Potential Undergraduate Research Topics

Potential Research Topics

• Trace Diagrams (abstract algebra, graph theory, some combinatorics) : Can you "draw" the determinant of a matrix? The trace? How are these related? Can funny pictures be more enlightening than equations? Answer: yes, using trace diagrams! Potential projects include (i) developing a theory of diagrams for certain kinds of matrix groups, (ii) using diagrams to understand trace identities, (iii) investigating the properties of "train track" graphs, and (iv) developing new proofs of traditional results in linear algebra. Trace diagrams are related to many other fields of mathematics and physics. This is a relatively unexplored area, so any results you obtain will most likely be new and possibly lead to publications.

• Mathematics and Economics of Cooperation (game theory, economics, social science): This research involves the application of subset team games to either real-world scenarios or theoretical constructs such as social networks or pursuit-evasion games. Subset team games provide metrics for both "altruism" and "selfish cooperation", and have the potential to provide a quantitative framework for traditionally qualitative topics.

• Pursuit and Evasion (Java programming, algorithms, game theory, differential equations, simulation): Cadets interested in programming or algorithms can contribute to the pursuit-evasion applet at http://www.dean.usma.edu/math/people/Peterson/applets/peg2/peg2.html. Research projects might include developing new algorithms/simulations, or studying the properties of existing ones. See http://www.dean.usma.edu/math/people/Peterson/research/Peterson-PursuitPoster-handout.pdf for a summary of recent work by cadets in this area.

• Mathematical User Interfaces (Java programming, algorithms): For those with a very strong ability and interest in Java programming, a research project might involve exploring the possibilities for new user interfaces for mathematical software, including the use of gestures, WII remotes, or other input devices.