Project Description
In mathematics, network science is often referred to as "graph theory". A graph is a set of vertices together with a set of edges that join pairs of vertices. A subset, D, of vertices is said to dominate the graph if every vertex in the graph is in D or is joined to at least one vertex in D. One research question is to consider special graphs where the size of a minimal dominating set is not known. Finding the size of a minimal dominating set is difficult. This parameter is called the domination number. There is a family of graphs that we will consider for the SURF project that are related to data analysis of grade assessment of students. The nodes of the graph we will consider correspond to all possible grade distributions. We will define two distributions to be joined by an edge if a single student grade can be moved one step up, or one step down to obtain the other distribution. The shortest path in this graph between two nodes is called the Earth Movers Distance. We call the graph EMD(g,n). We would like to obtain a simple way to compute the domination number for EMD(g,n).
Tasks and Responsibilites
Research mathematics involves making conjectures about the solution to a problem, and then attempting to either prove or disprove the conjecture by formulating counterexamples or writing rigorous proofs. If a proof is obtained then a more general conjecture is sought. If the conjecture is disproved then one wishes to add a hypothesis to rule out any counterexamples. Although it is not strictly necessary, the use of available mathematics software to code up examples is useful. The student will learn to program in Python, and then use platforms such as CoCalc or Sage. Once results are obtained, the student will learn to typeset in LaTeX. All of us will work on the project write-up and necessary literature searches using Google Scholar and MathSciNet. After a research article is produced, then the student will produce a slide deck so that they can speak at conferences about the results. A second phase of the project will involve using publicly available grade distributions to do data analysis. These data are often in spreadsheet format provided by Excel. The student will master Microsoft Excel, an essential skill for research involving data.
Desired Qualifications
None listed.