An abstract simplicial complex is called d-representable if it is the nerve of a family of convex sets in the d-dimensional Euclidean space. The research on d-representable complexes has a rich and fascinating history, starting with Helly’s theorem, that to a large degree shaped the modern discrete geometry.

A simplicial complex is d-convex union representable (or d-CUR, for short) if it arises as the nerve of a finite collection of convex open sets in R^d whose union is also convex. This is a fascinating and very natural subclass of representable complexes. In addition, the motivation for investigating such complexes comes from the theory of convex neural codes. While the class of d-representable complexes is well- and long-studied, the notion of d-CUR complexes was introduced only a couple of years ago. Consequently, our present knowledge about this class of complexes is almost null. For instance, while we know that CUR complexes are collapsible and so are their Alexander duals, we do not know the answers to such questions as: is the barycentric subdivision of a CUR complex also CUR? Is every shellable simplicial ball CUR? We are also very short on techniques allowing to construct many such complexes.

The goal of the project will be to further explore the class of CUR complexes and probe some of the open problems.

Recommended background:

• Some knowledge of discrete geometry and polytopes

There are a number of Schur-like bases for the space of quasisymmetric functions with combinatorial definitions reminiscent of the semi-standard tableaux definition of symmetric Schur functions. One such basis, the dual immaculate functions [2], can be defined as the generating function of a set of composition tableaux of shape α where the first column of α is filled with positive integers so that it is increasing and each row is weakly increasing. This basis expands positively in the Young quasisymmetric Schur basis [1]. Just as the symmetric Schur functions can be defined as the generating function of row-strict tableaux, with weakly increasing columns, there is a row-strict quasisymmetric Schur basis [3]. In this project we will define a row-strict version of the dual immaculate functions and then explore their combinatorics and algebra. Some directions for this research include finding an expansion in terms of the row-strict quasisymmetric Schur functions, identifying Pieri rules, and finding 0-Hecke modules with the row-strict dual immaculate functions as their quasisymmetric characteristic.

Combinatorialists encounter permutations in many settings, two of which will be the foundation of this group's work. First, the symmetric group is the finite Coxeter group of type A, and this consists of all permutations of a given size. As a Coxeter group, this can be endowed with a natural poset structure (the Bruhat order), and many interesting mathematical questions -- both answered and unanswered -- can be posed about this structure. Second, one can look at patterns occurring within permutations, examining both the characterizing and the enumerative properties that result from such occurrences. A permutation pattern is typically phrased as a binary property: containment or avoidance. However, recent evidence has shown that a higher-order approach can provide an important and influential framework for understanding various phenomena, with the question being a count of how many times a permutation contains a pattern, or in what ways that containment occurs. Connections between the Coxeter-perspective and the pattern-perspective have already been established, and this project aims to extend that relationship. Objects we will focus on may include ideals in the Bruhat order, commutation classes of reduced decompositions, and the RSK correspondence.

In this project, we will focus on the mysterious relation between alternating sign matrices and plane partitions. Alternating sign matrices had been introduced in the 1980's and it was soon discovered (conjecturally) that they are counted by the same simple product formula as two classes of plane partitions, namely descending plane partitions and totally symmetric self-complementary plane partitions. Very recently, a fourth class of objects (alternating sign triangles) has been added to the list of objects that are counted by this product formula. There have been many efforts to construct bijective proofs, but so far there only exists a very complicated (recent) construction that proves bijectively an identity that implies the equinumerosity of alternating sign matrices and descending plane partitions.

We have two concrete projects in mind. Alternating sign matrices include permutation (matrices) as a special case, and for this special case bijections have been constructed in the work of Ayyer, of Fulmek and of Striker. The first possibility is to extend this to the next level of complication, by introducing a small number of -1's. Recent work of Höngesberg [3] actually does include -1's for alternating sign triangles, so that this work will be a good starting point. The other project is around unpublished work (at the moment) concerning n+3 pairs of equivalent statistics on two new types of objects that extend alternating sign matrices and descending plane partitions, respectively. That's exciting because for alternating sign matrices and descending plane partitions only four such statistics are currently known.

The study of the combinatorics of convex polytopes has a long and interesting history, with connections to operations research, algebra and algebraic geometry. There are many open questions in the field; this project will focus on enumeration questions. The face vector counts the number of faces of each dimension; the flag vector counts the numbers of chains of faces (ordered by inclusion) of specified dimensions. Both are characterized for 3-dimensional polytopes, but there is not even a conjectured characterization for general 4-dimensional polytopes. The affine spans of the face vectors and flag vectors of polytopes of dimension d are known. Many inequalities are known. Characterization of both sets of vectors are known for simplicial polytopes, and some results are known for other classes of polytopes. Methods come from combinatorial analysis, but also by applying results in algebra, algebraic geometry, and algebraic topology to polytopes. Depending on the participants' interests and expertise, the project will focus either on a particular class of polytopes or on particular methods.

