings with bases”. We’ll look at a very simple example, the nil Hecke algebra, and its action on polynomials in infinitely many variables. The resulting dual basis is the “Schubert polynomials”. Surprisingly, these have positive coefficients, which we will compute by counting “pipe dreams”.

We start with the Bruhat decomposition of the full flag variety and the Borel presentation of its cohomology ring to introduce the Schubert polynomials, the main objects of interest for this seminar. We will then briefly discuss several topics that can be covered throughout the semester, including various monomial expansions of the Schubert polynomials, combinatorial algebraic geometry of the Schubert varieties, Gröbner geometry of matrix Schubert varieties and more.

We will be talking about the main ways to compute the Schubert polynomials from permutations, relationships between Schubert polynomials and RC-graphs, chute moves on RC-graphs, and the Monk’s rule.

We introduce combinatorial formulas of (double) Schubert and Grothendieck polynomials based on bumpless pipe dreams and give a combinatorial proof of Monk’s rule for Schubert and double Schubert polynomials using bumpless pipe dreams that generalizes Schensted’s insertion on semi-standard Young tableaux. We also give a bijection between pipe dreams and bumpless pipe dreams and discuss its canonical nature.

In this talk, we will present the relationship between the divided difference formula and pipe dream formula for Schubert polynomials via the work of Fomin-Stanley in the nilCoxeter algebra.

We will discuss several combinatorial characterizations of smooth Schubert varieties due to Lakshmibai and Sandhya, and Carrell and Petersen.

The Schubitope \(S_w\) is the convex hull of exponent vectors appearing in the Schubert polynomial associated to the permutation \(w\). All integral points in the Schubitope appear as the exponent vector of some term in the associated polynomial. We discuss a polynomial time algorithm, discovered by Adve, Robichaux, and Yong, for deciding if a given exponent vector is in the Schubitope of a given permutation. The proof relies on a tableau criterion that reduces to a tractable linear programming problem.

The singular locus of a Schubert variety \(X_v\) is determined by its smoothness at the \(T\)-fixed points indexed by permutations \(u\) less than \(v\) under the Bruhat order. We construct an affine open neighborhood around each \(u\) in \(X_v\), decompose it into a product of an affine variety with some affine space. We define a combinatorial relation on pairs of permutations called interval pattern embedding, generalizing the usual notion of pattern embedding. Finally, we show how the open neighborhoods associated with different pairs of \(u<v\) are related when interval embedding occurs. This helps us reformulate known results on singular locus of Schubert varieties.

In this talk I will introduce the weak Bruhat order \(W_n\) on \(S_n\) and the strong Sperner property of Posets and show that \(W_n\) is strongly Sperner with approaches from two different works. Stanley conjectured an order rising operator \(U\) on \(\mathbb{C}W_n\) and showed that if \(U^{r-2k}\) is invertible then \(W_n\) is strongly Sperner. In the first work, Gaetz and Gao constructed an order lowering operator \(D\) which led to an \(sl_2\) representation of \(\mathbb{C}W_n\) and used representation theory of \(sl_2\) to prove invertibility. In the second work, Speyer et al. proved a determinantal formula of \(U^{r-2k}\) conjectured by Stanley, through showing that the divergence operator acts on Schubert Polynomials as the operator \(U\) and this implies invertibility of \(U^{r-2k}\).

The polynomial ring has two distinct bases: Schubert polynomials and key polynomials. In this talk, we will first study the vector subspace spanned by Schubert polynomials in \(S_n\), and describe another basis consisting of key polynomials. Then we will explore the process of expanding a Schubert polynomial into a sum of key polynomials, demonstrating how these two bases are interrelated.

I’ll define double back-stable Schubert polynomials (in the form Dave Anderson and I gave a couple of years ago). I’ll show that they are characterized by the way that they count points in loci defined by rank conditions on matrices. I’ll state some of their main properties. And I’ll describe how they relate to the bumpless pipe-dream formula of Lam-Lee-Shimozono.

