Undergraduate Math Outside the Classroom

[f25274]

I am in my last year of high school. I do not know if I will become a mathematician or physicist. Right now, I enjoy physics more but I will double major in math and physics anyway. I know that research experience is important in undergraduate physics. However, how important is undergraduate research in math? I have a slight idea of what undergraduate physics research is like but I do not know anything about undergraduate research in math. I don't even know if it is needed for graduate school. So is there anything other than taking math classes that I need to do to be competitive for math graduate school? I am asking this because I might switch to considering math more seriously in college. I have not taken a pure math class (one involving proofs) yet. I am trying to teach myself to write proofs but I have not found a way to easily teach myself. It is hard and I don't know if my proofs are formal or if they're even correct. Would I be okay if I only do what is required of me in my math classes?

 

[hsetennis]

A little lengthy, but I hope it answers one of your questions: http://www.math.hawaii.edu/~lee/how-to.html

 

[Theorem.]

I am in my final year of an honours pure mathematics undergraduate and I can confidently say that all my fellow students who are thinking seriously about graduate school have participated in some form of summer research, or research during the year through classes offered by the department that allow students to focus on research. That being said, it is not a requirement. With this respect, it is very comparable to physics. Of course, the research can be very different. Most of my summer research scholarships so far have been dedicated to studying the structure theories of certain lie algebra representations. I have worked several times with the same professor, although many students work with various professors in order to experience the corresponding different areas of mathematics.

During the first summer I got a research scholarship, much of my time was spent actually learning material so that I would have a chance at contributing to a small research problem. This is common, since most problems in mathematics are not necessarily accesible at the undergraduate level. It was a neat experience since I had the opportunity to study a subject that isn't normally "available" to students at my level. I continued to work with the same professor and it turns out he will be my graduate supervisor now- and we already have a nice established relationship.

There are many benefits to doing this, but it really depends how seriously you are thinking about math. If you are planning on pursuing graduate studies in physics, then maybe it would be better to allocate summer research positions towards that interest. Of course, a mathematics research position could be enlightning for a physics student as well and expose them to some neat results. A lot of the stuff I work on has strong applications in physics (although, admitidly I am not too concerned with them, it is the math that has my heart). I have a few friends studying physics who have done summer research in mathematics at least one summer. They are more concerned with the theoretical side of physics though.

I hope this provides some insight.

 

[deekin]

I think what Theorem said is spot on. For either math or physics, if you intend to go to grad school, it helps significantly to have experience outside the classroom. Not only do you get to learn what research is like, but you also learn material you normally wouldn't have in one of the available classes and you (hopefully) develop a great working relationship with some of the faculty, which will help when it comes time for letters of recommendation. I don't know how often people do independent study (I haven't heard of anyone doing it in my physics dept, but I know a few that have done it for math), but this would be another way to develop a relationship with a faculty member and learn some cool stuff you wouldn't have otherwise.

You still have plenty of time though, enjoy your last year of high school!

 

[Theorem.]

Yes you still have lots of time : ) The best thing to do is to ask professors whose classes you enjoyed about their research once you get started at settled into university.

 

[Robert1986]

Well, your proofs probably suck, but don't be worried about that at all. You are thinking about proving stuff, which puts you heads and shoulders above nearly everyone else. Learning to write proofs is hard, and you will not be a great proof writer when you first begin writing them. Good pianists suck when they start out.

As for undergrad research in math, I am very sceptical about the "research" being done. I know many people whose summer "research" was essentially reading several papers. Now, this is important, and is 100% a part of any serious research program, but, from what I have seen, summer research programs are more like (perhaps advanced) reading courses. IMO, it is as important (if not more so) to get experience taking graduate level courses while you are an undergrad. Doing well in a grad level course is not easy, and if you can, you have shown that you can handle first year grad level course work.

 

[Theorem.]

Well, your proofs probably suck, but don't be worried about that at all. You are thinking about proving stuff, which puts you heads and shoulders above nearly everyone else. Learning to write proofs is hard, and you will not be a great proof writer when you first begin writing them. Good pianists suck when they start out.

As for undergrad research in math, I am very sceptical about the "research" being done. I know many people whose summer "research" was essentially reading several papers. Now, this is important, and is 100% a part of any serious research program, but, from what I have seen, summer research programs are more like (perhaps advanced) reading courses. IMO, it is as important (if not more so) to get experience taking graduate level courses while you are an undergrad. Doing well in a grad level course is not easy, and if you can, you have shown that you can handle first year grad level course work.

I don't necessarily agree with this last student. Even if for some reason it WAS just a reading course- It would be graduate level reading providing the same advantage as a graduate course. All the students I know in summer research actually do research. Sure they have to spend about half the first summer they do research reading- but this is incredibly valuable on it's own. After that they have the chance to actually participate in the research. I can say, that having completed several summer research scholarships in mathematics along with most students in my program- if not all. Most students do take graduate courses as well- I will be taking my 3rd this year- but research is stressed much more here, as it should be. Firstly it gives you insight into what actual research entails, provides you with a first hand experience, and also allows you to study things you wouldn't normally be able to grasp at that level (and many of the topics being studied are much more specialized then a lot of the graduate courses themselves, as they focus on a particular problem). Worst case scenario (and this is not very likely at least at my school), a student would spend 1 summer using their summer research position as a reading course. This alone is still a fruitful experience. And if they worked with the same professor later, it would allow for their full focus on the research problem at a later time.