A polyomino is a finite subset of Z2 , where each element is represented by a square cell. The combinatorial model is obtained by taking a connected polyomino and assigning a non-negative integer to each cell. By considering different shapes, convexity, symmetry, and content of the filling, we get a unified model that contains a hierarchy of combinatorial structures. Explicitly, we would consider the following families of polyominoes, including

1. square or rectangular shaped polyominoes whose fillings correspond to permutations and words;
2. Ferrers diagrams which are polyominoes given by an integer partition. Fillings of Ferrers diagrams correspond to matchings, set partitions, and general graphs;
3. stack polyominoes which are convex, intersection-free, and whose rows are arranged monotonically;
4. moon polyominoes which are convex and intersection-free;
5. more general shape of polyominoes with weaker convex properties.

For fillings, we consider two basic cases: the one using 0 and 1, and the one using arbitrary natural numbers. These fillings, together with the aforementioned polyominoes, allow us to use combinatorial operations and transformations to investigate special characteristics of each family. It is particularly convenient to study patterns and combinatorial statistics.

Another approach to fillings of polyominoes is to apply the tools developed in the study of symmetric functions and standard Young tableaux, for example, the Robinson-Schensted-like insertion/deletion processes, growth diagram, and the Knuth equivalence. Such techniques are particularly powerful in dealing with the maximal monotone substructures, such as increasing and decreasing subsequences in integer sequences, and the maximal crossings and nestings in various diagrams.

The objective of this project is to understand the interplay between the combinatorial transformations and the tableaux operations, and to characterize various combinatorial statistics in different families of fillings of polyominoes.

Background needed: Basic techniques in enumerative combinatorics; be familiar with permutation statis- tics. A basic knowledge of symmetric functions, STYs, and the RSK algorithm would be helfpul, (as in Stanley’s EC2.)

Do you have a favorite set of combinatorial objects S with a natural bijective action f:S->S? Then come investigate its dynamics with us!

We'll begin the project by discussing everyone's favorite objects and actions in the context of the questions listed below. We will then do some computational exploration, as described below each question. Based on the computations, we will concentrate on studying the most promising actions.

Questions to ask about your favorite objects and actions:

• Does your action have a predictable order?
  • If so, see if it exhibits the cyclic sieving phenomenon [5] (can predict orbit structure by evaluating a generating function at a certain root of unity).
  • If not, see if it exhibits resonance [2] (projects to another object/action pair with a predictable order).
• Does your object have important statistics?
  • If so, take a look at how those statistics interact with the action. It may be that they exhibit homomesy [4] (same average value on all orbits).
• Is your set of objects equivalent to a set of subsets?
  • If so, study its generalized toggle group actions (some examples of this are given in [9, Section 3]).
• Is your action equivalent to the composition of small local actions?
  • If so, characterize it as a toggle group action [10].

This project will involve computational aspects, so programming experience would be helpful. But it will also be a great way to develop your programming skills in a supportive environment, so prior experience is not necessary. References [8] and [6] are survey articles that give an overview of the area.

Chip-firing processes are combinatorial dynamical systems. Traditionally the dynamics are envisioned on a graph or network. A commodity (chips, sand, dollars) is exchanged between sites of a network according to simple local rules. Although governed by local rules, chip-firing systems result in compelling global behavior.

More recent work has sought to broaden the domain of chip-firing to other settings such as root systems, M-matrices and matroids. This project will focus on chip-firing in higher-dimensions. Higher dimensional chip-firing can be thought of as a flow-rerouting process on cellular complexes. A commodity (now thought of as a flow) is diffused along the faces of a cell complex. The diffusion is dictated by the structure of the complex or equivalently by a combinatorial Laplacian. We will investigate questions about the behavior of such systems, related for example to stabilization, confluence and expansion.

Preferred background: Comfort with basic programming in a language such as matlab or python will be helpful but is not necessary. Someone(s) with skills in visualization would be excellent.

The McKay matrix records the result of tensoring the finite-dimensional simple modules with a fixed finite-dimensional module V. In the case of a finite group G and its complex modules, the columns of the character table of G give a complete set of right eigenvectors for the McKay matrix determined by any finite-dimensional G-module V. The corresponding eigenvalues depend on V and are the values of the character of V on a set of conjugacy class representatives of G. This project involves investigating the analogous situation for supergroups, for small Lie superalgebras such as gl(1,1) and osp(1,2), and for their quantum analogues defined using roots of unity.

