# Talks

### Phased matroids and complex Grassmannians

#### Laura Anderson, Binghamton University

**Abstract:** Phased matroids are combinatorial objects that play the same role for the complex numbers that oriented matroids play for the real numbers. They were introduced in a 2012 paper as a potential tool for various problems in geometry and topology. I’ll discuss the state of the field, focusing on progress toward a combinatorial model for the Grassmannian *G*(*r*, **C^**n).

### Maker-Breaker Games on Random Geometric Graphs

#### Andrew Beveridge, Macalester College

** Abstract:** In a Maker-Breaker game on a graph *G*, Breaker and Maker alternately claim edges of *G*. Maker wins if, after all edges have been claimed, the graph induced by his edges has some desired property. We consider three Maker-Breaker games played on the Random Geometric Graph. For each game, we show that if we add edges between *n *points chosen uniformly at random in the unit square by order of increasing edge-length then, with probability tending to one as *n* → ∞, the graph becomes Maker’s win the very moment it satisfies a simple necessary condition. In particular, with high probability, Maker wins the connectivity game as soon as the minimum degree of is at least 2; Maker wins the Hamilton cycle game as soon as the minimum degree is at least 4; and Maker wins the perfect matching game as soon as the minimum degree is at least 2 and every edge has at least 3 neighboring vertices.

### Graphs with *Χ*=Δ have Big Cliques

#### Dan Cranston, Virginia Commonwealth University

**Abstract: **Let *G* be a graph with maximum degree Δ ≥ 3. Brooks’ Theorem says that if *G* has chromatic number Δ+1, then *G* has a clique on Δ+1 vertices; otherwise *G* has chromatic number at most Δ. In 1977 Borodin and Kostochka conjectured that if *G* is a graph with maximum degree Δ ≥ 9 and chromatic number Δ, then *G* has a clique on Δ vertices. For maximum degree Δ ≥ 13, we prove that if *G *has chromatic number Δ, then *G* has a clique on at least Δ-3 vertices. This is joint work with Landon Rabern.

### Parity, first-order logic, and gluing and cutting graphs

#### Amanda Redlich, Bowdoin College

** Abstract:** The zero-one law for random graphs states that the probability of any formula in first-order logic holding for a random graph is zero or one. Recently, Kolaitis and Kopparty extended this result to first-order logic with a parity quantifier. Here, I discuss extensions of this extension. The main technique used is a combinatorial game on graphs, together with analysis of the subgraph lattice and a two-step random process. This game for two players, Scissors and Glue, admits several variations with many open problems. Joint work with Robert DeMarco and Jeffry Kahn.

### Generalized splines

#### Julianna Tymoczko, Smith College

**Abstract:** Splines are a well-known algebraic and combinatorial construction originally developed for engineering applications but now used widely in algorithms in computer graphics, differential equations, numerical analysis, and other fields. Billera and others pioneered an algebraic approach to splines, using techniques from commutative and homological algebra, among others. Over the last fifteen years, geometers and topologists independently developed a way to construct equivariant cohomology rings for large classes of varieties that turns out to coincide with the ring of splines.

In this talk, we describe how to generalize the construction of splines to a more natural geometric and combinatorial setting, starting from a commutative ring and a graph *G*. We’ll also show how powerful this construction can be: how the ring of splines naturally decomposes in terms of splines for certain subgraphs of *G*, and how to construct generating sets for the generalized splines. We’ll end with a number of open questions with connections to Schubert calculus, geometric representation theory, and approximation theory.

# Posters

### Patterns in Signed Shifts

#### Kassie Archer, Dartmouth College

**Abstract:** We consider the left shift on infinite words on *k* letters. By equipping this set of infinite words with some (well-justified) total ordering, we can define patterns as the permutations in the same relative order as the orbits with respect to the left shift. These permutations have a nice combinatorial description and there are purely combinatorial corollaries that follow from this work.

### Patterns in Matchings and Rook Placements

#### Jonathan Bloom, Dartmouth College

** Abstract: **Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize 3-crossings and 3-nestings, have an interpretation, in the case of matchings, in terms of patterns in full rook placements on Ferrers boards.

