Inspired by Bruggesser-Mani's line shellings of polytopes, we introduce line shellings for the lattice of flats of a matroid: given a normal complex for a Bergman fan of a matroid induced by a building set, we show that the lexicographic order of the coordinates of its vertices is a shelling order. This gives a new proof of Björner's classical result that the order complex of the lattice of flats of a matroid is shellable, and demonstrates shellability for all nested set complexes for matroids.
We characterize the class of threshold matroids by the structure of their defining bases. We also give an example of a shifted matroid which is not threshold, answering a question of Deza and Onn. We conclude by exploring consequences of our characterization of threshold matroids: We give a formula for the number of isomorphism classes of threshold matroids on a groundset of size n. This enumeration shows that almost all shifted matroids are not threshold. We also present a polynomial time algorithm to check if a matroid is threshold and provide alternative and simplified proofs of some of the main results of Deza and Onn.
We introduce the poset of biflats of a matroid M, a Lagrangian analog of the lattice of flats of M, and study the topology of its order complex, which we call the biflats complex. This work continues the study of the Lagrangian combinatorics of matroids, which was recently initiated by work of Ardila, Denham and Huh. We show the biflats complex contains two distinguished subcomplexes: the conormal complex of M and the simplicial join of the Bergman complexes of M and M*, the matroidal dual of M. Our main theorems give sequences of elementary collapses of the biflats complex onto the conormal complex and the join of the Bergman complexes of M and M*. These collapses give a combinatorial proof that the biflats complex, conormal complex and the join of the Bergman complexes of M and M* are all simple homotopy equivalent. Although simple homotopy equivalent, these complexes have many different combinatorial properties. We collect and prove a list of such properties.
We provide multiple algorithms for computing the rational powers of monomial ideals. We also generalize rational powers to allow for real powers. We prove that given any monomial ideal, the function taking a real number to the corresponding real power is a left continuous step function with rational discontinuity points.
We create a program to play the card game cribbage. To do this, we implement and compare to different algorithms. One algorithm attempts to teach itself cribbage through reinforcement learning. It teaches itself poorly. The other algorithm uses a stochastic minimax algorithm to great effect.
An expository overview of the braid group created during the 2020 McNair program. We first introduce the symmetric group and braid group using many tikz diagrams. We then look at their applications to cryptography (it is non-commutative) and knot theory (Alexander's theorem).