The Mathematics & Statistics Department at UNC Asheville places a high value on assisting our students with undergraduate research in mathematics. Students have completed projects in a wide variety of mathematical areas. Some have been interdisciplinary, involving other departments or fields of study. Our students often present at conferences or have published results before they finish their undergraduate degree. Here are just a few examples of the exemplary work by students in mathematics doing undergraduate research.

## Senior Projects

Timothy Sawicki, “The Efficiency of the Dedkind-MacNeille Completion of a Partially-Ordered Set.” Tim investigated the behavior of a particular operation on posets, called the Dedekind-MacNeille completion. This operation begins with a poset P and, when the poset is finite, returns a more well-behaved order structure called a lattice. This operation is provably the most efficient means of constructing a lattice from P, and Tim’s motivating question was the following: in the “worst case,” how inefficiently does the Dedekind-MacNeille completion perform?

Katy Beeler, “The Moebius Function of Subgraph Lattices.” Katy sought to understand the structure of subgraphs of a given graph (network) by computing the Moebius function of its subgraph lattice. This function, which arises in many different areas of mathematics (including number theory, geometry, and topology), allows us to “invert” the relationship between two known functions, enabling much deeper understanding of those functions. Katy succeeded in computing the Moebius functions for the subgraph lattice of any path.

John Hedberg: “Numerical Analysis of a Family of Random Regular Graphs.” John worked to develop a computational understanding of several important structural measures of graphs (networks) created according to a certain random process. John wrote Mathematica code to generate numerous sample graphs and to estimate the diameter, connectivity, expansion ratio, and clustering coefficient of these graphs. Though John’s project was largely computational, his data will lead to a better theoretical understanding of the structure of the regular graphs he considered.

Jason Blanchard, Laura Tinney, Scott Spencer: “Breaking the Trifid Cipher”. This research project seeks to use modern mathematics to both describe what makes the Trifid Cipher so hard to break and then use even deeper mathematics to show how it can be broken nonetheless. The breaking of this cipher will bring us into the realm of modern mathematics, including number theory and abstract algebra. For example, in order to decipher the garbled message, you simply have to apply the Trifid permutation algorithm k times to the garbled message, where k is a divisor of the Euler phi function of 3n-1, where n is the length of the blocks in the garbled message. This links the Trifid Cipher to a branch of mathematics called number theory, a connection that has not been made before, since the introduction of this cryptosystem in 1901!