"All limits, especially national ones, are contrary to the nature of mathematics… Mathematics knows no races… For mathematics the whole cultural world is a single country" – David Hilbert, the foremost mathematician of the 20th century, speaking as the Nazis neared power in his native Germany.

The map of mathematics

Pretty good for a ten-minute summary.

If I were going to add anything, I’d say that the major omission is algebraic geometry (see below for summary description of the third-year course offered on that at Cambridge). But in ten minutes I’m sure I couldn’t do one-tenth as well to sketch a map.

Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory.

Initially a study of polynomial equations in many variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes at least as important to understand the totality of solutions of a system of equations, as to find some solution; this does lead into some of the deepest waters in the whole of mathematics, both conceptually and in terms of technique.

This course is an introduction to the basic ideas of algebraic geometry (affine and projective spaces, varieties), followed by a more detailed study of algebraic curves. We will develop the basic tools for understanding the properties of algebraic curves, and apply these at the end of the course to the beautiful theory of elliptic curves, which among other things played an essential part in the proof of Fermat’s Last Theorem.