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I am going into my sophomore year as an undergraduate and I would like to ask the more experienced folks a couple questions about learning math and related things. What are your experiences and advice concerning the following dilemmas?

Being limited to a rate of 4-5 courses per semester, I realize that I am certainly not going to be able to take all of the courses that I am interested in. I would like to get a build a broad and solid base of knowledge by studying all areas of math at least a bit, but this comes at a cost of being able to take the more advanced, deeper courses. My plan was to self-study measure theory/Banach spaces and topology this year so that I'll be able to immerse myself in the graduate-level courses, which I expect to be more challenging and interesting and rewarding. I was wondering if people had experiences/regrets/wisdom about whether or not this is a good idea? Do you think it's better to build up a broad foundation thoroughly or throw yourself beyond your comfort zone?

On a similar note, what is your advice concerning specialization versus developing a broad taste? In my very limited experience, I have enjoyed representation theory, algebraic number theory, and complex analysis a lot. But there are still so many areas that I've yet to sample: algebraic topology, differential geometry, more advanced real analysis, algebraic geometry, analytic number theory, combinatorics... What's a good balance between trying all the different fields of math and trying to quickly become an expert in one?

Stepping back a bit, let me pose this question for a broader context. I am rather interested in philosophy, psychology, computer science, physics, and economics in addition to mathematics. I would like to take courses in these subjects as well but I am worried that this will put me at a disadvantage should I choose to ultimately devote myself to math. To people who chose either path -- regrets? hindsight? And of course, to anybody -- opinions on this issue?

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I'm not sure this is the best forum for this question, but the advice I always give to undergraduates is to not specialize too early (there is plenty of time in grad school to specialize; don't waste the one time in your life when this isn't necessary!) and to take plenty of non-math courses in addition to the math courses. The latter is especially important if you aren't certain about what you want to do! –  Andy Putman Jun 24 '10 at 1:10
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I'd like to add that taking a few courses in computing, in particular programming courses including functional programming if you can take such a course, would be immensely useful. Many ideas can be explored and dismissed with a short program. –  supercooldave Jun 24 '10 at 8:17
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I'm afraid I'm having trouble even understanding the worry that taking courses in fields like philosophy, economics, etc. as an undergraduate could put someone at a disadvantage in their future, longterm study of mathematics. As an undergraduate, one can only take so many mathematics courses at a time without spreading oneself too thin and possibly burning out. Of course you should take courses in other areas besides mathematics (hint: try the humanities). Also you should speak to an advisor or professor about this: your position seems quite extreme. –  Pete L. Clark Jun 24 '10 at 9:49
    
I have a post ready to express my own opinion and take on the matter, but I find your stated background confusing. "I have enjoyed representation theory, algebraic number theory, and complex analysis" seems to indicate you've already mastered and taken courses in the foundational undergraduate curriculum in analysis (baby rudin), algebra (Artin) and topology (munkres). Is this the case? In general, self-study is a dangerous pursuit as an undergrad viewed through the eyes of grad admissions. –  Justin Curry Jun 24 '10 at 11:14
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For future reference, cs.lg.learning is for the sub-field of computer science called machine learning. It has nothing to do with this question. –  Willie Wong Jul 20 '10 at 20:35

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I think you should choose classes based on who is teaching them. If you go on to become a grad student you're going to learn all this material eventually anyway, and as you know more math you can learn more math quickly. So the reason to take one class instead of another class is because it has a teacher who you learn well from!

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@Noah: But, especially in a larger department, a first or second year undergraduate probably will not be familiar with many of the faculty members. (For instance, at University of Chicago, we had many year-long sequences. Excepting those, there were only a couple of professors that I had more than once.) Taking word of mouth advice from other undergraduates is not recommended. I would feel more comfortable advising a student to take core classes (algebra, analysis, geometry/topology) before moving on to more specialized classes or graduate classes. –  Pete L. Clark Jul 21 '10 at 4:45
    
Good point Pete. At Harvard we had a "shopping period" where you could try different classes for the first week. I'd forgotten that at many schools you have to decide before the semester and can't easily switch. –  Noah Snyder Jul 21 '10 at 5:36
    
Pete has made an important point. The answer to this question is related to where you study. It didn't occur to me that you would have to choose courses before attending them. Also, I attended courses I wasn't officially enrolled in, which might not be allowed everywhere. Still I like a lot Noah's advice. And one can always ask one's fellow students who the most recommended teachers are. –  Barbara Jul 21 '10 at 5:52
    
