[Undergrad Math Major Advice] What should be my next step?

[PieceOfPi]

Hi all,

I have just finished my second year at my university as a math major, and I am wondering what math classes to take next year. I will list my preferences and options so that you can give me some kind of advice.

Where I am now:

• I mentioned here a few times that I did not do so well in my real analysis class (Rudin's PMA level) last year that I ended up taking it pass/no pass., so this probably means I am still developing my mathematical maturity.
• I just finished my abstract linear algebra course (4-week, intensive summer course, text: Linear Algebra Done Right by Sheldon Axler. We covered about 80-85% of the text.), and while I struggled through it a few times, I really enjoyed the course, and I think I learned a lot from it. I ended up with a B+, which I think is a grade that I deserved, and I think there's still a room for some improvements.
• Besides math, I am currently interested in computer science as well, and planning to take a few upper-division CS courses (e.g. data analysis, algorithms, computational sciences, software engineering, etc.).
• I am not so sure of what I want to do with my degree yet... I am kind of interested in grad school, but that is still uncertain. On the other hand, mathematical modeling in fields of natural/social sciences seems kind of fun, but I have no experience with it yet.

As of now, I am currently interested in taking one pure-math sequence, and one applied-math sequence next year (this option can be changed if you suggest). Here are my options of pure/applied math courses that I can take next year:

Pure Math:

1. Real Analysis. Fall-Winter-Spring. Text: PMA by Rudin. Since I took it last year, and I still have the book with me, I feel like it's reasonable to take this course again. I know the professor well (he's a different one from the last year)--I had him for multivariable calc, and I enjoyed his teaching style quite a bit.
2. Abstract Algebra. Fall-Winter-Spring. Text: Abstract Algebra by Beachy and Blair. Because I enjoyed abstract algebra and I'm interested in CS, I feel like this one might be reasonable too (and quite frankly, this one seems a bit more interesting than analysis). However, I haven't heard much good stuff about the professor who's teaching this course--most of the complaints I have heard are from business calculus students (which isn't very helpful, since...well... what do they know about good math professors?), but I have heard some complaints from linear algebra students (which does scare me a little bit).
3. Topology. Fall-Winter (but many topology students take differential geometry in Spring). Text: Topology by Munkres. Topology sounds interesting, and I have heard a lot of good stuff about the professor who's teaching this class, but I have also heard that I should take topology after either analysis or algebra, since those two are more fundamental than topology for undergraduate math majors.

Applied Math

1. Probability/Statistics. (Fall-Winter-Spring, and the spring term is called "linear regression," where we deal with R-language or MatLab.)
2. Numerical Analysis. (Fall-Winter)
3. Complex Analysis. (Winter-Spring)

The only scheduling conflicts I have are that I cannot take statistics and topology at the same time, and I cannot take abstract algebra and numerical analysis at the same time.

Currently, I am signed up for real analysis and probability/statistics.

I am going to stop here since this post is getting quite long (I tried to give as many information as I could). Please let me know if you have any comment/suggestion/question.

Thanks!

 

[camilus]

Yes I smoke weed, and I've been smoking herb for like 8 years god. And I haven't failed to do exceptional in all my classes, which happen TO be the passions of my life, which are mathematics, physics and science, and philosophy.

Besides, all the BS I had to go though, it isn't the marijuana that's holded me back. In fact, I've done most of my best work almost immediately after getting high. This is an excerpt from another thread in the Academic Advice forum:

You're pretty much where I am at, just that I fked up my life, I am probably going to jail again... for smoking weed... its a fking shame because everyone who knows me knows that I have a gift for mathematics and science. In one year in High school, when I finally got my life on track, I skipped college algebra, trig, and precalc, and went straight to AP calc 1 and 2, as well as simultaniously taking physics I and II, and AP Physics the next semester. Aced every single one.

And its a shame because I had already began doing independant research on my own. I'd go to the Chair of the mathematics department, to speak to the most knowledgeable mathematician in the school, to ask em a simple question in complex analysis. I asked, how can I raise a number to a complex power, like what is 2^i?

The math department was stumped. two days later, not only did I know how to do such calculations, I independently derived the general case for any x raised to the power of a+bi. The math department was shocked.

In all my calculus classes, I was the only spanish kid, quiet, didnt have no friends, because all of the other kids were a bunch of racist white rich kids. They didnt like me much because I was the only kid in class for 2 straight semesters that got over 100% (like 104% final grade.) The next grade closest to mine was like a 94-95.

and what made these rich white kids hate on me, was that I'd openly talk about how I was a hardcore pothead, and I would get over 100% on an exam that I took literally blitzed out of my mind. and I didnt study at all, I smoked herb all day, while these kids studied their a$$es off and I still blew them out of the water. I am a pothead minority in my high school who knew more advanced mathematics than the whole department. Anyways, I went from a straight F student in 9th and 10th grade, got expelled from all Miami-Dade county public schools for getting arrested OUTSIDE of school on a fkin saturday night, and the beeotch a$$ cops went and snitched on me to the school.

