Opportunities for Undergraduate Research in Computational Algebra

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L^AT_EX and Lyx files for typing an Honors Thesis according to the Honors College guidelines.

What is the goal of undergraduate research?
How do I get started?
What are some possible projects or topics with Dr. Perry?
What are the benefits and costs?
What if I'm not that good at math (or algebra, or programming, or...)?

What is the goal of undergraduate research?

1. A risk-free ("not graded") opportunity to stretch yourself and make yourself more marketable to potential employers or graduate schools. The end result should be a well-written thesis that will summarize your study, and perhaps a publishable paper.
2. A chance to get a taste of how "real" mathematics is done.

How do I get started?

1. Look over the list of possible projects/topics below. If something strikes you as interesting, contact me.
2. If you don't find anything that appeals to you, take a look at these USM math department professors' pages instead:
1. Dr. Ding (numerical analysis, optimization, ergodic theory, dynamical systems)
2. Dr. Kolibal (fractals, differential equations, numerical analysis)
3. Dr. Lee (algebraic geometry, differential geometry, symplectic geometry, noncommutative geometry)
3. Make a habit of visiting the library every month and perusing math journals. Reading others' work often stimulates ideas. You probably can't understand everything in most articles, but you can certainly understand at least something in most articles; if you find an article that interests you, you can take it to a member of the department who specializes in that area and ask her or him to work through it with you. Two good journals that target undergraduates are:
• The College Math Journal
• Math Magazine
  
4. If you have your own ideas, you can talk to me (or any professor), and see if I am (s/he is) interested in working on it with you.
5. You can also try
1. Google's Summer of Code
2. AMS' listing of summer Research Experiences for Undergraduates (REUs)
3. AMS' listing of internship opportunities for undergraduates
4. NSF's listing of REUs

What are some possible projects or topics with Dr. Perry?

• Problems from MAA journals
  These tend to be relatively easy and short term (days or weeks). Solving them requires a bit of time and creativity. Once you finish a solution, you can submit it to the journal and see your name in print.
• Projects leading to original results and/or software, designed for math majors with some computer experience or comp sci students willing to learn a bit of math
• Problems in Linear Algebra and Modern Algebra I. Specific problems I have directed, or am willing to direct, include:
• Fixing Dodgson's method for matrix determinants
• Other fraction-free methods to compute determinants
• Techniques to triangularize a sparse matrix
• Methods to represent a polynomial on a computer: lists, matrices, geobuckets, ...?
These projects generally require at least a small amount of proof.
• Programming projects available for students able to work in C/C++, Eiffel (both ISE Eiffel and SmartEiffel), Java, Python (SAGE actually), or Maple. Such projects will be useful either for mathematical research (a problem I'm interested in) or for education (the common benefit of elementary and secondary schools).
• Projects that won't lead to original results, but will supplement or round out your undergraduate mathematics major (or minor)
• Galois Theory
• Applied Algebra
• The Fundamental Theorem of Algebra
• Theory of Computation
• Computer programming for mathematicians
• Note for students of the Honors College: The Honors College now discourages theses that do not include original results of some kind.

What are the benefits and costs?

• Benefits: You stretch your mind, gain a lot of experience with problem-solving outside the classroom, and end up with another item to add to your resume, CV, job application, etc. If you want to travel, you might have an opportunity to do that, too. REUs at other institutions have enough money to pay for your travel and lodging. Google's Summer of Code 2007 offers a $4500 stipend to students, and you don't have to travel. Unfortunately, I can't offer you any kind of stipend at the moment.
• Costs: All research takes a lot of time, patience, determination, and a willingness to deal with hard problems that have unexpected or even undesirable results. If you work with me, you also have to put up with my quirks.

What if I'm not that good at math (or algebra, or programming, or...)?

The only way to get better at it is to work at it and keep at it! I wasn't very good at them either once. I have memories of getting so frustrated that I was sure I'd never try programming or mathematical research again. Sometimes I still feel that way, but now I do it for a living and I enjoy it. (most days) You can too, if you're willing to plug at it hard enough.