You can do the 'shopping' for the next year. Like Pete Clark, I'm against taking advice from fellow undergrads, since I found that my preferences were radically different from those of the majority. –  rgrig Jul 21 '10 at 9:19

My best advice you any beginning student:

1) Learn as much as you can, as broadly as you can. Take as many hard core mathematics courses-especially proof-oriented ones-as you can. "As you can" is the most important part of this statement here,because mathematics takes a lot of time and discipline to learn.As a result,there are very real time constraints on what you can learn at one time in one semester.You also want to sample many diverse fields where mathematics has a role,which is just about everything. You want to be as well-rounded and flexible as possible in your base knowledge because that's what ultimately becomes your foundation as a researcher.

2) Know thyself and plan accordingly. It's a lot better to take only 3 hard core mathematics courses in one semester and get B's and above in them then to try and take 8 courses and barely squeak by. Maybe you're the type that can go days at a time without sleep running on caffiene and/or....other chemical assistance and fear without landing in a hospital and ace 9 courses.Even if you are of that priviledged few-it'll catch up with you sooner or later,trust me. I've buried a few friends that it caught up with just as they finished thier PHDs at Stanford or Oxford.It's better to do less and do it better for not only your grades,but your health. Quality not quantity. If you've got health problems and personal issues that will get in the way of a total commitment to your studies and undermine your performance-as was the case for me-you should consider taking a few semesters off to take care of that. Otherwise,you could be a very sorry student later. Trust me.

3) Grades Matter. I know,you're probably like,duh? But this really hits on your question about the comfort zone-because throwing yourself outside it to prove yourself a badass is risking your GPA. Graduate level mathematics is serious buisness. Indeed,a number of such classes will be Moore method type courses where you're basically on your own and will have to prove basically all results yourself and a good portion you'll be graded on. And if you do poorly-it could yank your GPA down destructively. Getting into as prestigious a graduate school as you can manage determines how successful you'll be early in your career and whether or not you'll be banished to obscurity in some community college. My point is you don't want to risk that being a badass,it's not worth it.I would strongly advise you audit such courses at first.That way,you learn the high-level material and your performance doesn't affect your grades.Once you've gotten your feet wet-by all means,try and take ONE graduate course.Then depending on how it goes-take more.

4) Talk to People. Develop good bonds with willing professors and graduate students.This way,you learn about the field from many other people's perspectives as well as learning about thier experiences. It's also good to hang out with people who enjoy what you do!

5) Start Reading Journals. This is probably something you're not really going to be able to do until your last year before graduate school simply because you don't have enough background. But there are a few journals that are written at a low enough level for undergraduates to understand-like the American Mathematical Monthly. But you definitely should try and get a feel for active,living mathematics,even if most of it goes over your head at first.Go to seminars as often as you can.

Good luck!

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I really hope I know this before I went to Moscow. Or before my undergraduate year starts... –  Kerry Jun 24 '10 at 3:31
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"Banished to obscurity in some community college"? I find that statement a bit offensive. People don't become professors in community colleges because they are failed researchers. It's really a distinct avocation from being a research mathematician -- I have some friends who are faculty at community colleges, and they would probably describe me as a "failed educator" rather than describe themselves as "failed researchers". –  Andy Putman Jun 24 '10 at 3:59
    
@Andy Firstly,please don't be offended-I've been suspended enough. My point is that's a horrifying distinction,Andy.I believe teaching should be just as important.I know many rising young PHD's and grad students who think being horrific teachers is a mark of thier talent & success as mathematicians."Those who can't,teach."Don't your own words betray that underlying-if unfair-perception? Because you're percieved as a "failed educator",you're at Rice-a relatively prestigious private school.Would you trade that for being perceived as a better teacher and perhaps less of a researcher?Honestly? –  Andrew L Jun 24 '10 at 4:35
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@Andy Continued: The sad part of the whole ridiculous culture is that you may in fact be quite a good teacher in addition to your published research-and it doesn't matter. You're on faculty for your output as a researcher-and if that dried up,chances are the university would in a year or so make a change if you're not tenured there. The tragedy isn't that you're a bad teacher.It's that it doesn't mean a damn thing as far as your career advancement goes. Meanwhile,outstanding teachers are turned into paper pushers who never get promoted or tenured. And that's very sad. –  Andrew L Jun 24 '10 at 4:49
    
This is rapidly getting off-topic, so I'm not going to respond. If you want to continue this conversation, my email address is easy to find. –  Andy Putman Jun 24 '10 at 5:04

In my country, future mathematicians usually take only mathematics courses (with a bit of physics and computer science thrown in) from the beginning of their undergraduate studies. Some universities are tougher than others, and students end up learning advanced material pretty early. Some of those students burn out, some have great fun and learn a lot of stuff. On the other hand, a respectable part of working mathematicians comes a longer way, learning something else first (physics, engineering, economics,...).

