Good areas in math where I can make some contributions?

[A.MHF]

I know this sounds like a weird question, but I'm interested.
So I've always loved math, especially pure mathematics. I spend a lot of time reading about theorems, mathematical proofs, and I try to come up with my own proofs. Recently I had the idea that maybe I can spend my time reading some high mathematical theories and try to come up with something new.
So my question is, are there some specialised areas that you would recommend in mathematics, preferably pure mathematics, that I can put my focus on, and maybe make a contribution like discovering something new or proving a theory or whatever? I know that these stuff happen at random, but I'm wondering if someone could maybe have an idea or anything, or just an advise on what to focus on.
Thanks.

 

[Dr. Courtney]

Without knowing your background or strengths, the question suggests you may be an inexperienced researcher looking for low hanging fruit.

Most more senior researchers like to save their ideas on low hanging fruit for their own students. I recommend looking for mentors and research advisors at your school and getting ideas from them.

Even so, my experience is that productive avenues open to less experienced researchers are often more applied than pure. Look to grow your skills in areas that may lead to more pure math pursuits, but don't eschew more applied questions in the process.

 

[micromass]

This is definitely not random. A lot of skill, talent and hard work is involved.

Anyway, research in mathematics can be done in virtually any area of mathematics. You just need to go deep enough. So pick an area you like, start studying hard. Eventually, you'll reach the point where you can make contributions.

 

[mathwonk]

you might try reading some articles in a publication like the American Mathematical Monthly, and see if something discussed there appeals to you. You could also go to the math library and browse some more advanced journals, but be forewarned the articles will mostly be very hard to grasp. Or you could take some of the theorems you have learned and proved and ask yourself if they can be modified slightly to yield new ones, or if not, why not, and try to find counterexamples to modifications. I myself consider even a new or more understandable or more elementary proof of an old theorem to be a contribution. And to cite George Polya and others, once you have found a new proof of an old theorem, the new ideas involved may then yield a new theorem if you look for one.

 

[A.MHF]

Thank you for your answers, you were helpful.