Like many professional mathematicians, I am active in mathematical research. People outside academia often seem surprised to learn that that there are still unsolved problems in mathematics, questions for which no one (yet) knows an answer. Some assume that since humanity has been studying mathematics for thousands of years, we ought to have it all figured out by now; others seem surprised that there are even questions in mathematics in the first place.

The second type usually suffers, I find, from the misapprehension that mathematics is "all numbers and calculations." On the contrary, we mathematicians prefer to leave the long boring calculations to accountants or, better yet, computers (since they don't get bored). Mathematics is not about long tedious calculations; rather, it's about proving theorems.

Some are surprised that mathematicians haven't yet proved all the theorems that there are, or at least all of the important ones. However, the number of theorems is infinite, and out of these, the number of interesting and important ones is almost certainly infinite as well. Moreover, as anyone who has participated in mathematical research knows, during the course of answering one question we nearly always end up raising three more. The more we learn, the better we understand how much more there is still to learn.

Starting as early as my first summer of college, and continuing through graduate school and up to the present, I've been very active in mathematical research. As an undergrad at the University of Wyoming, I participated in the McNair scholars program for three summers, with my last year including a trip to the national McNair conference at the university of California at Berkeley, where I gave a plenary session talk about symmetry in chaotic dynamical systems.

In recent years, students from my classes who were interested in doing mathematical research have approached me about doing projects, many of which have led to published papers with the students as coauthors. I keep a list of project ideas from which interested students can select a project that they find interesting. Once a research student has selected a project, I usually give the student some background reading and some assignments starting at the student's current level of background, building up to solving the main problem(s) of the project, with regular meetings to discuss our progress. Once the problems are solved, we work together to write up our results as a paper, which we then submit to a journal for peer-review and, hopefully, publication. In my three years at UCR, I wrote ten papers with students.

Research in mathematics consists mostly of proving new theorems as well as finding new and better proofs for old theorems. A proof for a theorem is an explanation of why the theorem is true. A good proof should be clear, as easy to understand as possible, and should be a rigorous (i.e., a sound deductive) argument. New research is written up in the form of articles which are then submitted to academic journals; before publication, these journal articles are sent to experts (called "referees") who evaluate the results for correctness, importance and originality, a process known as peer review. New research is also communicated through talks given at conferences, seminars and colloquia. Peer review can be a lengthy process, and professional mathematicians who want to stay current in their research fields cannot wait for papers to appear in journals; instead we read preprints, or not-yet-published manuscripts, obtained either directly from our fellow researchers or from preprint servers like www.arXiv.org. These papers have not yet necessarily passed peer review, so they sometimes contain typos and other errors; thus, it is best for non-experts to stick to peer-reviewed published articles.

Peer review is not to be confused with reviewing services such as Zentralblatt MATH or MathSciNet, where summaries of published papers are made available to help researchers find which articles they need to read.

In the past, mathematical research has been done by all sorts of people, not just professional mathematicians; one very famous mathematician, Pierre de Fermat (perhaps you've heard of his "Last Theorem"), was a French lawyer in the 1600s who did mathematics as his way of unwinding in the evenings after a long day, for example. It is probably true that the majority of the "easy" theorems have already been proved, and that most mathematical research these days requires a fairly extensive amount of background. Today, mathematical research is done mostly by professional mathematicians, including math professors as well as researchers at private and public research institutes, and by graduate students, who are required to do a certain amount of original research before they can obtain their PhD degrees. However, it is not uncommon for mathematicians to work with undergraduate students on research projects, which can lead to published papers, as we've seen above.

The benefits of doing mathematical research are almost exclusively intellectual as opposed to, say, financial; journals do not pay article authors for their contributions, and speakers at math conferences not only do not get paid but generally have to pay fees themselves to attend the conferences. Why, then, do we continue to do research? It is true that a certain amount of published research is required for professional advancement in mathematics, e.g. to complete a Ph.D. or to get tenure; however, the majority of high-quality research is not done by those who are merely trying to minimally satisfy job requirements.

Contrary to popular belief, research is not even generally done for fame and prestige. Even the giants of research in a particular area are generally unknown to those outside of the field. Everyone's heard of Einstein, but how many non-mathematicians are aware of the work of giants such as Milnor or Poincare, or even Gauss or Euler (that's pronounced "oiler", NOT "yooler", for the non-German speakers)? If you're looking for fame, mathematical research is unlikely to produce it, except in the limited sense of "respect among your peers". New theorems are proved every day by people all around the world.

Research does not typically lead to fame or money, and most research is not even required by our jobs. Rather, we mathematicians do research because we're curious about the answers to the problems, and once we find the answers we get excited about explaining them to others who are also interested. In other words, mathematical research is not merely a means to an end; it is a goal in and of itself.

Some of my research has been in the fields of virtual knot theory and quandle theory. These are both relatively new areas of mathematics in which a lot of work is being done and a lot of work remains to be done. You can read my papers on www.arXiv.org, if you're interested. Math students who are interested in working with me on a research project should feel free to drop by my office or send me an email or instant message. Depending on my teaching load and the number of students already working with me on research projects, I may have a project idea that you can help with.