If you decide to work hard, a good rule to avoid burnout is to check your choice at regular intervals: if you get bad marks and/or need large amounts of coffeine you're doing it wrong. Make sure you always have a social network around you.

I personally spent several years concentrated on mathematics. I had a group of fellow students and we discussed what we were learning in espresso bar patios, during aimless walks and long afternoon teas. I learned a lot and enjoyed it. You should find out what fits you.

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In my opinion the undergraduate years are good for two things: 1) Developing your abilities to do mathematics with your bare hands and 2) exploring as many different topics (in or out of math) as possible. For the former, make sure you don't shortchange all the standard core courses in analysis, algebra, topology. I would recommend combinatorics and probability, too. In principle, you should be able to learn a lot of this on your own by doing all the problems in a book, but it is also rather important to get feedback from an instructor, both to check your work and to make sure you are presenting your work clearly. Beyond that, I would advise avoiding taking too many specialized graduate math courses. Instead, take all those non-math courses you are interested in. You are going to be totally immersed in math while you're a graduate student, so your undergraduate years are your last chance to explore non-math topics.

ADDED (and inspired by Willie's comment below): And, most importantly, don't worry about any of this too much. It's not as if you have only one chance to do it right. Even if you do it all wrong, you'll learn a lot from your mistakes. It can be argued that those of us who tried too hard to plan carefully and avoid mistakes missed something important.

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+1. It is better to learn the few classes you do take well, then to venture too deep or too broad as an undergraduate. I've always regretted trying to do five years worth of classes in my four years of undergrad time (I paced myself to try an cram in both the specialized graduate classes and non-maths courses). Sure I made the grades and it looked good on paper, but my fundamentals in anything besides Riemannian geometry and analysis of PDEs are shaky as a result, and I had to work really hard for quals in graduate school. –  Willie Wong Jul 20 '10 at 21:19
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Willie, but it seems like whatever you did has worked pretty well, no? You're doing pretty well, as far as I can tell. –  Deane Yang Jul 20 '10 at 21:28
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There is one sense in which you DO have only one chance to do it right: putting together a strong profile for grad admissions. I'm sure there are many who need the advice you have given. But speaking as someone whose grad admissions did not go according to plan (despite what was considered a very strong profile at my undergrad institution), I wish someone had given me different advice when I was a freshman/sophomore. I wish someone had laid out a very ambitious study/course plan for me, told me which profs to get to know, and so on. –  Mike Benfield Jul 21 '10 at 2:57
    
I wish my father's cancer hadn't begun it's agonizingly slow downward spiral towards his grave just as I'd discovered my talent for mathematics and dragged me down with it.But life is horribly unfair sometimes. Which is why the best advice I can give any student is never-EVER-delay until tomorrow what you can do today. –  Andrew L Jul 21 '10 at 3:09
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Mike and Andrew, you two are of course completely right. But I would encourage you two to try to make the best of it. I know too many mathematicians who did not graduate from top schools but hung in there and are now quite successful and clearly better than others (like me) who graduated from better schools. But you do need to be flexible, thick-skinned, and relentless. –  Deane Yang Jul 21 '10 at 3:22

General advice: set a goal, make a plan, follow the plan as far as is reasonable/comfortable/non self-destructive, take notes regarding your progress along the way, re-evaluate both your progress and the sensibility of the goal along the way, take time off to do the moral equivalent of smell the roses. Repeat for as long as feasible and for as long as you have joy of life.

I could talk about taking time to attend lectures of interest in other fields, go to teas or gatherings in other departments, and have you set yourself the task of allocating time between some balance of in-depth focus on a few subjects with making a fleeting acquaintance of a wide array of other subjects that may or may not inspire you to change subjects or adopt a different view of these subjects. I'll instead say that I should have attended more special lectures in subjects outside of mathematics, so that I might have a better appreciation for different perspectives.

