So you want to be an Applied Mathematician

The way of the Applied Mathematician is one full of challenging and interesting problems. We thrive by association with the Pure Mathematician, and at the same time with the no-nonsense, hands-in, hard-core Engineer. But not everything is happy in Applied Mathematician land: every now and then, we receive the disregard of other professionals that mistake either our background, or our efficiency at attacking real-life problems.

I heard from a colleague (an Algebrist) complains that Applied Mathematicians did nothing but code solutions of partial differential equations in Fortran—his skewed view came up after a naïve observation of a few graduate students working on a project. The truth could not be further from this claim: we do indeed occasionally solve PDEs in Fortran—I give you that—and we are not ashamed to admit it. But before that job has to be addressed, we have gone through a great deal of thinking on how to better code this simple problem. And you would not believe the huge amount of deep Mathematics that are involved in this journey: everything from high-level Linear Algebra, Calculus of Variations, Harmonic Analysis, Differential Geometry, Microlocal Analysis, Functional Analysis, Dynamical Systems, the Theory of Distributions, etc. Not only are we familiar with the basic background on all those fields, but also we are supposed to be able to perform serious research on any of them at a given time.

My soon-to-be-converted Algebrist friend challenged me—not without a hint of smugness in his voice—to illustrate what was my last project at that time. This was one revolving around the idea of frames (think of it as redundant bases if you please), and needed proving a couple of inequalities involving sequences of functions in \( L_p \)—spaces, which we attacked using a beautiful technique: Bellman functions. About ninety minutes later he conceded defeat in front of the board where the math was displayed. He promptly admitted that this was no Fortran code, and showed a newfound respect and reverence for the trade.

It doesn’t hurt either that the kind of problems that we attack are more likely to attract funding. And collaboration. And to be noticed in the press.

Alright, so some of you are sold already. What is the next step? I am assuming that at his point you own your Calculus, Analysis, Probability and Statistics, Linear Programming, Topology, Geometry, Physics and you are able to solve most known ODEs. From here, as with any other field, my recommendation is to slowly build a Batman belt: acquire and devour a sequence of books and scientific articles, until you are very familiar with their contents. When facing a new problem, you should be able to recall from your Batman belt what technique could work best, in which book(s) you could get some references, and how it has been used in the past for related problems.

Following these lines, I have included below an interesting collection with the absolutely essential books that, in my opinion, every Applied Mathematician should start studying:

Everyone should begin with something in the line of the following five texts: Torchinsky’s Real Variables; Dummit and Foote’s Abstract Algebra; Do Carmo’s Differential Geometry; Hoffman and Kunze’s Linear Algebra; and Ahlfors’ Complex Analysis. [I included a collection of links to the books below for your convenience] I know, I know: there are much better sources than these out there. The ones I presented have an advantage over any other, in my opinion: a huge selection of problems. If you patiently solve every single one of them, none of these topics will hold secrets for you. That is a promise.

Where do we go from there? It will depend on what you want to focus, but no matter what you choose, you need knowledge of the following five: partial differential equations, Theory of Variations, Theory of Distributions, Sobolev spaces and Functional Analysis. The list of books on these topics is long, and I chose the following representative titles: John’s Partial Differential Equations; Gelfand and Fomin; Friedlander and Joshi; Adams and Fournier; and Brezis.

You also need to know the basics of computer programming. Also, more importantly, what wavelets are and how they are used. You need to have a firm grasp of numerical analysis, signal processing, how to handle matrices properly, some image processing, finite element methods, Approximation Theory, the Radon transform, and how it is used to solve simple inverse problems, for instance.

Alberto Torchinsky
Real Variables
Dummit & Foote
Abstract Algebra, 3rd Edition
Manfredo P. do Carmo
Differential Geometry of Curves and Surfaces (See all Differential Geometry Books)
Hoffman & Kunze
Linear Algebra (2nd Edition)
Complex Analysis (See all Functional Analysis Books)
Fritz John
Partial Differential Equations (Applied Mathematical Sciences) (v. 1) (See all Differential Equations Books)
Gelfand and Fomin
Calculus of Variations (See all Mathematical Physics Books)
Friedlander and Joshi
Introduction to the Theory of Distributions (See all Mathematical Physics Books)
Adams and Fournier
Sobolev Spaces, Volume 140, Second Edition (Pure and Applied Mathematics) (See all Differential Equations Books)
Functional Analysis, Sobolev Spaces and Partial Differential Equations (Universitext) (See all Differential Equations Books)
Abelson and Sussman
Structure and Interpretation of Computer Programs, Second Edition (See all Programming Language & Tool Books)
Ten Lectures on Wavelets (CBMS-NSF Regional Conference Series in Applied Mathematics) (See all Waves & Wave Mechanics Books)
Numerical Analysis
Brenner and Scott
The Mathematical Theory of Finite Element Methods (Texts in Applied Mathematics)
The Radon Transform (Progress in Mathematics) (Vol 5) (See all Electricity Principles Books)
The Mathematics of Computerized Tomography (Classics in Applied Mathematics) (See all Discrete Mathematics Books)
DeVore and Lorentz
Constructive Approximation (Grundlehren der mathematischen Wissenschaften)
Matrix Analysis (See all Probability & Statistics Books)
The Scientist & Engineer's Guide to Digital Signal Processing
Geometric Partial Differential Equations and Image Analysis

If you are in the field, and feel that there is a missing text in my list, please make me know. I focused mostly on those which I needed at some point to progress on my research, which is basically oriented to mathematical imaging. So, five cents anyone?