This project will explore the properties and structures of various types of magic labelings applied to directed graphs. We will investigate conditions under which directed graph components (vertices, directed edges, or both) can be assigned unique and consecutive labels from the positive integers such that ingoing and/or outgoing sums are consistent across all vertices (or edges). The goal is to discover new results on a certain type of magic labeling for directed graphs and to  generate additional magic or antimagic labeling questions for directed graphs that faculty and undergraduate students can further explore. 

The game of best choice (also known as the secretary problem) has been studied since at least the 1950's and was widely popularized in a 1960 column of Martin Gardner. In the classical setup, a player conducts "interviews" with a fixed number of "candidates." Immediately after each interview, the player ranks the current candidate against all of the candidates that have been considered so far. The player must then decide whether to accept the current candidate and end the game or, alternatively, whether to (irrevocably) reject the current candidate and continue playing in the hope of obtaining a better candidate in the future. This problem features an exploration–exploitation tradeoff: each interview reveals more information about the candidate pool, yet acting on this information (by selecting a candidate) ends the game.

We are interested in optimal strategies for these games as well as the probability that a player selects the best candidate. Crucially, most of the classical analyses assume that candidate rankings are uniformly distributed (so each possible permutation occurs with probability 1/n!). In this project, we use a new combinatorial perspective on the game that allows us to relax this assumption and discover new facets of the best choice game.

The well-known trace map on matrices can be generalized to a map on any module, M, over a commutative ring, R. Its image, the trace ideal of the R-module M, is the ideal generated by the homomorphic images of M in R. Within commutative algebra, these trace ideals have recently experienced an uptick in attention, in part, because the naturalness of their construction ensures the ubiquity of their applications. In this project we will specifically explore the relationship between various notions of closure, for example, being a trace ideal, reflexive, integrally closed, or stable. We will focus on settings in which these notions coincide, for example in Arf rings, with an eye to characterize these and related rings that have a well-established history within singularity theory.

In the early 1500's, work of Scipione del Ferro, Gerolamo Cardano, Nicolo Tartaglia illuminated a way to express the roots of a cubic polynomial in terms of square and cubic roots. Nearly 200 years later, Paolo Ruffini showed in 1799 while Niels Henrik Abel showed in 1823 that similar results do not hold for quintic polynomials. Indeed, we cannot express the roots of a quintic in terms of radicals. However, Charles Hermite showed in 1858 how to express the roots of a quintic using hypergeometric functions, completing a circle of ideas first initiated by Ernst Kummer in 1836. A coherent theory to solve all polynomials of degree at most five was completed by a series of authors including Hermann Schwarz in 1873, Lazarus Fuchs in 1876, and Felix Klein in 1884.

In this project, we explore Hermite's construction using the modern language of Belyi maps. We will explore how to relate roots of polynomials using Belyi maps; express the roots of polynomials by inverting such maps through the use solutions to hypergeometric differential equations; and create animations of Galois groups of the polynomials using monodromy.