Research

In addition to working with Kutztown faculty, mathematics students can apply to paid national prestigious summer research internship programs, such as Research Experiences for Undergraduates (REU) programs.  Here are some examples (in parentheses are the Kutztown students who participated and when they did so):

• Lafayette College (Christina 2024; Grant 2018; Shannon 2017)
• Moravian University (Phil 2023; Sam 2022)
• Muhlenberg College (Alex 2018; Jake 2017; Nate 2016)
• Oak Ridge National Laboratory (Vanessa 2019)
• Rutgers University (Katelyn 2014)

Stories about Undergraduate Mathematics Research

A few sample research projects of interest to our faculty

• Combinatorics and Algebra: Algebraically Defined Graphs

Combinatorial graphs are essentially networks: they contain nodes (called vertices) and edges.  These graphs are called algebraically defined if edges are created according to whether given algebraic conditions are satisfied.  Certain important properties of these graphs may be discovered by studying when special equations (or systems of special equations) have solutions.   Questions related to this project have applications to incidence geometry.  Dr. Kronenthal is interested in this topic, and has collaborated with undergraduate students to study related questions.

• Graph Coloring Games

Coloring the vertices or edges of combinatorial graphs to satisfy certain conditions is a popular research area.  Dr. Wong worked with Diego Manzano-Ruiz, an undergraduate student, on a coloring problem in the context of a two-player game.

• Graph Theory and Network Theory: The Cable-Trench Problem (CTP)

The name "Cable-Trench" comes from the fact that a physical application of the CTP is the problem of minimizing the cost to create a campus network in which each building on a campus is connected to a central server with its own dedicated cable. There is one cost to dig trenches between buildings and a separate cost to purchase and lay the cables. In their 2002 paper, Dr. Vasko and his co-authors proved that there is no known efficient algorithm that finds the optimal solution to CTPs.  The CTP has been applied to numerous real-world applications, such as medical research, cancer detection, and identifying blood clots.  Dr. Landquist, Dr. Lu, and numerous Kutztown students have worked on this and closely related projects.

• Jaeger's conjecture

Let A be an invertible n-by-n matrix over a prime field of size at least 5. Do there always exist two vectors x and y of length n such that all entries of x and y are nonzero, and y=Ax?

Dr. Wong is very interested in this open conjecture. Partial progress has been made, and he is looking for collaborators to further his findings.  

• Metaheuristic Research for Solving Combinatorial Optimization Problems

Metaheuristics are solution strategies (as opposed to specific algorithms) that can be used to generate very good solutions (perhaps optimal) to problems in reasonable computing time.  These problems are so difficult that finding an exact solution could require days, weeks, months or even years.  Metaheuristics that can generate high quality solutions using little or moderate computer resources are very desirable because of their wide industrial and business applications.  Dr. Lu and Dr. Vasko are especially interested in this area of study.

• A probabilistic take-away game

Alice and Bob are playing a very simple game. Each of them starts with a pile of n chips, and they take turns to lose p chips or q chips randomly and independently with equal probability. The first player who removes all chips from their pile is the winner. We are interested in finding the winning probability for Bob. This project is an extension of the research paper that Dr. Wong and Ms. Jiao Xu coauthored. Ms. Xu graduated from Kutztown University in Spring 2017.