Classical Invariant Theory

Glen M. Wilson, Mathematics and Statistics (on right in photo)

Faculty Mentor: Dr. Thomas Hagedorn

              Since January 2007, Dr. Thomas Hagedorn, Jeffrey Hatley and I have been investigating the classical theory of invariants.  Classical Invariant Theory was pursued by mathematicians all around the globe during its glory days in the nineteenth century. We see such famous names as Cayley, Gordan, Lie and Sylvester all working together on understanding invariants. A simple definition of an invariant is a function that depends on an object which has the property that its value remains the same after transforming the object in some way. The study of these invariant functions was crucial to the development of mathematics. Many of the ideas used in trying to solve problems in invariant theory subsequently created much of the modern mathematics today. As a result, the interest in computational invariant theory got redirected to developing the theories which it helped create. The computational questions have been left partially solved or incorrect since Sylvester worked on them in the late 1800’s. It was our original goal to correct some of the computational questions in invariant theory and try and solve some that were not attempted. We have written a few computer programs using Mathematica which have verified the solutions to some of the computational questions and are working to improve them to get new results as well. In addition to the computation work we have done, we have also investigated theoretical methods which allow us to represent the invariant functions in a way that is more computer-friendly and also provides a different perspective on some old questions. We are currently working on a new algorithm and approach to calculating invariants which will be easier to implement in a computer program than those which we have been using.

Personal Statement

              The Summer Undergraduate Research Program has helped me organize my goals and aspirations for a career in mathematics, as well as provided a great opportunity to get to know the faculty at TCNJ and other students. When I first began to study mathematics, I knew I loved the subject, but I didn’t have any idea of what I wanted to do with it. This program gave me a great opportunity to get first hand experience on conducting research. I learned how to approach problems and what research in mathematics is all about. I really enjoy performing research in mathematics, and I think the Summer Undergraduate Research Program has definitely gotten me excited to pursue mathematics in graduate school. Not only did it help me academically, it helped me feel more at home at TCNJ. The program made it possible for me to spend time working with professors in the mathematics and statistics department at TCNJ as well as other faculty and students in the Summer Undergraduate Research Program. I really enjoyed being able to work with Dr. Hagedorn on a problem and seeing how he approached it and thought about mathematics in general as well.  The Summer Undergraduate Research Program provided so many opportunities and challenges that I would not have been able to find anywhere else and I am so thankful that I was able to participate in such a great program.