EDIT: As Vladimir Nesov points out below, don’t just study what I recommend; this is a list of things that you might look up outside of classes to help stay motivated while doing a degree in mathematics, more than a list of things you should study to learn mathematics outside of school.
Also refined point (7).
I am a mathematics graduate student; I currently focus on number theory and arithmetic geometry. So here are a few areas I’d recommend, coming from different goal structures I have:
1) if you are interested in learning things that are really cool and beautiful I would recommend elementary number theory, for example from Hatcher’s visual approach. This doesn’t require much heavy grounding and is absolutely awesome and has neat pictures. If you want to continue on this path, p-adic numbers are the place to go.
2) if you are interested in studying advanced mathematics, I’d recommend studying representation theory and category theory; these seem to have lots of applications in almost every area of mathematics, including algebra, number theory, mechanics, geometry, and topology. Maybe less in analysis or logic, although more logicians’ perspectives on category theory seems valuable to me. Also complex analysis.
3) If you are interested in abstract concepts that feel like they have universal applicability (I don’t know how much the metaphors I draw from these actually help me but I draw them almost constantly): linear algebra, group theory, and basic real analysis. Symmetry and distance are every day concepts; seeing them mathematically derived was very cool to me.
4) if you want to make lots of money, I think calculus and dynamical systems lead most directly into financial modeling; I’m not really sure.
5) if you want to do interesting research with real life applications, you might be better sticking to statistics and probability theory; although dynamical systems have their applications in game theory the impression that I get is that the difficulty mostly comes from differential equations and computation complexity, not from mathematical insights.
6) You probably shouldn’t study algebraic geometry. I do a little, but it is filled with technical definitions and complex terminology and it has a reputation for taking people a very long time to be able to understand at all. If you want an intellectual challenge for yourself maybe it’s appropriate, but if you want to learn a field quickly and use the insights it is probably more trouble than it is worth, at least until some amazing new text book comes out on it which I doubt may ever happen. It is too late for me, save yourself!
7) if you are in school at a university, I’d suggest looking up math professors on ratemyprofessors, the ratings aren’t perfect but it does look like they correlate well with my experiences with my professors. Requiring slightly more effort but giving much better information would be asking other math majors or TAs about different professors’ teaching styles. And then, just take courses from good professors. This is probably worthwhile in any subject; better professors are going to mean more than good classes. Taking a class with a good professor means you will probably enjoy the class, taking it with a bad professor means you probably won’t. I don’t think this is the context of your question but it is probably relevant to others asking similar questions.
I don’t believe specialization on the level you imply is sustainable. You’d get lost even on upper undergraduate level if all topics outside most of your recommended sets are completely left out. The mathematical maturity that allows you to imagine limiting your study to just a few topics came from having studied the others.
I probably didn’t make this clear, but I do agree with you. If you want to study mathematics you need to study lots of areas which is why they have general requirements like that for getting degrees in mathematics.
But if you are an undergraduate looking for a particular subject to get you in the “mathematics groove” I think (1) is a good recommendation for independent study alongside classes, (3) is really a way to lay a foundation for mathematical maturity; certainly a class on proofs and logic would be necessary before almost any of my recommendations.
Again: I didn’t mean my list to be a list of “only study these things” so much as “if you want more math, these might be good places to look.”
I’d suggest looking up math professors on , the ratings aren’t perfect but it does look like they correlate well with my experiences with my professors.
My favorite professors are all rated 4.5 or higher, and my least favorites are rated 3.0 or lower; ones I’d rate in between are commonly rated higher than my favorites, but it contains substantially greater than 0 information relative to my school. Apparently YMMV!
I would guess talking to TAs or other same-major students is going to get most LWers a more useful perspective than RMP, but website reviews are still better than nothing.
Only detailed reviews. If you don’t know what the criteria are, they can easily be worse—counting as negative a factor you would consider positive, for example.
I would suggest that if you know nothing about the rating system, it is still likely to positively correlate because of universal factors like speaking clearly. In the case of RMP, I’d suggest that you’d expect an even better correlation because easiness and attractiveness are asked about separately. It’s still possible for this not to work out because of what you suggest, but it seems less likely to me on average.
I think RMP is more for undergrads who want to coast through their degree with the least amount of effort.
RMP is also very useful if you want to sign-up for classes taught by attractive professors. I like to have something nice to look at while not doing course-work, lol!
I agree with you, I was just trying to help magfrump.
