Gifted Issues Discussion homepage Forums Recommended Resources How to Become a Pure Mathematician

http://hbpms.blogspot.com/
How to Become a Pure Mathematician (or Statistician):
a List of Undergraduate and Basic Graduate Textbooks and Lecture Notes - the blog

The site recommends books, some of which are free online, in the following areas. I think the author was a graduate student in math at School of Mathematics and Statistics, UNSW, Sydney, Australia. The fear sometimes expressed that if gifted students study algebra or calculus early, they will run out of math to learn is unfounded.

Stage 1
Elementary Stuff
Introductory Discrete Mathematics
Introductory Algebra
Introductory Calculus

Stage 2
Linear Algebra
Introductory Higher Algebra
Calculus (Introductory Real Analysis, Several Variables Calculus, Vector Calculus, etc)
Complex Variables (Introductory Complex Analysis)
Differential Equation
Probability and Statistics

Stage 3
Introductory Analysis
Abstract Algebra
Introductory Number Theory
Introductory Topology
Differential Geometry
Mathematical Modelling (optional)
Statistical Inference (optional)
Probability and Stochastic Processes (optional)
Statistical Computing (optional)

Stage 4
Foundations and Discrete Mathematics:
Foundation, Logic, Set Theory, etc, Graph Theory, Combinatorics, Cryptography, Coding and Information Theory.
Analysis Functional Analysis, Measure Theory, Hilbert Spaces, Real and Complex Analysis, Fourier and Harmonic Analysis.

Algebra:
Advanced Linear Algebra, Groups and Lie Algebras, Rings, Fields and Galois Theory, Modules and Representation Theory, Commutative Algebra, Homological Algebra and Category.
Number Theory Algebraic NT, Class Field Theory, Analytic NT, Riemann Zeta Function and the Hypothesis, Modular Forms, Elliptic Curves, etc.

Geometry:
Algebraic Geometry, Differential Geometry, Riemannian Geometry, Fractals.

Topology:
Geometric Topology, Algebraic Topology, K-theory, Differential Topology.

Further Calculus (optional):
Ordinary DE, Partial DE, Calculus of Variations.

Mathematical Physics (optional):
Mathematical methods in physics, Relativity, Quantum Mechanics, Quantum Field Theory, String Theory, Chaos.

Probability (optional):
Probability built upon Measure Theory, Stochastic Processes, Stochastic Analysis.

Statistics (optional):
Statistical Models and Regression, Multivariate Analysis, Bayesian Statistics, Simulation and the Monte Carlo Method, Nonparametric Statistics, Categorical Data Analysis, Data Mining, Time Series.

Biostatistics (optional):
Statistical Methods in Epidemiology, Design and Analysis of Clinical Trials, Longitudinal Data Analysis, Survival Analysis.

The Math Olympiad training program of AOPS http://www.artofproblemsolving.com/School/woot.php is sponsored by

DE Shaw
Two Sigma
Jane Street
Citadel
Dropbox

the first four of which are hedge funds or firms that trade financial derivatives. Presumably they are interested in hiring students who are good enough at pure math to do well on a Math Olympiad.

Brilliant! Thanks for this, Bostonian. I admit I was worried about my son starting algebra so early. He is in year one and is learning year six algebra, and zipping through that rather quickly. I'm not sure what a kid is meant to do with all the knowledge, tbh.

"Meant to do"?

I've noticed that kids have a tendency to become adults.

Hmmm. Yep.

Nice breakdown Bostonian. I kept trying to come up with something you missed and then would find where it would fit in. The only things I wonder if you would add separately (though they may fit under another topic) are Linear and Nonlinear Programming and Game Theory. You have some things listed very specifically and others pretty broad (Data Mining)

I don't think people realize the how much of the amazing "computer" programs, simulations, models etc are mathematical models just programmed into a computer. Somebody has to understand the math that drives those things.

There, fixed it for you

ETA actually, the extreme competitiveness characteristic of WOOTers may be of as much interest as their cleverness. At any rate, I don't think one has to posit deep applications of Euclidean geometry puzzles to finance to explain the interest! Seriously, while some aspects of pure maths do have (ugh!) applications in finance, it would be a real stretch to apply the stuff you have to be good at to do well in an Olympiad. Success here is a proxy for what they're interested in, which I think is a combination of analytical insight and performance under pressure.

I'd also argue that statistical mechanics and biophysics applications are missing in the list above-- both use fairly specific mathematical tools in some unusual ways.

Actually, when you get right down to it, every STEM field has its own version of applied mathematics, and in the case of biochemistry, chemistry, and physics, multiple subdomains. There is certainly overlap with pure mathematics, and applications from other fields, but each domain has its own thing.

For example, the geometry and modeling that goes into receptor-ligand models and their kinetics/mechanisms. Or protein folding. Or screening libraries for structure-activity relationships.

Most awesome collision of fields ever for me as a polymath, by the way. My spouse would say the same thing about thin-film materials science instead, though, so there are a LOT of those little niches in STEM.

I realize that this gets away from "pure" mathematics, but honestly, most of the other things aren't pure mathematics, either, when you get right down to it.

Being able to construct a model that predicts market fluctuations on the basis of a set of inputs isn't that different from being able to construct a model that predicts the diffusion of a radioactive contaminant that is partitioning between soil particles and moving groundwater. KWIM?