# What's a good order to take upper division math classes?

Good afternoon everyone, I'll be starting my trek into upper division mathematics next year (applied math major) and since the perquisites are essentially just multivariable calculus and linear differential equations, I can pretty much take them in any order. However, I know that I should probably take Linear Algebra first and plan on doing that, but after that, I'm at a loss.
Here are the classes I plan on taking (also if you have suggestions to switch them to something else, input would be great):

Linear Algebra
Real Analysis
Complex Analysis
Mathematical Modeling
Linear/Nonlinear Systems of DE
ODEs
PDEs
Applied Numerical Methods
Optimization
Probability Theory and Stochastic Process
Combinatorics
Methods of Applied Mathematics
Algebra for Applications
Fourier Analysis

Also, don't wanna ask too many things (and this might not even be in the right forum) but will these courses make me, I suppose, "Marketable" out in the real world? I definitely want to pursue a master's in some sort of math/computer science field, and I wanna make sure these classes are good for getting a job. Sorry if that was weirdly worded

The order you wrote them in seems good.

It's unusual that a university offers applied abstract algebra, I don't mean to be invasive but are you going to a public university in the US? None I have looked at offer such a course.

oh well, I guess I got lucky with the order then heh heh.
Actually I was a little surprised as well, I just found about it. I was trying to find room for taking abstract algebra since I feel the need to take it as a math major (applied or not) and then found the course.

Here's the course description:
117. Algebra for Applications. (4) Lecture, three hours; discussion, one hour. Requisite: course 115A. Not open for credit to students with credit for course 110A. Integers, congruences; fields, applications of finite fields; polynomials; permutations, introduction to groups.

versus the normal abstract algebra route:
110A. Algebra. (4) Lecture, three hours; discussion, one hour. Requisite: course 115A. Not open for credit to students with credit for course 117. Ring of integers, integral domains, fields, polynomial domains, unique factorization. P/NP or letter grading

Furthermore, the page stated that the applications course was the "fast version" of abstract algebra

Are you planning to go to grad school? If so, I would suggest taking a real Algebra sequence (ie Algebra I and Algebra II) and also the whole Real Analysis sequence (I'm guessing there is a RAI and RAII) and also I would suggest taking a class in topology. The reason I say this is because if you go to grad school (even if you just focus on applied math) you will probably have to take Algebra and Analysis and then pass comps. on both of them. Similarly, topology seems to be important in many areas of math. I am starting grad school in math in August, and I didnt have undergrad topology. I'm am working through a book now, but it would be better if I could spend that time going over the new stuff I'll be doing in August.

If you don't plan to go to grad school, I'd still suggest taking a complete Algebra and Analysis sequence. These are topics that a math major should know. But, the topology class in this case might not be as important.

Not the OP but the grad programs I have looked at let you choose what 3 areas you want, algebra is completely avoidable depending on the university.

Not the OP but the grad programs I have looked at let you choose what 3 areas you want, algebra is completely avoidable depending on the university.
Where are you looking? That is kind of surprising to me. I can't imagine a Math Ph.D. not knowing algebra.

I definitely plan on getting a Master's in applied math, but I'm a bit unsure of the PhD mostly because of the job prospects. I mean I don't mind making not so much as a professor, but even getting those jobs are a challenege.
If I take the algebra/analysis series I'd have to get rid of a few of those courses, is there any recommendations on that?
I would think:
Stochastic process
Fourier Analysis or PDEs (but I like the idea of PDEs)
Optimization
Methods of Applied Mathematics (not sure on this)

Well, I'm not expert, but this is what I think I would do.

To add an analysis and algebra class, you need to get rid of two classes you have listed, right? There are some classes you have listed which will probably have lots of overlap:

Applied Numerical Methods
Mathematical Modeling
Methods of Applied Mathematics

You could probably just keep one of these and get a good intro to this subject. Then, when you go to grad school, you can take the grad-level version of these courses and do just fine.

Another to consider cutting is Linear/Nonlinear Systems of DE because you will probably get some of this stuff in ODE and then, like before, you can take them at the grad level. (of course, nothing prevents you from buying the book and reading it with out actually taking the class.)

Also, I would consider dropping the Probability and Stoch class and taking a more general, calc-based intro to stats sort of class.

