What is research in mathematics?
Mathematical research goes beyond reading someone else's discoveries in a textbook or journal, and beyond solving ready-made problems. Mathematical research involves first determining the right questions to ask, and then attempting to answer them. To varying degrees, this involves solving problems that nobody has solved before, often with little idea of what might work, and sometimes without a clear question.
Solving mathematical problems never before seen involves intuition as much as knowledge, and the unstructured nature of mathematic research is often intimidating - whether you are an undergraduate, doctoral student, or professional mathematician. Students are usually introduced to research through a carefully supervised apprenticeship process. Research skills and insight develop through close interaction with a faculty mentor.
If you are a mathematics major, or are thinking about becoming one, you already know that learning mathematics is exciting. Your professors can assure you that discovering mathematics is an even more exciting activity. You can participate in research programs like independent study courses, the Honors Program, summer internships, and participation in summer Research Experiences for Undergraduates (REU) programs. Writing an Honors Thesis or participating in summer REU programs can give you a preview of mathematics doctoral study, if that path interests you.
Undergraduate mathematical research at W&M often results in professional publications. Look for faculty members who research areas that interest you, or who have taught a course you particularly enjoyed. Ask if there is an opportunity to work with them.
The National Science Foundation (NSF) and other federal granting agencies support about two dozen REU programs around the U.S. Most of these programs are small, involving about ten students so that students can work closely with a faculty member on a research problem. REU participants typically receive modest stipends and free on-campus housing for eight to ten weeks in a summer. Different REU sites focus on different research areas, and a list of REU sites is available from the NSF. Admission to REU programs is competitive and requires letters of recommendation from faculty members.