LIFE AFTER CALCULUS

Where do you go from here? (i.e., post-calculus)

Calculus is not the end-all of mathematics: it’s rather a beginning, an entry into a thousand worlds. You’ve learned some answers, enough to ask (and eventually answer) far more questions. Here’s some of the paths you might follow.

1. Mathematically

1.1. Algebra. Wanna solve the 3 great problems of antiquity?

• can’t trisect an angle

• can’t double the cube

• can’t square the circle (or round the square) … using just compass and straightedge.

Wanna solve cubics and quartic, and (more interesting), see why there’s no formula (just taking roots) to solve the quintic? See how complex conjugation generalizes to other fields!

Group theory is the study of symmetries: it generalizes the Platonic solids (cube, tetrahedron, icosahedron, dodecahedron, octahedron) and tilings to classify all possible symmetries!

Representation theory is the study of how to realize these symmetries geometrically^{1}, as we do with the Platonic solids – but this requires some linear algebra.

It’s also related to Fourier series!

1.2. Linear algebra. In “real life”, you have functions of several variables, and need to use linear algebra and matrices to work with them.

It also allows you to do high dimensional geometry.

1.3. Multi-variable calculus. Differentiating and integrating R ) R^{n} was easy; doing calculus for R^{m} ) R or R^{m} ) R^{n} is more difficult and requires linear algebra.

This is a key tool in applying calculus to “real life” questions, and also allows you to do differential geometry.

Also, this lets you apply calculus to manifolds.

1.4. Differential Geometry. I.e., geometry using calculus.

Make sense of non-Euclidean geometries, like “positively” and “negatively” curved space!

1.5. Complex Analysis. Power series make sense over the complex numbers, every polynomial has a root (zero) over the complex number, and functions that are complex differentiable are all analytic: much of the theory of complex functions is much nicer than for real functions, and it’s much more geometric.

1Well, there’s other ways to realize these symmetries, but representation theory is an important one.

1.6. Real Analysis.

• make (more) sense of Fourier series

• measure theory be able to integrate χ_{Q} and many other functions.

• probability: this is a special case of measure theory, and allows you to apply calculus to probability and statistics

• function spaces and distributions: make sense of all those topologies also,

learn to take the derivative of any continuous function! Also, understand the techniques used to solve differential equations.

1.7. Others. We didn’t do much of these in calc, but they’re whole fields of math: • discrete math/combinatorics: counting things, theoretical computer science • logic: study the “higher infinities” and prove meta-theorems, e.g., prove

that it’s impossible to prove something!

• Topology: The circle, line, sphere, torus, Möbius strip and Klein bottle are just the beginning of the vast menagerie of manifolds.

2. Applications

Being a pure mathematician, I don’t know as much about how this stuff is used as others do; here’s some things I know.

Note that you often won’t be really told the math that’s going on when you’re working in other fields, but it’s underlying much of your calculations (and even if you recognize some math, there may be other math underneath, making it work)

– if you recognize the math, you’ll often have a better understanding of what’s happening.

2.1. Physics. Virtually everything in math is useful in physics, somewhere. It’s a bit scary.

2.2. Differential equations. These show up everywhere in modeling; you’ll likely learn countless tricks for solving these.

2.3. Optimization. You often want to maximize something, especially profit: calculus gives you a first tool to do this, but you’ll also use numerical methods to approximate solutions.

2.4. Statistics. To really understand statistics, you need to do linear algebra, measure theory, and other real analysis.