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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.

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I am quite sure that math can lead to more math, which can lead to more math, ad infinitum (but not ad nauseam, for the most part). I have heard tell that it can lead to other pleasant careers as well, but I have no personal experience with this. Seriously, this is yet another question that you should ask in person to someone at your particular institution. The anonymous internet is a poor substitute. –  Pete L. Clark Mar 4 '11 at 16:01
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You sound kind of directionless, could you edit your question and tell us some things you like or are interested in? Perhaps things you don't like? –  Ryan Budney Mar 4 '11 at 16:02
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The first question you need to ask yourself is whether you want or need to make a lot of money. The second question might be whether you like real-world applications. If the answer to both of those questions is no, then you can pretty much go anywhere you want to go. –  Qiaochu Yuan Mar 4 '11 at 16:03
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@Qiaochu: if we ever have the great fortune to hire you in my department, remind me never to nominate you for union rep. (Wait, did you edit that part of your answer? It seems more balanced now, and this comment correspondingly less funny.) –  Pete L. Clark Mar 4 '11 at 16:04
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@chaire: It would help if you tell us (i) what courses you have taken and really enjoyed; (ii) what other courses you have taken. Yes, there are specific directions within math, and subfields within fields, etc. But to put them in context, it would help to know where you are (not just where you want to go) before we start giving you directions. –  Arturo Magidin Mar 4 '11 at 16:56
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6 Answers

There are definitely particular directions you can take. I can see several ways in which this could play out. You might come across a mathematician whose style of mathematics is really interesting to you, you read all his papers and then you set yourself the goal of learning what you need to know to get into that area. You might become impressed with a particular applied problem like modeling climate change or prediction of the stock market and then you try to learn all the mathematics you need to effectively work on that problem. You could start by reading what people have written on the topic and learning enough to understand the approaches they are all using.

Since you seem to have broad interests, try to get involved in some undergraduate research project that seems interesting. If there is nothing in the mathematics department at your school, you can try other schools in your area or the Physics, Chemistry, Computer Science or Economics department. It would be good for your CV and it might have the side benefit of helping you get a sense of what it feels like to be more focused.

A life in research is pretty boring in the abstract. You pick some really small sub-problem that very few people are interested in. You work at it and make marginal successes that might have applications 100 years after you are dead. On the other hand, it's fun for some people and its a clean, safe, low-intensity way to make a living.

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Are you looking for direction in mathematics, or in the real world of life and career? Different questions, different answers.

Within Math, in my experience, you need to be a generalist though a couple years in grad school because you will have a required body of knowledge that spans Analysis, Algebra, etc. In that time, you will/should develop an interest in some pretty narrow question for your research. The choice probably depends on a mix of what's interesting to you, and who is interesting to work with.

In life, Math is divided into the academic and the practical. You see academic mathematicians all the time. The practical ones are spread thin as actuaries in insurance companies, "quantitative" types on Wall Street, management scientists in advertising, operations analysis for the military, etc, etc, etc. My personal bias is that computer programming is the black hole that sucks in failed mathematicians (like me). Relative to other fields, it's easy to learn, easy to get work, easy to make decent money, but hard to get out of.

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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.

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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.

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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!

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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.

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