New mathematics

As a school student, most of the maths you've encountered has been known for hundreds, if not thousands, of years. Perhaps you're wondering what more there is left to study and discover? 

One great revolution in the history of mathematics was the 19th century discovery of strange non-Euclidean geometries where, for example, the angles of a triangle don’t add up to 180°, a discovery defying 2000 years of received wisdom. To experiment with this, try drawing a triangle on a sphere. You can see what straight lines look like by looking at flight data (since a straight line is the shortest distance between two points).

In 1931 Kurt Gödel shook the very foundations of mathematics, showing that there are true statements which cannot be proved, even about everyday whole numbers. This problem arose from a list of 23 (then unsolved) problems that David Hilbert published at the start of the 20th century. Remarkably there are still four that have yet to be solved (including the Riemann hypothesis) and several others that have only been partially resolved.

A decade earlier the Polish mathematicians Banach and Tarski showed that any solid ball can be broken into as few as five pieces and then reassembled to form two solid balls of the same size as the original. This completely contradicts our intuitions about geometry (I can't turn break a £50 note into five pieces and then reassemble them to form two £50 notes, otherwise our monetary system would break down!). There's a rich array of ideas and surprises to be found in recent mathematics, which shows no sign of abating. 

Looking through any university’s mathematics prospectus you will see course titles that are familiar (e.g. algebra, mechanics) and some that appear thoroughly alien (e.g. Galois Theory, Martingales, Communication Theory). These titles give an honest impression of university mathematics: some courses are continuations from school mathematics, though usually with a lot more proof and a change in emphasis, whilst others will be thoroughly new, on a topic which you previously thought mathematics had nothing to say whatsoever.