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\title{Life after Calculus}

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\maketitle

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

Calculus is \emph{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.

\section{Mathematically} % -----------------------------------------------------

\subsection{Algebra} % --------
Wanna solve the 3 great problems of antiquity?
\begin{itemize}
 \item can't trisect an angle
 \item can't double the cube
 \item can't square the circle (or round the square)
\end{itemize}
\dots 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!

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

\emph{Representation theory} is the study of how to realize these symmetries
geometrically\footnote{Well, there's other ways to realize these symmetries,
but representation theory is an important one.},
as we do with the Platonic solids -- but this requires some
\emph{linear algebra}.

It's also related to \emph{Fourier series}!

\subsection{Linear algebra} % ---------
In ``real life'', you have functions of several variables, and need to
use \emph{linear algebra} and \emph{matrices} to work with them.

It also allows you to do \emph{high dimensional geometry}.

\subsection{Multi-variable calculus}
Differentiating and integrating $\bR \to \bR^n$ was easy;
doing calculus for $\bR^m \to \bR$ or $\bR^m \to \bR^n$ is more
difficult and requires \emph{linear algebra}.

This is a key tool in applying calculus to ``real life'' questions,
and also allows you to do \emph{differential geometry}.

Also, this lets you apply calculus to \emph{manifolds}.

\subsection{Differential Geometry} % -----
I.e., geometry using calculus.

Make sense of non-Euclidean geometries, like ``positively'' and
``negatively'' curved space!

\subsection{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.

\subsection{Real Analysis}
\begin{itemize}
\item make (more) sense of \emph{Fourier series}
\item \emph{measure theory} be able to integrate $\chi_\bQ$ and many other
  functions.
\item \emph{probability}: this is a special case of measure theory,
  and allows you to apply calculus to \emph{probability} and \emph{statistics}
\item function spaces and distributions: make sense of all those topologies
  also, learn to take the derivative of any continuous function!
\end{itemize}
Also, understand the techniques used to solve \emph{differential
equations}.

\subsection{Others}
We didn't do much of these in calc, but they're whole fields of math:
\begin{itemize}
 \item \emph{discrete math/combinatorics}: counting things,
 theoretical computer science
 \item \emph{logic}: study the ``higher infinities'' and prove
 meta-theorems, e.g., prove that it's impossible to prove something!
 \item \emph{Topology}:
 The circle, line, sphere, torus, M\"obius strip and Klein bottle are just the
 beginning of the vast menagerie of \emph{manifolds}.
 \item \emph{Algebraic Geometry}:
  Can you do geometry just using polynomials (and rational functions)?
  We enumerated all the Pythagorean triples using a certain rational
  map: can we do this for other equations?

  This also explains why you can't take certain integrals in closed
  form: their (algebraic) geometry is more complicated.
\end{itemize}

\section{Applications} % ------------------------------------------------------------

Being a pure mathematician, I don't know as much about how this stuff
is \emph{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 \emph{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.

\subsection{Physics}
Virtually everything in math is useful in physics, somewhere.
It's a bit scary.

\subsection{Differential equations}
These show up everywhere in modeling; you'll likely learn countless
tricks for solving these.

\subsection{Optimization}
You often want to maximize something, especially profit: calculus
gives you a first tool to do this, but you'll also use \emph{numerical
methods} to approximate solutions.

\subsection{Statistics}
To really understand statistics, you need to do \emph{linear algebra},
\emph{measure theory}, and other \emph{real analysis}.


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