We enumerate 312-avoiding matchings and partitions, obtaining algebraic generating functions, in contrast with the known D-finite generating functions for the 321-avoiding (i.e., 3-noncrossing) case. Our approach also provides a more direct proof of a formula of Bóona for the number of 1342-avoiding permutations. Additionally, we give a bijection proving the shape-Wilf-equivalence of the patterns 321 and 213 which greatly simplifies existing proofs by Backelin–West–Xin and Jelínek, and provides an extension of work of Gouyou-Beauchamps for matchings with fixed points. Finally, we classify pairs of patterns of length 3 according to shape-Wilf-equivalence, and enumerate matchings and partitions avoiding a pair in most of the resulting equivalence classes.

### Multiplication Rules for Peterson Schubert Classes

#### Elizabeth Drellich, UMass Amherst

**Abstract: **A Peterson variety is a subvariety of the flag variety $Gl_n(G/B)$ which appears in construction of the quantum cohomology of partial flag varieties. Each Peterson variety has an $S^1$ action and its $S^1$-equivariant cohomology has a basis of Peterson Schubert classes. Given such a module basis, Schubert calculus asks how to multiply basis elements within the ring. In type $A$ Harada-Tymoczko gave a positive Monk formula, and Bayegan-Harada gave a Giambelli’s formula for multiplication in the cohomology ring of the Peterson variety. Using excited Young diagrams we give multiplication rules for Peterson varieties of all Lie types.

### Rook Placements and Colored Permutations

#### Jonathan Godbout, Worcester Polytechnic Institute

The Robinson-Schensted correspondence gives a bijection between permutation and pairs of standard Young tableau of the same shape. The Bruhat partial-order describes the containment of the closure of GL(V ) orbits of flags over C. It is known that permutations index these orbits while coloured permutations give an analogous ordering in a larger space. In hopes of understanding this larger flag space, Roman Travkin expanded on Robinson-Schensted correspondence with the Mirabolic RSK algorithm, giving a bijection between colored permutations and augmented tableau. Permutations have an obvious bijection between rook placements of size *n* on an *n*×*n* chessboard and permutations. Using a weighting induced by the Mirabolic RSK algorithm, we give a weighted bijection between rook placements of any size on an *n*×*n* chessboard and coloured permutations of length n.

### Some Properties of Generalized Schur Functions

#### Jennifer Harnish, Dartmouth College

** Abstract:** Schur functions are well-known to be a useful tool in representation theory. Generalized Schur functions can be used similarly. My research is based on trying to generalize facts we know about Schur functions, in particular about the Kronecker product of Schur functions, to generalized Schur functions. In this poster I will be giving a result about the width and height of the Kronecker product of two generalized Schur functions. After that, I will talk about a stability property of Schur functions and state a conjecture I am currently trying to prove relating to the stability of generalized Schur functions.

### Tree-Width of the Triangular Grid Graph

#### Brett Smith, Wesleyan University

**Abstract:** In their series of papers, Graph Minors, Robertson and Seymour introduce a tree decomposition of a graph. This definition leads to a useful graph property called tree-width. The *n*×*n*-grid

graph is the classical example of a planar graph of tree-width *n*. We prove that the (*n*+1)-triangular grid graph has tree-width *n.* The techniques in this proof can be used to find a proper minor of the *n*×*n*-grid graph that has tree-width *n*.

### Separation of Variables and the Computation of Fourier Transforms on Finite Groups

#### Sarah Wolff, Dartmouth College

**Abstract:** We present a general diagrammatic approach to the construction of efficient algorithms for computing the Fourier transform for a function defined on a finite group. Our approach is a continuation of earlier work which first made use of Bratelli diagrams in the construction of Fast Fourier Transform algorithms. In this work we make more explicit use of the path algebra connection to the construction of Gel’fand-Tsetlin bases and work in the setting of general semisimple algebras and quivers. We relate this framework to the construction of a configuration space attached to a Bratelli diagram. In this setting the complexity of an algorithm for computing a Fourier transform reduces to the calculation of the dimension of the associated configuration space.

We give explicit counting results to find the dimension of these configuration spaces, and thus the complexity of the associated Fourier transform. Our methods give improved upper bounds for the general linear group, the special linear group, the classical Weyl groups, and homogeneous spaces while also recovering the best known algorithms for the symmetric group and compact Lie groups.