Looking to round out my [math] classes.

[StevenK]

First off I want to apologize if there is a topic about this type of thing already that I missed. I went back through the last 5 pages and didn't see anything that seemed to fit it and am unsure of what I would need to search for to look through that function.

Past that, I am a Mathematics major at a small public college in Nebraska (CSC) and I wanted to ask about what I should take that isn't offered at my college to better my understanding of the field and to hopefully be prepared to enter a masters program elsewhere.

I have taken...
Calc 1, 2, 3
History and Foundations of Mathematics (light intro to logic and set theory)
Number Theory (included basic principles of cryptology)

I am taking:
Differential equations
College Geometry (geared towards high school math teachers)

I will take:
Probability and Statistics
Elementary Linear Algebra
Modern Algebra
Introductory Analysis
Operations Research (game theory and the like, statistics subset)
and possibly a statistics course on sampling techniques as it seems potentially useful.

This is every single math class that is not teaching certification oriented that the college offers except for theory of statistics (which I may pick up for the fun of it anyway).

Also I am currently talking with one of my teachers about doing independent graduate study in partial DE's which isn't a standard offering by the school but which he specifically studied.

Is there anything I should be looking to take elsewhere to fill in gaps here? I know that there are always 'more' things to study in Math but I am not in a position to be aware of what I should or should not have moving forward.

Anything that I could not take through some sort of online course (which I have done for math classes before due to my work (number theory and calc 3)) would have to wait until after I finish my degree. But considering the fact that I am 30 years old and have full-time employment there is no pressing need for me to jump from undergrad to graduate work if I need to take a semester off to self study. My only concern for that is in it not appearing in a transcript when I apply to another school.

And from out in the middle of a corn field in Nebraska... Thanks.
(Well in the middle of town, but town is surrounded by corn!)

 

[Robert1986]

Well, I can list some topics, but other people will have more to say (the following is in roughly order of importance with the most important at the top; but lots of people will probably disagree and I might agree with their disagreements once I see them):
complex analysis
more analysis an algebra
topology
combinatorics
differential geometry

 

[jgens]

Your current program has absolutely no topology in any of its delicious flavors. While admittance to a masters program is certainly not going to be contingent upon taking topology, it is one of the pillars of modern mathematics. So if you want to understand the different directions the field has taken, then learning some topology is an important step.

 

[StevenK]

Thanks to both of you for the responses, they are exactly what I was looking for. If the 'top three' would be complex analysis, a further study in that field, and topology I can definitely see about getting those bases covered. To be perfectly honest if I hadn't had to write a paper on Mary Cartwright (dealt heavily with topology) I would have simply assumed that all 'major' subsets were being dealt with by the school, small though it may be. I'm glad that I had the opportunity to notice otherwise.

 

[jgens]

I would have simply assumed that all 'major' subsets were being dealt with by the school

This seems to be a general trend actually. I am currently at a major research university and while algebra and analysis are required topics for graduation, the courses in topology are nonessential electives. Part of the problem is that topology borrows heavily from both algebra and analysis, so without a decent grounding in both of those subjects, learning any topology beyond the very basics is difficult.

 

[Hercuflea]

Sounds like you attend a relatively small math department. Mine was like that too when I started, but it has grown significantly in the last few years, now that I am a senior. The number theory course you took, how proof heavy was it, was it a graduate course?

Anyways, definitely take modern algebra, but seeing that you seem to like applied math (or maybe that is the only thing you have had the opportunity to take?) such as partial differential equations and operations research, you might not like modern algebra which is very abstract and not very applicable in the real world (at least not in most physical applications).

Anyways the main courses that are a solid preparation for a mathematics graduate degree are Real Analysis, Complex Analysis, Modern/Abstract algebra, an advanced course in linear algebra, and topology. If you can, try and talk to one of your professors about proposing new courses in these areas, that maybe they would teach only once a year or two. Some of my professors told me that if I could find enough students to commit to a class, they would make one and put it on the schedule, but that is the hard part.

Is there any other college near your home which have a larger math department?

 

[IGU]

I am a Mathematics major at a small public college in Nebraska (CSC) and I wanted to ask about what I should take that isn't offered at my college to better my understanding of the field and to hopefully be prepared to enter a masters program elsewhere.

The easiest way I know of to answer this sort of question is to fire up your web browser and take a look at the requirements for a math major at such schools as Stanford, Princeton, Harvard, Berkeley, etc. Also look at the differences for an Honors designation. This tells you what sorts of classes your competition at top colleges will have taken.

Notice that none of them have anything like "College Geometry" in their lists. Ditch that. You want to be taking differential geometry after you have a bit of analysis and topology under your belt. Sure, take an online class or two in things your school doesn't have, but get a professor to oversee your work, grade your exams, and see that you get credit toward your degree. Then it will appear on your transcript, maybe as a "reading" or "independent study" class. This is not uncommon. I know a guy who did this at Caltech, as they didn't have a class on something he wanted to learn. He used OCW, but there are many other resources available now.

 

[StevenK]

The number theory course you took, how proof heavy was it, was it a graduate course?

Is there any other college near your home which have a larger math department?

The class was not a graduate course and spent a good amount of time doing what I would consider applied proofs (use induction to prove x => z, or similarly rigorous methods). Also where I am (west central Nebraska) does not have much in the way of options. It would be well over an hour drive each way to attend any other four year school. In the time it will take me to finish my degree my work may have me at another location, but I am not willing to wait to see for a variety of reasons.

Modern Algebra is one of the classes I will be taking, and the only reason I am taking so much applied is because that is what the school offers. Partial DE's is an exception to that, but it is either that or something completely outside of the Math department to meet my upper level requirements.

The easiest way I know of to answer this sort of question is to fire up your web browser and take a look at the requirements for a math major at such schools as Stanford, Princeton, Harvard, Berkeley, etc.

Notice that none of them have anything like "College Geometry" in their lists. Ditch that. You want to be taking differential geometry after you have a bit of analysis and topology under your belt. Sure, take an online class or two in things your school doesn't have, but get a professor to oversee your work, grade your exams, and see that you get credit toward your degree.

While I am looking to advance I am not in a position where attending a school like Stanford, Princeton, Harvard is in the realm of possibility. I simply want to be well rounded enough that (insert random, generally ignored Mid-West college here) doesn't have me needing to take 5 more undergrad classes to have enough of a background to begin graduate work.

Also, an axiomatic study of euclidian (with some non-euclidian) geometry really doesn't seem very out of place considering the amount of impact the Elements has had on that field. Is it basic in a sense? Yes. But it still offers valuable lessons on rigor and what can be accomplished (or kept from being accomplishable) depending on the starting points of our assumptions.

But past that, the college has expressed a willingness to let me take anything I want from other institutions so long as I am meeting their own requirements. It may be a small school but I have had excellent luck in working with them on a number of levels.