I think to give yourself a solid undergrad math education that will prepare you for math graduate studies (again, I am no expert, and some of what I will list are courses I didn't take but wish I had) you should take:

Linear Algebra
ODEs
PDEs
Calc-Based Stat
Combo
Analysis I
Analysis II
Algebra I
Algebra II
Complex Analysis
Topology

And then pick 3 other classes to take. Although, as I think about it, you might do fine only taking one algebra class or only taking one analysis class.

Thanks again for the replies.
Robert1986:
My university requires that I pick two of the three two-quarter sequences, which are:
Applied Numerical Methods A&B
Probability Theory A&B (Or their statistics counter parts)
Linear/Non-linear and ODEs
I'm also required to take mathematical modeling for my major, and I -hear- that having stats classes is good for the job market. I also really enjoyed my lower division linear diff. class which is why I wanted to further expand my knowledge on it. In addition, I'm going to get this "specialization in computing" which requires 4 programming classes and then the 2 courses on Applied numerical method. However I do really want to take the algebra series (ours is 3 classes...yay quarters, but I can just take 2 of them).
So I think Robert's list would probably be the most ideal for me and then the other 3 would include numerical methods and the modeling. I guess I just really wanna learn all the math I can and be able to take classes the jobs would like.
Sorry for the wall of text.

I know someone from UCLA that got into a top 10 applied math grad program without taking algebra (or any of 134-136 either), so it's certainly possible. I would also suggest taking the honors sequences that are available. I don't think the order is a big deal for most of those classes.

Even so you'll almost certainly have algebra requirements in the applied math phd program. The grad math department at my school is top 20, and both the pure and applied math students have an algebra requirement.

Algebra takes a bit of time to get used to, so you should learn it now.

"almost certainly" may be a bit strong. Here's what I found when looking at a few applied math programs wrt algebra (not linear)

Required
MIT
Berkeley

Not Required
UCLA
Washington
Caltech
NYU
Harvard (SEAS)
Princeton
Brown
Duke

I kind of "feel the need" to take at least one algebra course, which was why I was leaning towards algebra for applications. I also feel the same for topology, but I'm not sure if that's an absolute necessity as an applied math major. And UCLA would be my number one choice for a Master's (I go there for undergrad), but I'm just a wee bit turned off by the M.A. Probably not a huge deal anyway.
I want to take analysis I in the Winter, but I've heard from many that it is the hardest undergrad math course offered at my school, so I'm a little skeptical about it. I've started reading a free textbook by William Trench, so hopefully that gets me off to a good start. I probably will end up taking the course sometime junior year though.

Well, I am currently at a high-ranked school that is top 20 in applied math, and all of us have to take a comp in Algebra and Analysis.

However, even if algebra is not technically a requirement, this does not mean it isn't a de facto requirement. For example, at CalTech, you have to take a comp in two of Alanysis, Algebra or Topology. However, it is my oppinion that every math Ph.D. should know information from each of these areas. Perhaps you'll know less algbrea than topology or analysis, but you should know enough. Also, if you are doing applied math, then algebra is a MUCH better thing to be proficient at than topology.

Well, I am currently at a high-ranked school that is top 20 in applied math, and all of us have to take a comp in Algebra and Analysis.

However, even if algebra is not technically a requirement, this does not mean it isn't a de facto requirement. For example, at CalTech, you have to take a comp in two of Alanysis, Algebra or Topology. However, it is my oppinion that every math Ph.D. should know information from each of these areas. Perhaps you'll know less algbrea than topology or analysis, but you should know enough. Also, if you are doing applied math, then algebra is a MUCH better thing to be proficient at than topology.
It isn't a defacto requirement. How is algebra more usefull then topology?

It isn't a defacto requirement.
I disagree. I think as a mathematician, one is expected to know a fair amount of algebra analysis, and topology. Now, like I said, you might know some areas more than others, but I find it hard to imagine a mathematician who is not familuar with these areas (again, I am not a mathematician.) Because of this, people should take algebra at the graduate level.

How is algebra more usefull then topology?
Well, I can think of ways that algebra is useful outside of math. For example, if you want to go into some sort of information security type of field, then the stuff from algbra is certainly important. If you want to go into computational type math (as it looks like the poster wants) then you are going to be dealing with a lot of polynomials and algebra is of course useful for that.

Topology, on the other hand, doesn't seem that aplicable to areas other than math - though I could be wrong. And of course I am not suggesting that topology should be ignored. I just think there are some areas of math that modern mathematicians should know.

For example, at CalTech, you have to take a comp in two of Alanysis, Algebra or Topology.
That's for pure math, not applied.