Well sorry for getting off topic.

Out of the courses you listed, I suggest, analysis, complex analysis (Im a big fan of analysis, I have a like a natural talent for it), and topology, which I am DYING to take.

It depends also, how hardcore of the pure mathematician you are. I'm a Platonist along the lines of G.H. Hardy, I don't care about "applications", I love number theory for the simple fact that its the most beautiful, original creation of the human spirit, like Whitehead said.

 

[PieceOfPi]

It depends also, how hardcore of the pure mathematician you are. I'm a Platonist along the lines of G.H. Hardy, I don't care about "applications", I love number theory for the simple fact that its the most beautiful, original creation of the human spirit, like Whitehead said.

That's a tough one to answer. Personally, I do care about applications a little bit, and I find it beautiful that some of the mathematical concepts can have applications to the other area of the subject. But of course, like you've said, find some of the theorems and results in mathematics to be beautiful as well. So personally, I want to learn both the theories and applications of mathematics.

 

[camilus]

My Holy Bible is basically 'THE BOOK' of which Paul Erdos spoke about. A book that God always carried arround him containing all the most beautiful, creative, and interesting and deep theorems.

Thats my holy grail, if hypothetically* that BOOK existed, I would honestly trade my life for a full understanding of all the most beautiful, creative, original, interesting and deep theorems ever. These are my life's passions, without them, my life would be almost meaningless... I would still have my family though, that the other most important aspect of my life.

 

[rodigee]

Why do you have complex analysis in the "Applied" grouping?

Anyways, if you already took Real analysis I don't see much point in taking it again. You're going to want to have seen Real, Complex, Algebra, and Topology before you graduate so I recommend you go for two (or three :P) of those.

Are you interested in taking any grad courses before you graduate? If so, then you could base your selection of sequences for next year on what grad sequences you want to take during your senior year.

 

[VeeEight]

Some schools have an 'applied' version of complex analysis which is geared towards physics, engineering, and applied math undergrads.

From what you have listed I would suggest Abstract Algebra and Topology. Abstract Algebra is essential for a math student and serves as a foundation for later, also important, Algebra courses like Group Theory, Ring Theory, and Field Theory. Topology is usually taken after Real Analysis (or Metric Space Analysis) and is important if you want to take courses in Algebraic Topology, Differential Geometry, etc.

Numerical Analysis is a good course to take but it is more beneficial for an applied math student rather than a pure math student.

Complex Analysis is a good course also. It is similar to Real Analysis and is used later on in Functional Analysis.

Other good undergrad math courses are Differential Equations, Elementary Number Theory and Combinatorial Math.

 

[PieceOfPi]

Why do you have complex analysis in the "Applied" grouping?

At my school, the class is rather called "Functions of Complex Variables," and I heard it's more computations than proofs so that physics majors can take it.

Anyways, if you already took Real analysis I don't see much point in taking it again.

I forgot to mention; I took real analysis Pass/No Pass last year, and that's why I want to take it again so that I can get a letter grade.

You're going to want to have seen Real, Complex, Algebra, and Topology before you graduate so I recommend you go for two (or three :P) of those.

I would agree with you if I am going for pure math. However, I am not so sure about that yet... currently I am looking for both pure and applied to see which one I like better. In the case if I picked applied, would those courses still be helpful? (Either it's helpful or not, I am interested in taking those courses, though.)

Are you interested in taking any grad courses before you graduate? If so, then you could base your selection of sequences for next year on what grad sequences you want to take during your senior year.

As of now, I don't think I have enough talent/knowledge to even think about taking a grad course.

From what you have listed I would suggest Abstract Algebra and Topology. Abstract Algebra is essential for a math student and serves as a foundation for later, also important, Algebra courses like Group Theory, Ring Theory, and Field Theory. Topology is usually taken after Real Analysis (or Metric Space Analysis) and is important if you want to take courses in Algebraic Topology, Differential Geometry, etc.

I am interested in abstract algebra as well--I am just scared about the professor who's teaching that course next year. But who knows--people's opinions are subjective, so he might not be as bad as people say. I think I will attend the first few days of the class to see what he's like.

And as I have mentioned above, I haven't completed real analysis yet... I only took the first quarter for pass/no pass, and decided not to take the later sequences. So I guess I should take topology after real analysis?

Numerical Analysis is a good course to take but it is more beneficial for an applied math student rather than a pure math student.

I actually don't know if I am a pure math student or an applied math student; I just know that I am a math student. This course might be good since I am also looking for something in computer science.

Complex Analysis is a good course also. It is similar to Real Analysis and is used later on in Functional Analysis.

Other good undergrad math courses are Differential Equations, Elementary Number Theory and Combinatorial Math.

I'll consider complex analysis in winter term.

