Where to go with Mathematics?

I am currently a math major in college and my main problem is that it feels directionless. My college offers little in term of variety in undergraduate math so I moved on into taking graduate courses and I am actually loving it. But at this point I am just randomly taking courses that looks interesting and was wondering if there was specific directions to take within math. And even past that where will it lead.

Thanks

EDIT
After reading the comments I decided to add extra info. On the courses that I like, I have taken Abstract Algebra and Linear Algebra and really enjoyed them and also Number theory. I have also taken the typical three semester Calculus courses, Differential Equations, Discrete Math, Probability, Numerical Analysis and Real Analysis. On these courses I like them though I am not as enthusiastic about them as I am for the other courses I have mentioned. Though I find their applications very interesting.

And more specifically on the question it is on what paths are there to take in school and also what paths are there to take after graduation.

Solution 1:

My advice to you is to take whatever kind of mathematics you find interesting. The subject is vast and profound, filled with diverse ideas and methods, each of which can be pursued further than one might imagine. For any course you take, there is a further course, and another and perhaps another and then published research work carrying those same ideas further and further into distant abstract lands. So just follow the path where it leads you. As long as you have talent and find it interesting, you will do well. The fact that you have enjoyed abstract algebra, linear algebra and number theory suggests that you might enjoy more pure mathematics courses, and surely there are plenty. The various subject areas of mathematics have a structure that will become apparant to you as you learn more, and there is little need at first to grasp the whole structure.

At the elementary level, such as in high school, mathematics can often seem to follow a linear progression---one might have elementary algebra, then geometry, then trigonometry, then pre-calculus, then calculus, etc.---but of course at higher levels, mathematics does not follow a linear progression. Rather, it branches out into diverse areas, each of which develop the structure of its own ideas.

When I was an undergraduate student, I had taken a sophomore course (Math 6 at Caltech) that had included a semester on mathematical logic, which I thoroughly enjoyed. Later, when looking at the catalogue to register for the next semester, I realized that there was an entire subject of mathematical logic, with many courses and an entire research program. I was enthralled!

Solution 2:

unless you really truly see yourself getting a phd in math and eeking out some academic career, you should problem take a bunch of applied math, study engineering of some sort, or learn a bunch of programming so you can actually get a job.

Solution 3:

I would say math is more a tool that you have to apply to the real world to make any practical use of it. For example, learn some physics, computer science, or engineering, then you will find yourself applying Mathematics in many real world examples.

A great example: if you master the priniples of physics (which requires a mastery of mathematics), then you can create the most realistic physics inside video games.

Math is only a framework from which to describe the real world. Staying strictly in mathematics is something I'd recommend not doing unless you want to teach mathematics. Find the area in the sciences that you like, and apply mathematics to it.

Solution 4:

While this is purely anecdotal (and added months after the question), perhaps it will help. I got my degree in Mathematics, and while my university did not offer specialization at the undergraduate level I had definitely focused on the theoretical over the applied.

I had wanted to go into research, but that virtually requires a PHD and my financial situation at the time prevented me from working on graduate studies at that moment. So, after exploring options, I joined the Army and got a commission in Military Intelligence. While I never used any math more advanced than High School Calculus, the rigorous, logical thinking I had learned in my studies both helped me to get into Intelligence in the first place and helped me extensively int he analysis I was called on to do there. While the military is certainly not for everyone, I found it to be very rewarding and I am glad I had an opportunity to serve.

I later made the decision not to continue my time in the military for family reasons. Instead, with a little extra studying mostly done on my own and with the help of a couple of a good mentors in the field, I moved into computer programming. This calls for the direct use of mathematics more often, but again little of it (at least int he type of programming I am doing currently) is much beyond what would be learned by the advanced high school student. But the mind set I developped in studying mathematics has been tremendously helpful.

Currently, while still programming, I have entered law school. I have been told that while they accept any degree mathematics is considered one of the more favored degrees partially for the mindset it fosters and partially simply due to its relative rarity (good law schoools go out of their way to get a student body from a diverse background).

So, I think even a relatively directionless study of mathematics can help prepare a student for a variety of jobs and mathematics generally provides a solid foundation from which someone who is undecided can move in a variety of directions.