Math major looking for some course guidance

[altcmdesc]

I'm currently a freshman math major and registration for spring semester has got me thinking about sophomore year already. I'm currently taking a year long honors course which treats multivariable calculus and linear algebra at a theoretical level.

For an idea of what this course is like, we covered:

basic vector/matrix stuff
linear transformations
epsilon-delta definitions of limits and continuity in the multivariable context
basic topology (open, closed, compact sets, bolzano weierstrass theorem)
the derivative as a linear map
derivatives of matrix functions
continuity and differentiability
linear independence, span, bases, subspaces and vector spaces
images and kernels of linear maps
rank-nullity theorem
abstract vector spaces (space of polynomials, matrices, etc)
eigenvectors and eigenvalues
Newton's method

in the first semester. In semester two I believe we discuss manifolds, integration, the generalized stokes theorem and differential forms.

So, my question is, what will I be ready for come next year? I'm thinking about taking an honors algebra course (another two semester long sequence) along with a semester each of differential equations and complex analysis. Would you say that this is an overload, just right, or could I take on more?

Also, should I try to take an intro. topology course before I tackle analysis? I've heard the topology in the course can hurt you if you're not experienced.

Thanks

 

[dans595]

All the things you've mentioned are essential.

I recommend a first course in real analysis before topology, perhaps just because that is the way I did it. I think that topology is more abstract, and I argue that it is more canonical to see concrete cases before abstract cases.

Honors algebra sounds like a very good idea.

Two to four math classes is average/reasonable, depending on what the courses are and on what else you're taking.

Do you have any idea what type of math appeals to you? If not that is perfectly fine, you have to take basic courses for a while, so it will give you time to be exposed to things that particularly interest you.

 

[PowerIso]

So that is 2 math classes per semester for your sophomore year? Sounds about right. As for if you should take complex analysis before or after topology, that really depends. Each school teaches that course differently. At my school, it depended on real analysis. If people at your school say , "hey topology will help you out" then you should listen and take topology.

 

[altcmdesc]

I really enjoyed the linear algebra we covered in my current course much more than the differential multivariable stuff. The proofs we much more clear to me and the subject matter a bit more interesting. It seems to me as if analysis requires clever tricks to prove a lot of theorems.

I also liked the topology we did as well. I really don't think I'd go for anything in applied mathematics or analysis, but we'll see.

I'll have a talk with my current TA about complex analysis and topology, but I think honors algebra next year is a sure thing.

Also, is a combinatorics course really necessary to have? My school offers both enumerative and nonenumerative combinatorics at the upper division level - if combinatorics is something I should be exposed to, which one would be best?

Thanks for your help

 

[dans595]

Combinatorics likely isn't essential, but that also depends on what your goals are with your degree.

If you're trying to gauge what an undergraduate "should" know: Does your program have degree requirements? These are likely a good gauge for "essential undergraduate material," though not fool-proof in general. You can look at various programs to see what the standard undergraduate curriculum is. Typically stuff like: abstract algebra, linear algebra, real and complex analysis, topology, differential geometry in pure math.

I seem to recall that Mathwonk posted something about "So you want to be a mathematician?" or something. It is longer than the Bible, so you'll have to look for the specific information you need, but it likely gives a thorough answer to any question on curricula that you have.

People around here seem to recommend quantum physics to people who liked linear algebra, but I myself have not studied quantum so I can't say for sure.

 

[chiro]

I really enjoyed the linear algebra we covered in my current course much more than the differential multivariable stuff. The proofs we much more clear to me and the subject matter a bit more interesting. It seems to me as if analysis requires clever tricks to prove a lot of theorems.

I also liked the topology we did as well. I really don't think I'd go for anything in applied mathematics or analysis, but we'll see.

I'll have a talk with my current TA about complex analysis and topology, but I think honors algebra next year is a sure thing.

Also, is a combinatorics course really necessary to have? My school offers both enumerative and nonenumerative combinatorics at the upper division level - if combinatorics is something I should be exposed to, which one would be best?

Thanks for your help

Do you like or have you been exposed to discrete math? There's a lot of ripe ground for investigation as far as computational topics are concerned. Since you have had an honours introduction into calculus linear algebra i'll assume you have worked with some elementary concepts of proofs.

If you like this sort of thing you can possibly set yourself for some higher levels of logic like the multivalued logics and the lambda calculus type logics. There's also a lot of applied work out there dealing with logics such as coming up with programs to prove mathematical results and tonnes more (I last looked at this kind of thing a long time ago so please forgive my ignorance).

If you like quantum mechanics and physics there's a tonne of stuff you can get into there. If you study non-commutative geometry, there's a tonne of stuff that is being worked on in relation to physics. Or you could do what Von-Neumann did and work on the area of operator algebras and further that field. Actually what I found more interesting was his work on continuous geometry but that's another story.

Honestly there is that much mathematics out there to keep a person busy for possibly the next thousand years or so which I think is a fantastic thing.

I guess you should ask yourself what you really want to get out of your mathematics learning. If you want to explore a particular area and try and prove theorems then its a safe bet you would speak to your advisers/lecturers on what areas are like and what's happening with them.

Also quite a lot of work is still going on which is trying to take the mathematics we have and evaluating that in a new light whereby we can make more use of it. Since a lot of people don't typically invent new mathematics this is a more common approach which still solves a great deal of problems.

Plus why not take a crack at the millenium problems if you got the time and need some spare cash ;)

 

[altcmdesc]

Just a few more questions:

1. Is taking a course just on Logic (or Set Theory) necessary?
2. Complex Analysis before, after or concurrent with Real Analysis?
3. A theoretical course on ODEs before, after or concurrent with Real Analysis?

 

[dans595]

1. Is taking a course just on Logic (or Set Theory) necessary?

It is a good idea to learn axiomatic set theory if you're a math major. Whether you do it in a course or not is up to you.

2. Complex Analysis before, after or concurrent with Real Analysis?

I don't think it matters unless the specific courses are structured such that one depends on the other (e.g., one may introduce topological spaces, compactness, connectedness, etc, while the other assumes you've taken the first and already are familiar with said concepts). My school does real first, complex second. I think I've seen schools that do it the other way around.

3. A theoretical course on ODEs before, after or concurrent with Real Analysis?

I'd say after.

 

[HallsofIvy]

We don't know your interests or abilities nor have you given the prerequisites for the courses. The best people to give you information on these matters are the faculty- those that are teaching these courses and those that have taught you.

 

[altcmdesc]

Well, I'm not too interested in applied mathematics. I haven't picked out anyone area of mathematics quite yet, although I enjoyed linear algebra more than calculus. Because of that, maybe I can get away with a less theoretical treatment of differential equations?

Maybe you guys could give your undergraduate schedules so I have a vague idea?

I'm just trying to get a very general feel for what people think.