We will discuss Richardson varieties and introduce R-polynomials as their finite field point counts. We will then use these to define Kazhdan-Lusztig polynomials and finish with a survey of some results and open problems about these.

Schubert polynomials of vexillary (2 1 4 3 avoiding) permutations are well known to have tableau and determinantal formulas. The goal of thistalk is to extend this to general permutations, by writing a generalSchubert polynomial as a sum of products of such determinants. Thisis based on the bumpless-pipe-dream formula of Lam, Lee, andShimozono. After reviewing some basic properties of back-stableSchubert polynomials, we’ll discuss determinantal and tableauformulas (some of which seem to be new), and then show how to usethese to decompose general Schubert polynomials. Anything originalis joint work with Dave Anderson.

I’ll give an introduction to the dimer model and its connections with statistical mechanics, cluster algebras, total positivity, integrable systems etc.

We first set up some preliminaries on graph homology and explain how to generate a discrete surface from a dimer covering. Then we prove Kasteleyn’s theorem, which expresses the partition function of a planar dimer model as a determinant.

As we have seen, the set of dimer covers on a graph is in bijection with several other interesting objects. In this talk we will discuss a bijection between dimer covers on a graph and spanning trees on a related graph, a generalization of Temperley’s classic bijection. The powerful tools developed for spanning trees, like Wilson’s algorithm for selecting a uniformly random spanning tree can then be used on dimer covers as well.

I will talk about the dimer model for a bipartite graph embedded on a torus. We will define the partition function and discuss some of its properties.

I will talk about the totally nonnegative grassmannian and a parametrization of this using a dimmer model.

I will review Kasteleyn’s method for planar graphs, its adaptation to graphs on a torus, the notion of slope of a tiling, and the spectral curve of a graph on a torus. I’ll explain how the spectral curve controls the asymptotic behavior of periodic tilings as the period domain expands. I’ll then state Kenyon and Okounkov’s theorem that the spectral curve is always a Harnack curve, and that all Harnack curves are spectral curves.

Cancelled for MCAIM.

I will discuss tilings with convex polygons and their relationship to the dimer model. I will introduce certain types of tilings called T-graphs and discuss connections to Dehn’s theorem about tiling a rectangle with squares and Menelaus’ theorem from Euclidean geometry.

In the coming two talks we will discuss a construction of a cluster variety with a Poisson structure, coming from bipartite graphs on the torus. This week will start with some definitions and motivation for Poisson varieties, and proceed to a construction of a cluster variety from a (Newton) polygon in the plane. We will see that every bipartite graph has a natural algebraic torus attached - the moduli space of line bundles on the graph. These tori glue together along mutations of the graph, giving a cluster variety.

Last time, we discussed how the coordinate ring of the moduli space of line bundles on a graph provided the right object to give gauge invariant functions on bipartite graphs. We then discussed how to glue together such moduli spaces to get a global modified partition function on the graph and how to associate a Newton polygon to it. In this lecture, we will explore how to recover a family of graphs whose moduli spaces of line bundles are all related by a cluster mutation. We will also define a Poisson structure of these varieties and discuss the construction of special Casimir elements with respect to the Poisson structure. Finally, time permitting, we will discuss how this leads to a set of Hamiltonians corresponding to the interior points of the Newton polygon.

I will talk about the spectral transform, which is the torus-analogue of the boundary measurement map to the Grassmannian, and how it identifies the cluster integrable system with the Beauville integrable system.

In this talk, I will introduce Ergodic Gibbs Measures (EGM) for infinite periodic bipartite graphs and the definition of phases based on height fluctuations wrt this measure. I will present a sketch of how to construct EGMs with fixed slopes as well as how phase classifications can be done using information about the amoeba of characteristic functions of the graph. I will illustrate the theory using honeycomb dimers.