I could talk about how spending a lot of time concerned about abstractions and reading books can lead to a contemplative life, possibly with opportunties to engage in discussions that stimulate and excite the mind, and the thrill that occurs when you embrace a new perspective and new vistas of intellect are revealed. I could add the health issues that might arise from an excess of such behaviour (asthma, less robust immunity, allergies, less stamina from insufficient exercise, a poor body-mass index). I will instead say that through trial and error, you will adjust your life so that you find an appropriate balance of physical, social, and mental activities.

Something that might be worth considering: volunteerism. If you can donate some of your time toward your church or social group, you may find a way of sharing your math with them. It may also help you decide what is important. Then you can set goals, make plans, follow the plans ...

Gerhard "Ask Me About System Design" Paseman, 2010.07.21

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Why are you taking courses? Don't take courses if you don't need it.

Some psychologists match children development to human history. We have a period in which we play with dirt, a period of wars and fights, periods of obscurantism, renaissance,... You can let yourself be guided by history (math history) to study math. The most important to study first are those topics appearing first in history, the classics. Take this in a broad sense, after learning about the problem of squaring the circle you can read the proofs of the irrationality of Pi right away without waiting a proportional time to the one human kind waited to know them.

Don't be too eager about the "hard core" courses. Most of the time, what is hard core about them is an overwhelming number of definitions to learn. eg. Much more useful than an advanced algebra course, in which you learn the (should I least some) huuuge number of definitions that they will give you, is to solve the same number of high school problems in algebra. If you let the definitions come in some osmotic-historical-like way that will be enough and you get a better grasp of them than after a year of being drowned with a list of definitions and theorems that most of them are exercises (and most of them exercises simpler than the ones you would be solving if using the time in a different way). A key point is that what is important is not "what" but "how". It doesn't matter if you run of swim, what is important is to either run of swim a lot and with the finest technique, to keep the muscles trained. It is the same, with math.

Courses serve as orientation and motivation. They tell you what is important (if it is being taught in the course it should be important then) in the area and motivates you to solve problems (they give you homework) but, there are alternatives to courses to find orientation and motivation. Reading courses. Many of your professor would be willing to point to some sections in some books and tell you some names of theorems and concepts that are important and from that you got all the orientation that a course can give you. Join two more friends take a book recommended (maybe by some professor), a book with lots of problems and sit down with a fork and a knife and eat it like a gourmet pizza, solve each and every single problem.

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Many (a few? most?) people learn better in a course. –  Barbara Jul 21 '10 at 5:41
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Indeed. "Don't take courses if you don't need it" is, strictly speaking, sound advice, but how are you supposed to know whether you need a course or not? I am starting to appreciate that my own feelings on this issue are relatively conservative, but: I firmly believe that most undergraduates will not make as good choices as to which courses to take or skip as a faculty adviser will make for them (or even the standard curriculum of undergrad analysis followed by undergrad algebra)... –  Pete L. Clark Jul 21 '10 at 6:49
    
@Pete L.Clark Sadly,such "advisers" are encouraged in this behavior by the fact that they can steer these talents into publishing sooner and that in turn pads thier reputations as the mathematician who "discovered" so and so and was thier "advisor."Of course,for every superstudent that survives such a charge to the top,there are 50 students who wash out who could have been solid mathematicians with the right amount of foundational training and a slower pace. –  Andrew L Jul 21 '10 at 7:40
    
@Pete continued: It also leads to a graduating class that lacks fundamentals at the top schools,where they'll need them the most.I knew a student once at Stanford who was publishing algebra papers in major journals at 20-but never heard of generalized convergence with filters in topological spaces,the Cantor set or symplectic geometry.It's a little alarming and makes you wonder what kind of training is actually going on there. –  Andrew L Jul 21 '10 at 7:43
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Andrew L took my second comment in a direction which made me uncomfortable, so I decided to simply delete my comment. All I had meant to say was that to my taste, some undergraduate programs would benefit from faculty being more vigilant in saying "no" and "take the undergrad course in X before you take the grad course in X" to their undergrad math majors. This is a matter of taste, and the faculty at departments (like Harvard) who allow students to register for almost any course they want to certainly know what they are doing and have no ulterior motives: they just feel differently than I. –  Pete L. Clark Jul 21 '10 at 8:59

I have a small advice: you need to be really good with the basics, that is, the first year of math courses should be crystal clear.

The best way to repeat and learn these courses is to teach, so I suggest you to do some private tutoring or being a TA. Trying to explain concepts for other people will really enlighten you on the subject.

This has helped me greatly, since all further courses really depends on the basic mathematics. A good foundation is everything!

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