Preparation for an applied math career

[icarus-II]

I'm a current math major and I'm interested pursuing a graduate degree in applied math. My question:

Is it better to take as many pure undergrad math classes (real analysis, abstract algebra, topology, etc.) as possible for preparation? Or should I focus on classes like differential equations and statistics? I've been reading that it's better to take proof based math in order to prove to graduate programs that I can handle the more "rigorous" topics even if I want to get a master's or a PhD in applied math. What do you guys think?

 

[S.G. Janssens]

In the preface of his nice book "Analysis for Applied Mathematics", the late Ward Cheney (University of Texas, and also industry consultant) wrote:

A look at the past would certainly justify my favorite algorithm for creating an applied mathematician: Start with a pure mathematician, and turn him or her loose on real-world problems.

This is a nice summary of my opinion, as well.

As an aside, but perhaps also to underline the above point: I think courses like differential equations and statistics can be taught in many way, and some of them require a good understanding of topology, geometry, and functional analysis (differential equations) or measure theory and functional analysis (statistics).

Most important are your own preferences. You can ask yourself various questions. What is your style? Do you enjoy applying knowledge from "pure" classes to "real-world" problems? Do you already have an idea of what field in applied mathematics you fancy? Browsing SIAM Undergraduate Research Online could be inspiring, but also look further into articles in other journals. Read articles written by authors from different localities. (What is understood to be "applied math" differs between, for example, the US, UK, and countries in continental Europe.) Do you have a prof. or group in mind to work with? Then browse through their publications to see if it is a fit.

In any case, and here I am strongly opinionated, the adjective "applied" should never be used as an excuse for "less rigorous", although I often see it being used for that.

 

[Dr.D]

Well, yes, I like the quote above from Cheney, but ...

I was briefly a pure math major (at UT-Austin), and I took the pure math course in Differential Equations (not from Cheney, but another famous mathematician). We spent an entire semester proving properties of the solution of x'' + K^2 x = 0, although we never got so far as actually solving the ODE. I changed majors the next semester.

 

[S.G. Janssens]

We spent an entire semester proving properties of the solution of x'' + K^2 x = 0, although we never got so far as actually solving the ODE.

Why would you need to solve an equation if you have already proved all its properties? ;-)

A bit more seriously: If I would teach an introductory ODE course to math students (pure or applied, does not matter), I would probably explain that $x''(t) + K^2x(t) = 0$ is a very special case of the linear non-autonomous homogeneous system $u'(t) = A(t)u(t)$ with $A$ a real or complex matrix-valued function. I would discuss inhomogeneous perturbations and variation-of-constants on the one hand, and the autonomous case (constant $A$) and the relation with Jordan normal form on the other hand. Various classes of oscillators could be used as illustration.

If you really spent an entire semester on the harmonic oscillator, then something went wrong, or your prof just really liked harmonic oscillators.

 

[Frabjous]

If you really spent an entire semester on the harmonic oscillator, then something went wrong, or your prof just really liked harmonic oscillators.

I bet they resonated with him.

 

[Dr.D]

If you really spent an entire semester on the harmonic oscillator, then something went wrong, or your prof just really liked harmonic oscillators.

It was Prof. Wall, a close associate with Moore. Moore was my advisor, and when I said I wanted to take Physics II, he said, "My students don't take physics." I took Physics II any way.

@caz is definitely correct!