# What pure mathematics foundations should an applied mathematician have?

I'm studying mathematics, with some statistics also, and I've always chosen applied courses. I'm getting to the point where I'm studying 3rd year undergraduate to graduate level material.

My first problem is that I have the feeling that some of the time I'm just hacking, without really being sure of why I'm doing what I'm doing or whether it's even reasonable. For example with Taylor series expansions or limits I'm not really comfortable about convergence or discarding smaller terms.

Moreover a bunch of techniques seem to crop up over and over in different guises, decomposition of a function into basis functions being a case in point -- I only noticed this was a common theme the second or third time it came up however. I keep hearing terms like subspaces but don't know what they mean.

I'm starting to think that unless I want to just be a hacker there are some fundamentals from pure maths I ought to look at. Limits being one. Vector spaces, subspaces and bases being another. An example of something that wouldn't be relevant is number theory. Problem is I don't know what I don't know so it's hard to make a list of what I should know!

My question is, what areas should an applied mathematician know at the end of a typical undergraduate course in order to have solid foundations (or at least a good intuition) for using the applied maths they have learnt?

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What is your intent? What are you ultimately trying to accomplish? There are lots of different types of "applied mathematicians." – Thomas Andrews Jan 24 '14 at 20:39
"An example of something that wouldn't be relevant is number theory." Number theory is of paramount importance in, for example, cryptography and information security, one of the most prominent parts of applied mathematics today. It doesn't seem like there's a whole lot of an undergraduate curriculum that wouldn't be useful for applications of some sort. – Julien Clancy Jan 24 '14 at 20:42
@ThomasAndrews At the moment it's quite wide, but I'd say data analysis to start off with, so statistics, and techniques such as PCA etc and gradient descent / optimization. Also scientific computing, so solving ODEs, PDEs, series approaches to said, plus numerical methods such as finite differences, integral transforms. Especially anything to do with computing + maths, because computer science is my original background. – TooTone Jan 24 '14 at 20:48
@JulienClancy I stand corrected, thank you. – TooTone Jan 24 '14 at 20:49
For example, complex analysis is very useful in certain areas of applied math, but not others. Depends on what you want to apply it to. In any event, for the stuff you are currently struggling with, real analysis is definitely a good idea. – Thomas Andrews Jan 24 '14 at 21:10

## 1 Answer

Linear algebra, for sure. Real analysis and "advanced calculus" also for sure. These concepts form the basis for numerical analysis, which is critically important for many applied fields. Even if a mathematician isn't directly working on numerical problems, there's typically a desire to develop concepts in a way that they are numerically solvable.

Probability is another field. Uncertainty quantification is hugely important in applied fields, and the interplay between probability, statistics, linear algebra, and dynamical systems cannot be overstated.

Lately, I am of the personal opinion that general algebra and things like representation and category theory are underutilized in applied fields. I have no formal basis for this, aside from my everyday work of research in engineering fields. A great many (i.e. almost all) papers completely ignore abstract formulations of the problems at hand, and they go through extreme procedural distortions to realize only moderate gains in solvability, accuracy, or computability... and so often the example still fail to represent real-world problems of interest in any meaningful way.

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with respect to the underutilized in applied, i would like to think that we are about to see the role of them (abstract maths) in the current time and in the short term. – janmarqz Jan 24 '14 at 21:24
thanks, by advanced calculus do you mean fluency in the techniques of solving 1 and 2 order ODEs (including Euler's and Bernouilli's equations etc), or other stuff like calculus of variations or solving using transforms (Fourier etc) or something else? I'm interested in what you say about uncertainty having studied it somewhat but without really sound theoretical foundations: do you think measure theory is essential here? Finally, I hadn't come across representation and category theory before, and they look interesting-- in fact I've heard of covariance / contravariance in programming. – TooTone Jan 24 '14 at 22:10
I don't know if this is what Arkamis had in mind, but at my school, "advanced calculus" meant really digging in and starting to prove some calculus-type theorems. I don't recall much, but I know we used Kenneth Ross's "Elementary Analysis: The Theory of Calculus" as the text. I mostly remember a lot of $\lim\sup$'s and sequences of functions, but I don't think that's a fair representation. – pjs36 Jan 24 '14 at 22:40
@TooTone By "advanced calculus", I generally mean calculus on manifolds and multivariable calculus, including, but not limited to, Stokes' Theorem and vector calculus. – Arkamis Jan 24 '14 at 22:52
with vector analysis at hand try to pursue to differential geometry, manifolds, bundles, connections, Lie groups and algebras – janmarqz Jan 24 '14 at 23:32