The centralizer algebra of transformations that commute with the action of the (super)group or (super)algebra on tensor powers of the module V often have significant connections with diagram algebras and braid groups. Another goal is to investigate the centralizer algebras that arise from studying these super objects and their representations.

This project studies a symplectic version of the Berenstein-Kirillov group, BKC, and its relationship with the combinatorial internal action of the cactus group Jsp2n, constructed by Halacheva [HKRW20] using the Dynkin diagram I = [n] of the root system in type Cn, on a normal crystal of type Cn via partial Sch ̈utzenberger-Lusztig involutions.

We focus on the type Cn crystals of Kashiwara-Nakashima tableaux [KN94, L07]. They are compatible with a plactic monoid, bumping algorithm and symplectic jeu de taquin [L07]. Considering KN = KN(λ, n) to be the crystal of Kashiwara-Nakashima tableaux of straight shape λ on the alphabet [±n], a full evacuation algorithm [S21], whose construction depends on the long element of the Weyl group W = Bn (the hyperoctahedral group), also exists and agrees with the Lusztig involution on the set KN. For 1 ≤ q < n, the crystal KNq, the restriction of the crystal KN to the connected subdiagram [1, q] of the Dynkin diagram I, is of type Aq with Weyl group Sq+1. Partial evacuation algorithms agreeing with the Lusztig involution on the set KNq are not known. The action of the cactus group Jsp2n on KN makes manifest the action of the Weyl group Bn on that crystal but symplectic Bender-Knuth involutions for Kashiwara-Nakashima tableaux are not known. However, the action of certain elements of the cactus group Jn = Jgln on a straight shape semistandard Young tableau (or Gillespie-Levinson-Purbhoo shifted tableau) crystal is (shifted) Bender- Knuth like [BK95, R21]. We hope that a similar behaviour of the action of certain elements in the cactus Jsp2n on KN gives an indication on what a symplectic Bender Knuth involution for the Kashiwara-Nakashima tableaux should be.

The Berenstein-Kirillov group or Gelfand-Tsetlin group, BK, studied by Berenstein and Kirillov [BK95], is the free group generated by the Bender-Knuth involutions, modulo the relations they satisfy on semistandard Young tableaux of any shape. Fixing the subgroup BKn, it has an alternative set of generators whose action on the straight shaped semistandard Young tableau crystal SSYT(λ, n) agrees with the partial Sch ̈utzenberger-Lusztig involutions, and, therefore, coincides with the internal action of the cactus group Jn on SSYT(λ, n), studied by Halacheva [H20]. Chmutov, Glick and Pylyavskii [CGP20] also related the Berenstein-Kirillov group and the cactus group Jn using semistandard growth diagrams, and determined implications between subsets of relations in the two groups, which yields a presentation for cactus groups Jn in terms of Bender-Knuth generators. In the vein of Halacheva and Chmutov-Glick-Pylyavskii, a shifted version of the Berenstein-Kirillov group has been studied in [R21] considering shifted Bender-Knuth involutions and an action of the cactus group Jn on a Gillespie-Levinson-Purbhoo shifted tableau crystal [GLP17]. A similar presentation of the cactus group Jsp2n in terms of symplectic Bender-Knuth generators also makes sense to be explored.

Suggested background: Combinatorics of crystal graphs and tableau combinatorics along the lines of D. Bump and A. Schilling’s book ”Crystal Bases. Representations and Combinatorics”.

Splines are a fundamental tool in applied mathematics and analysis, classically described as piecewise polynomials on a combinatorial decomposition of a geometric object (a triangulation of a region in the plane, say, or a convex polytope) that agree up to a specified differentiability on faces of codimension one. Generalized splines extend this idea algebraically and combinatorially: instead of certain classes of geometric objects, we start with an arbitrary combinatorial graph; instead of labeling faces with polynomials, we label vertices with elements of an arbitrary ring; and instead of applying degree and differentiability constraints, we require that the difference between ring elements associated to adjacent vertices be a multiple of a fixed ring element labeling the edge. Billera showed that these two characterizations coincide in most cases of real-world interest.

In this project, we’ll try to make headway on the so-called upper bound conjecture from analysis, which asks for the dimension of the collection of splines on certain triangulations in the plane. We’ll think about generalized splines

1. on planar trivalent graphs
2. whose edges are labeled by two-variable polynomials of the form (ax+by)^2
3. and whose vertices are labeled by polynomials of degree at most two

Our goal is to conjecture and/or prove formulas for the dimension of the space of splines in as special or general context as we can, whether in terms of combinatorial features of the graph, or for certain families of trivalent graphs, or by developing algorithms that crawl through the graphs in particular ways or using other strategies that we come up with.