What exactly constitutes math research ?

[Anonymous217]

What exactly constitutes "math research"?

I'm an incoming second-year, majoring in pure math, and I'm trying to get involved in research this upcoming Fall.

I recently asked my advisor and he said the only option during the school year would be an Independent Study, where you have a faculty sponsor/supervisor, and it will basically be a self-paced reading course.
Would you consider this "research", or "research with a professor"? Isn't it basically the same as a standard course, with the exception of the self-pacing?

I'm doing an REU next summer, but as far as research during the school year, the only option is the Independent Study. So I'm considering either the Independent Study or a grad course (maybe PDEs), in addition to 3 grad courses as my Fall schedule. Which would you guys recommend? I'll have plenty of grad courses by the time I graduate, and having an Independent Study could potentially help my research aspect. But would you really consider it "research"? Maybe I could try producing a paper/project by the end.

 

[micromass]

I'm an incoming second-year, majoring in pure math, and I'm trying to get involved in research this upcoming Fall.

I recently asked my advisor and he said the only option during the school year would be an Independent Study, where you have a faculty sponsor/supervisor, and it will basically be a self-paced reading course.
Would you consider this "research", or "research with a professor"? Isn't it basically the same as a standard course, with the exception of the self-pacing?

I'm doing an REU next summer, but as far as research during the school year, the only option is the Independent Study. So I'm considering either the Independent Study or a grad course (maybe PDEs), in addition to 3 grad courses as my Fall schedule. Which would you guys recommend? I'll have plenty of grad courses by the time I graduate, and having an Independent Study could potentially help my research aspect. But would you really consider it "research"? Maybe I could try producing a paper/project by the end.

Math research involves a lot of reading. You'll need to know a lot before you can actually start solving problems. The only difference between your "independent study" and "research" is that you have no concrete goal to work on.

You say that you will do an REU next summer. Maybe you should ask yourself what you want to work on. Then you can ask your future advisor what exactly you need to know for the REU. You can already study that during the independent study. This is good, because this way you'll have the studying out of the way, and you can focus on the research itself during the REU.

So in a way, you can use the independent study as something that can help you with the research.

 

[Anonymous217]

Math research involves a lot of reading. You'll need to know a lot before you can actually start solving problems. The only difference between your "independent study" and "research" is that you have no concrete goal to work on.

You say that you will do an REU next summer. Maybe you should ask yourself what you want to work on. Then you can ask your future advisor what exactly you need to know for the REU. You can already study that during the independent study. This is good, because this way you'll have the studying out of the way, and you can focus on the research itself during the REU.

So in a way, you can use the independent study as something that can help you with the research.

I never thought of that. Thanks for the advice!
My advisor also suggested to complete a second honors thesis, so maybe I could use the Independent Study to read/prepare in Fall, complete the actual thesis in Spring, and then do an REU all within the same field.

I guess the Independent Study would ultimately narrow my interests into a more specific line of work.

 

[Anonymous217]

So as of now, I'm considering either algebraic number theory, elliptic curves, or graph theory/combinatorics. However, I'm really having a hard time deciding on a topic since I'm not that aware of the particular topics in Mathematics (beyond a general scope). I've just been staring at the list on arXiV.

Do you guys have any recommendations, or any particular topics you personally enjoyed? I'm just curious.

 

[chiro]

So as of now, I'm considering either algebraic number theory, elliptic curves, or graph theory/combinatorics. However, I'm really having a hard time deciding on a topic since I'm not that aware of the particular topics in Mathematics (beyond a general scope). I've just been staring at the list on arXiV.

Do you guys have any recommendations, or any particular topics you personally enjoyed? I'm just curious.

I think if you're going on to second year, you should get a taste of what is out there first, if you don't have a complete idea of what you want to get into.

If you have been thinking about something in your head though, I would run it by your professor/lecturer and ask them advice on where to start reading. You could do it yourself though and if you need help, I'm sure people here would be more than willing to help you.

If you have applications of math in mind, rather than being biased towards pure math, then finding some materials that talk about applied math in that area is also a good way to go.

My advice to you is try and narrow down an area and then spend any free time you want to dedicate to learning the ins and outs of that area. You don't have to become an expert straight away, just get a feel for the area and you'll end up getting a feel for specific areas of investigation.

Also just want to give you some advice on math research: take a mix of classes: everything is connected in weird and wonderful ways. Be aware of what is out there as it may help you unexpectedly in the future. You will have to specialize your focus if you want to do research, but knowing what is out there allows you to be aware of other tools that you may need at some point.

I'm sorry I can't really add anything to ideas for your research since I myself are not familiar with those areas specifically, but some of those have been covered in a cryptography course I took, so maybe if you are interested, you could look into that.

 

[micromass]

So as of now, I'm considering either algebraic number theory, elliptic curves, or graph theory/combinatorics. However, I'm really having a hard time deciding on a topic since I'm not that aware of the particular topics in Mathematics (beyond a general scope). I've just been staring at the list on arXiV.

Do you guys have any recommendations, or any particular topics you personally enjoyed? I'm just curious.

Algebraic number theory and elliptic curves requires algebraic geometry (most of the time). So you should probably study that if you want to go into these fields

Try to read this:
http://mathcircle.berkeley.edu/BMC4/Handouts/elliptic/elliptic.html
and read the book "rational points on elliptic curves" by Silverman and Tate. This will give you a taste on what research of elliptic curves is all about

 

[Anonymous217]

I think if you're going on to second year, you should get a taste of what is out there first, if you don't have a complete idea of what you want to get into.

...

I'm sorry I can't really add anything to ideas for your research since I myself are not familiar with those areas specifically, but some of those have been covered in a cryptography course I took, so maybe if you are interested, you could look into that.

Thanks for the advice. I've been slowly narrowing my field of study, but it's still pretty vague. By now though, I know it's not applied math/analysis or probability. That still leaves algebra, geometry/topology, and logic. I guess I still have a long way to go before I can decide on a field.

Algebraic number theory and elliptic curves requires algebraic geometry (most of the time). So you should probably study that if you want to go into these fields

Try to read this:
http://mathcircle.berkeley.edu/BMC4/Handouts/elliptic/elliptic.html
and read the book "rational points on elliptic curves" by Silverman and Tate. This will give you a taste on what research of elliptic curves is all about

Thanks for the helpful references! I've been pretty curious on number theory in general, and what tools are used to solve its problems. Berkeley has this extremely intensive algebraic geometry course, but I won't be able to take it until next year. So I guess it would probably be better to try out another topic. Doing a topic in combinatorics could be helpful for REUs since most seem to be based on combinatorics.