Posts Tagged ‘research’

Draft of an Undergraduate Research Philosophy

April 30, 2014

I am working on establishing a sustainable undergraduate research program. I want to record some ideas that I have here.

First, I think that this might mean a shift toward searching for problems that undergraduates can understand the research question. This is actually pretty close to what I have been doing anyway, although I hope to more consciously seek out easily-understood problems. I had lunch last weekend with Andy Rundquist (#brag), and he told me that he changed his research focus so that it would be easier for undergraduates to work with him. Fortunately for me, he did this in part because of non-academic considerations (lasers are expensive), but I understand that his main focus was to allow students to work with him immediately after their freshman year. I still will study group theory, although I will see if there are questions that students could quickly understand after just having had one or two semesters of calculus. In particular, I might start learning something about finite fields, which I think could be accessible (it is just like the real numbers, only there are only a finite number of points!).

If I can find good questions (that is part of my goal for my sabbatical next spring), then I would like to form a research group. I hope to work with several students at once. The model I have in mind is that each student will work on solve the research question for a specific case—likely a specific family of groups. The students will be able to talk to each other, since each knows the question being asked, although not every student will know the structure of each, say, particular group.

Essentially, these students would be working on the examples that I would do myself if I were trying to solve the problem on my own. After the students complete their work, they perhaps write a thesis on the problem and I see if I can use their work to solve the entire problem.

This gives students a chance to do undergraduate research, gives them a chance to do it in a more collaborative manner (they get to work with a research team), and it gives me a chance to kill two birds with one stone—the undergraduate research is actually supporting my own research, so time spent with undergraduates is really time spent on my own research.

Do you have anything thoughts on this model? Do you have any alternative models for undergraduate research that work?

Conferences are important

June 2, 2011

I am not a great mathematician. I have many deficiencies. The deficiency I am going to focus on today is that I am not very good at generating my own research questions. But I have some ideas on how to get better below.

(Note: I am not a great mathematician, but I am an okay mathematician. This is because I have some amount of the most important quality for a mathematician: tenacity. I am not work efficiently and I may not have the background I should, but I am willing to sit down most every day and work. This is really huge).

First, I went to the Zassenhaus Group Theory Conference last weekend (I am one of the giants in the back of the picture). I was reminded how important these conferences are. For one, I got a lot of ideas for research questions from listening to other presenters speak. Some of these ideas were given to me directly by the speaker, and other ideas were tangential to what the presenter was actually discussing. But I also was a presenter, and I presented on a problem that I am stuck on. I received several great suggestions on how to proceed.

Second, I have been working with a collaborator for the past semester on a problem. Something clicked today about one problem I have: I search for a solution a little too directly. But reading my collaborator’s ideas, I realize that he plays with the ideas much more, collecting a bunch of ideas that may or may not be useful to the problem at hand. This seems like it would be enormously useful, and I am shocked that I do not do it already.

My goal is to play more. This is what I ask my students to do, and they sometimes just do not get it. Apparently, neither do I. I am hoping that

  1. I can work past this and
  2. I can learn what it took for me to work past this, so that I can help me students to learn to play more.

Problem Posing

January 22, 2010

My college is now accepting applications from its students for summer research. I met with my first student today, and I fortunately have a handful of research questions that would work for him. However, I find this to be a very difficult task; heck, I find it difficult to come up with my own research questions.

Last semester, I started trying to fix my lack of questions by creating one question every other day. The question does not need to be good, but it has to be something that could theoretically be a research question. I was not as successful as I would have liked, although I did come up with one third of the questions I had hoped for. I used a couple of them with my student.

This is one of my weaknesses—I am not very good at creating questions. This makes me think that I should nurture this quality in my students, lest they end up like me. I think I will re-read The Art of Problem Posing, which was recommended to me by Juliana Belding.

Tip of the Day

October 12, 2009

My thesis advisor gave me a great piece of advice: never leave the office without giving your computer something to do.

Since then, I have tried to have a program running overnight whenever possible. This is not the greenest thing to do, but it is productive. Here are two things that I do regularly.

  1. I use the computational algebra system GAP in my research, and I try to have a program running whenever I am not at the computer.
  2. I participate in the Great Internet Mersenne Prime Search (GIMPS) to use my computer to find large primes.

Frankly, I haven’t been very good about having GAP programs run. However, it is very easy to search for primes–GIMPS has programs that are foolproof.

How to Write a Lot

September 23, 2009

I was a little skeptical that writing a professional weblog would be of any benefit to anyone, but it has actually already paid off for me. As a response to my post on Getting Things Done, two friends/colleagues wrote to me about How to Write a Lot by Paul Silvia. I read this over the weekend, and enjoyed it.

The book roughly says, “To write a lot, schedule time each day to write.” This might seem simple, but it can be a little difficult to implement. Before reading the the text, I had scheduled myself time to do math for 70 minutes every other day. So far, this has not paid off a lot. However, I think that I might change that to daily starting next week. When this happens, I hope to see a rise in productivity.

A second thing that I am doing is to apply for a fellowship. This fellowship will release me from one course next year, giving me time to work on a textbook I am writing. I have to imagine that this, combined with a schedule, will help me to “write a lot.”

A second benefit of the book is that it spelled out the difference between psychology research (Silvia’s field) and mathematics (mine). Silvia wrote that most psychology professors have a backlog of data, and they could produce many papers if they could find time to sort through the data. Basically, psychology professors are not lacking for things to write about. I find this not to be true in mathematics. My struggle is to create original mathematics that people care about. I would love to have the problem of having too many ideas stashed away in my filing cabinet. Because of this, I expect that most of my “writing time” will actually be “thinking time.”