EDIT: As Vladimir Nesov points out below, don’t just study what I recommend; this is a list of things that you might look up outside of classes to help stay motivated while doing a degree in mathematics, more than a list of things you should study to learn mathematics outside of school.
Also refined point (7).
I am a mathematics graduate student; I currently focus on number theory and arithmetic geometry. So here are a few areas I’d recommend, coming from different goal structures I have:
1) if you are interested in learning things that are really cool and beautiful I would recommend elementary number theory, for example from Hatcher’s visual approach. This doesn’t require much heavy grounding and is absolutely awesome and has neat pictures. If you want to continue on this path, p-adic numbers are the place to go.
2) if you are interested in studying advanced mathematics, I’d recommend studying representation theory and category theory; these seem to have lots of applications in almost every area of mathematics, including algebra, number theory, mechanics, geometry, and topology. Maybe less in analysis or logic, although more logicians’ perspectives on category theory seems valuable to me. Also complex analysis.
3) If you are interested in abstract concepts that feel like they have universal applicability (I don’t know how much the metaphors I draw from these actually help me but I draw them almost constantly): linear algebra, group theory, and basic real analysis. Symmetry and distance are every day concepts; seeing them mathematically derived was very cool to me.
4) if you want to make lots of money, I think calculus and dynamical systems lead most directly into financial modeling; I’m not really sure.
5) if you want to do interesting research with real life applications, you might be better sticking to statistics and probability theory; although dynamical systems have their applications in game theory the impression that I get is that the difficulty mostly comes from differential equations and computation complexity, not from mathematical insights.
6) You probably shouldn’t study algebraic geometry. I do a little, but it is filled with technical definitions and complex terminology and it has a reputation for taking people a very long time to be able to understand at all. If you want an intellectual challenge for yourself maybe it’s appropriate, but if you want to learn a field quickly and use the insights it is probably more trouble than it is worth, at least until some amazing new text book comes out on it which I doubt may ever happen. It is too late for me, save yourself!
7) if you are in school at a university, I’d suggest looking up math professors on ratemyprofessors, the ratings aren’t perfect but it does look like they correlate well with my experiences with my professors. Requiring slightly more effort but giving much better information would be asking other math majors or TAs about different professors’ teaching styles. And then, just take courses from good professors. This is probably worthwhile in any subject; better professors are going to mean more than good classes. Taking a class with a good professor means you will probably enjoy the class, taking it with a bad professor means you probably won’t. I don’t think this is the context of your question but it is probably relevant to others asking similar questions.
I don’t believe specialization on the level you imply is sustainable. You’d get lost even on upper undergraduate level if all topics outside most of your recommended sets are completely left out. The mathematical maturity that allows you to imagine limiting your study to just a few topics came from having studied the others.
I probably didn’t make this clear, but I do agree with you. If you want to study mathematics you need to study lots of areas which is why they have general requirements like that for getting degrees in mathematics.
But if you are an undergraduate looking for a particular subject to get you in the “mathematics groove” I think (1) is a good recommendation for independent study alongside classes, (3) is really a way to lay a foundation for mathematical maturity; certainly a class on proofs and logic would be necessary before almost any of my recommendations.
Again: I didn’t mean my list to be a list of “only study these things” so much as “if you want more math, these might be good places to look.”
Did you mean to link to Rate My Professor?
On a lark I looked at my favorite professors (that is, the ones I felt taught me the most) and all of them have ratings below 3.0.
I think RMP is more for undergrads who want to coast through their degree with the least amount of effort. (This makes a certain amount of sense.)
My favorite professors are all rated 4.5 or higher, and my least favorites are rated 3.0 or lower; ones I’d rate in between are commonly rated higher than my favorites, but it contains substantially greater than 0 information relative to my school. Apparently YMMV!
I would guess talking to TAs or other same-major students is going to get most LWers a more useful perspective than RMP, but website reviews are still better than nothing.
Only detailed reviews. If you don’t know what the criteria are, they can easily be worse—counting as negative a factor you would consider positive, for example.
I would suggest that if you know nothing about the rating system, it is still likely to positively correlate because of universal factors like speaking clearly. In the case of RMP, I’d suggest that you’d expect an even better correlation because easiness and attractiveness are asked about separately. It’s still possible for this not to work out because of what you suggest, but it seems less likely to me on average.
RMP is also very useful if you want to sign-up for classes taught by attractive professors. I like to have something nice to look at while not doing course-work, lol!
I agree with you, I was just trying to help magfrump.
I did; fixing the link.