1. Lucky colorings of graphs; Led by Dr. Michael Tait
Label the vertices of a graph with positive integers, and for each vertex v let S(v) be the sum of the labels of all of its neighbors. A labelling is called lucky if for every edge uv, S(u) is not equal to S(v). The lucky number of a graph is the minimum number of labels necessary to produce a lucky labelling. In this project we will investigate this graph parameter.
Pre-requisites: MAT 2600 and any previous experience with graph theory or combinatorics.
2. Extremal graph theory; Led by Dr. Michael Tait
Extremal graph theory seeks to quantify the statement "when a graph gets large, it contains structure somewhere" or "complete disorder is impossible". In this project we will consider Ramsey and Turan problems for graphs and hypergraphs.
Pre-requisite: A course in graph theory.
3. Measuring distance between permutations; Led by Dr. Katie Haymaker and Dr. Alexander Diaz-Lopez
In mathematics, it is common to measure the distance between objects. Some common examples are to measure the distance between two points in a cartesian plane and the (Hamming) distance between vectors. In this project, we will explore combinatorial objects called permutations and different ways to measure the distance between permutations that share similar properties. The project is exploratory and in its initial phase. In short, this means we will explore many examples and use these examples to spot patterns, make and (hopefully) prove conjectures, and look for connections to other interesting problems.
Pre-requisites: Foundations of Mathematics (MAT 2600) is required and Modern Algebra (MAT 3600) is preferred but not required.
4. Multi-Sample Rank Tests; Led by Dr. Jesse Frey
Rank tests are statistical tests where only the ordering of the observations matters. Such tests require fewer distributional assumptions for validity than do tests such as the two-sample t test and the F test. Two well-known examples are the Wilcoxon rank sum test and the Kruskal-Wallis test. This project will involve becoming familiar with existing rank tests and learning to compare rank tests by doing power studies in R. The eventual goal is to create a new multi-sample rank test based on a new nonparametric likelihood. This is intended for one undergraduate.
Pre-requisites: STAT 4310. Additional statistics courses and R experience are also helpful.
5. Fair Funding and Educational Outcomes in Pennsylvania; led by Dr. Bruce Pollack-Johnson and Dr. Michael Posner.
Pennsylvania passed a law implementing an excellent Fair Funding Formula to ensure all public school districts get fairly funded based on economic circumstances, geography, taxing capacity, etc. But this is being phased in very slowly (only 10% of funds currently). In this project, you will be trying to understand better why it is that districts whose funding is closest to the formula get the best educational outcomes, as measured by enrollment in post-secondary education programs, even compared to those that are over-funded according to the formula. This project will involve manipulating a big data set, looking for models of the relationships between the variables, and doing statistical tests.
This would be possible for both an undergrad or a grad student.
Pre-requisites: STAT 4310 (or STAT 7404) and STAT 4380 (or STAT 7500). Experience with statistical packages such as R or SAS is essential.
6. Assignment Pebbling Graphs; led by Dr. Andy Woldar
The standard pebbling game is an impartial two-player game played on a graph G. Initially, pebbles are randomly placed on the vertices of G. A pebbling move consists of removing two pebbles from a vertex v, while simultaneously adding one pebble to a neighbor of v. The loser of the game is the player who is unable to make a legal pebbling move.
Related to the initial pebbled graph G is a rooted graph [S_G], which we call the pebble assignment graph. The root node r of [S_G] is the initial state of the pebbling game, and nodes x and y of [S_G] are adjacent provided y arises from x (or vice versa) via a single pebbling move. It turns out that [S_G] is always a connected, bipartite graph. In particular, the girth of [S_G] must be even (or infinite).
Curiously, for every even integer k not equal to 8, we are able to construct an assignment graph of girth k. We conjecture that girth 8 is not possible.
7. The Kelly Criterion in the presence of maximum payouts; led by Dr. Klaus Volpert
Suppose you get to play a game repeatedly where your chance of winning is 60% each time. Your fund starts with $100. What percentage of your fund should you bet each time to optimize your expected outcome after, say, 100 games? The Kelly Criterion, established in 1955, says that you should bet 20% of your fund each time.
But what if there is a maximum payout in place, say $1000? And what if one is at $900 after only 50 games? Surely one would not bet 20% in the next game! Not even 10%. It is easy to demonstrate that in this situation there is a better strategy than the Kelly criterion. But what is the best strategy possible?
Pre-requisites: basic calculus and statistics. Willingness to learn programming in R or Maple or Python.