I have already taken elementary number theory and combinatorics (and I liked both of those!). I took the basic differential equations (like the one you take after single-var calc), but I can't take the upper-division diff eq (ODE, PDE, and Fourier Analysis)this year due to scheduling conflicts. I might take that next year, though.

Thanks for your comments!

 

[rodigee]

I would agree with you if I am going for pure math. However, I am not so sure about that yet... currently I am looking for both pure and applied to see which one I like better. In the case if I picked applied, would those courses still be helpful? (Either it's helpful or not, I am interested in taking those courses, though.)

I don't know if they'd be helpful because I don't know anything about applied math :P. You should check out the websites of some applied math grad programs and see what they expect you to know.

And as I have mentioned above, I haven't completed real analysis yet... I only took the first quarter for pass/no pass, and decided not to take the later sequences. So I guess I should take topology after real analysis?

Ah so that's why you're looking at doing analysis again. By all means, go for it. However, I don't think you need to take analysis before topology. I didn't and was fine.

I have already taken elementary number theory and combinatorics (and I liked both of those!). I took the basic differential equations (like the one you take after single-var calc), but I can't take the upper-division diff eq (ODE, PDE, and Fourier Analysis)this year due to scheduling conflicts. I might take that next year, though.

If you're interested in applied math, definitely take a rigorous ODE and PDE courses. I know I said earlier that I don't know anything about applied math. However, I do know that!

 

[crystalentity]

First, talk to your advisor and your professors. Your basic analysis, algebra, and topology courses are somewhat complementary, but all fundamental and valuable. I would recommend taking a one year course in both real analysis and algebra, in any order. I agree that you must retake analysis, especially if you have any plans for grad school. I would probably take topology after analysis, but since you've at least passed analysis, there's no reason not to take topology now. The applied stuff, especially probability, is useful, although I wouldn't take more than a semester unless you were really interested in it. The applied mathematics majors are called physics, engineering, etc. I would consider any undergraduate math degree a pure math degree, and would go with a double-major with an actual applied math area such as CS or economics if you are interested in applications in addition to math.

I hope this helps.

 

[qspeechc]

Let me just add that I hated every moment of numerical analysis that I did. I enjoyed the elementary statistics course I took. That's all. :)

 

[PieceOfPi]

If you're interested in applied math, definitely take a rigorous ODE and PDE courses. I know I said earlier that I don't know anything about applied math. However, I do know that!

Thanks. I just realized that I can still take the first term of rigorous ODE in the fall quarter (topics covered: System of first order linear DE, existence and uniqueness, stability, Liapunov's second method, periodic solutions, chaos, etc), so I might consider that.

First, talk to your advisor and your professors. Your basic analysis, algebra, and topology courses are somewhat complementary, but all fundamental and valuable. I would recommend taking a one year course in both real analysis and algebra, in any order. I agree that you must retake analysis, especially if you have any plans for grad school. I would probably take topology after analysis, but since you've at least passed analysis, there's no reason not to take topology now.

Yes, I'd certainly like to take all of those, and I think I am a little more confident about taking pure math courses. I think I will attend the first few lectures of both analysis and algebra in fall to see which one I'd like to stay (of course, if I liked both of them, I will consider taking both of them). I won't be able to take topology unless I drop probability/stat, but that can be an option for next year.

The applied stuff, especially probability, is useful, although I wouldn't take more than a semester unless you were really interested in it.

At my university, the first two quarters are really the standard introductory "Probability and Statistics for Math Major" course, and the third quarter is what's rather called "Regression Analysis and Analysis of Variance," where we get to use statistical software to do some projects. So yes, I think I will take the last quarter only if I really enjoyed the topic, but I think the first two quarters are pretty important.

The applied mathematics majors are called physics, engineering, etc. I would consider any undergraduate math degree a pure math degree, and would go with a double-major with an actual applied math area such as CS or economics if you are interested in applications in addition to math.

That kind of explains why my school has so many pure math professors and only two or three applied ones. While there is "Applied Math" track to complete B.S. (or B.A.) in mathematics, I used to think it's important for any math major to know the theory well before one gets to the application. I used the terms "used to" because that's why I took analysis last year, and got a little disappointed with myself after doing less than mediocre job in that class and after seeing some of the really bright students in that class, and I started to think maybe pure math is not for me :P But then I took abstract linear algebra this term, and I am starting to think that I can learn pure math if I work harder.

As far as the application goes, my main interest now is CS, so I will take a few more courses. If I liked it, I think I will complete minor since I doubt I can complete the full major if I'm going to take all analysis, algebra, and topology (unless, of course, if I decided to go for an extra year). Economics sounds interesting, but it seems like I need to go through a lot of boring courses (e.g. intro to microeconomics) before I even get to the part where I get to use math. Physics also seems interesting, but after taking a few lab courses, I no longer wanted to become a physics major :P

Let me just add that I hated every moment of numerical analysis that I did. I enjoyed the elementary statistics course I took. That's all. :)

What made you hate every moment of numerical analysis and enjoyed the elementary statistics?