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).