What Are the Best Research Fields in Mathematics for Beginners?

[kurt.math]

Hi,

What would be the best fields of mathematics to do research in. I have about 5 years of mathematics under my belt and would like to start some research.

 

[TheForumLord]

The answer to your question is very individual...
There are many beautiful subjects in math, such as (for my opinion): Game theory, Additive number theory, Knot theory, algebraic geometry, algebraic structures, combinatorics, complex analysis, etc...
Maybe if you'll be able to tell us what are you interested at, we'll be able to give you names of fields related to your answer...

 

[kurt.math]

The answer to your question is very individual...
There are many beautiful subjects in math, such as (for my opinion): Game theory, Additive number theory, Knot theory, algebraic geometry, algebraic structures, combinatorics, complex analysis, etc...
Maybe if you'll be able to tell us what are you interested at, we'll be able to give you names of fields related to your answer...

Im interested in differential geometry (as well as algebraic geometry) and all forms of topology mostly, but i don't know whether I will be capable of research because i am unaware of what has not been researched already and if there is anything to find.

 

[JD88]

Have you tried to read through the relevant literature out there.

 

[qspeechc]

There are books with titles such as Open Problems in Topology and Unsolved Problems in Geometry, which obviously a collection of problems in the relevant fields that have not yet been solved (duh!), and maybe are a place for you to look if you are really stuck for ideas. Maybe not the greatest idea because these problems have been around pretty long and have repelled the attacks of some excellent mathematicians, but would make good research problems, I guess.

 

[Landau]

If you have studied mathematics for 5 years, you must have learned that the question 'what is the best field of mathematics to do reasearch in'? is not very well posed. If there would exist a 'best' field, everyone would do reasearch in that field.

 

[TheForumLord]

Hmmmm... If you have 5 yrs. of experience with math, they don't involve any Msc. degree? In your advance degrees, you have a supervisor that knows excatly what the open questions are and you are doing research while studying for the degree...

 

[boboYO]

combinatorics.

also you didn't specify 5 years of what maths.

 

[Landau]

also you didn't specify 5 years of what maths.

especially since you seem to be 17 years old, could you be more specific about what your math level is right now? What courses have you taken, what degree do you have?

 

[kurt.math]

It is true that i am 16, but it is true that i have learned 5 year of mathematics. I have taught myself the first 3 years of university math, honors mathematics (pure) and 1st year graduate work.

Because i am applying for university at the end of next year, i would like to write a paper to use as substantial evidence to put me through an accelerated program because of course doing at least the first 3 years of mathematics again would be horrifying (but note that i don't want to write a paper only for the single aforementioned reason but because i am passionate about mathematics and would like to be apart of the immortal race to learn more about this beautiful and elegant subject).

I have learned low dimensional topology, differential topology, algebraic topology, algebraic geometry, differential geometry, commutative algebra, lie algebra, galois representations, PDEs, functional analysis etc

Could anyone tell me with that kind of knowledge if research is possible in anyone of those fields mentioned?

 

[story645]

Because i am applying for university at the end of next year, i would like to write a paper to use as substantial evidence to put me through an accelerated program because of course doing at least the first 3 years of mathematics again would be horrifying

a) Some schools let you test out of pre-reqs; contact the ones you plan to go to.
b) first 3 years is a subjective assessment - you're basically saying you've studied all there is to know in math well enough that you only need a year of coursework for a degree. I don't see admissions people taking you seriously unless you can really back that up. A paper isn't going to do it. (I have a friend who won one of most prestigious high school awards in the country for research in chemistry, but he still had to take chem 102 with the rest of us.)
c) Dunno what city/state/country you're in, but look into doing research with a professor if possible. If you're in the US, look at the Intel Science Talent Search program.

 

[zpconn]

I think your very best bet is to try to get into contact with a professor, explain your whole situation, and see if he'd be willing to work with you.

You may need to do research to figure out which professor would be best to contact. Some professors are much more open to requests like this than others. You could try emailing some graduate students, briefly explaining your situation and asking which professors they recommend. (If you do this, you may want to email a couple of graduate students focusing in different areas, since they will probably be best affiliated with different faculty members.)

By the way, it may be easier if you focused on publishing an expository paper first. For instance, I know a few high school students have published in the Harvard College Math Review. [On another note, the website for this seems to be down!]

Good luck!

 

[eof]

Are you really sure that you know the stuff that you've taught yourself? If you know that stuff you should be able to do qualifying exam problems for any major university. Can you solve any of the following problems:

http://www.math.harvard.edu/graduate/quals/qs09.pdf

There is usually a major difference between just knowing something and actually knowing how to use it. I mean you can know a bunch of theorems from algebra, but how well can you apply them when needed? It's pretty easy to think you understand something when you actually don't. That's why I am pretty skeptic about your claims that you have taught yourself all of the above stuff.

 

[Tiger99]

You might get some interesting responses if you ask this question at mathoverflow.net
It's worth a try anyway.

 

[bwebster]

Doing this on your own is simply not a realistic plan, as is clear from your confusion about the whole subject. Even people who've done 5 years of math in a university course and who have proper advising are usually not really ready to do good research; it's actually a very difficult transition to make.

If you insist on it, the only reasonable thing to do is find a professor to work with, who can suggest problems, but I wouldn't even recommend doing this. You've got the rest of your life to be in the "immortal race" of mathematics research; be glad that you can focus now on understanding what other people have done. If you actually find yourself as a research mathematician, you'll never have enough time for that, and will wish you had done more later.