# Mathematics at University

## What do mathematicians do?

The clearest change of emphasis in mathematics at university is in the need to prove things. Much mathematics is too abstract or technical to simply rely on intuition, and so it is important that you can write clear and irrefutable arguments, which make plain to you, and others, the soundness of your claims. This ability to think logically and argue your thoughts clearly is incredibly valuable, both in mathematics and in the wider world.

Mathematicians usually work in either pure or applied maths (although recent advances mean that even topics traditionally considered to be pure maths are seeing applications in medical imaging, art history, and e-commerce). An academic mathematician will be part of a research group; these groups can be very specific (e.g. Analytic Topology) or inter-disciplinary (e.g. Mathematical Geosciences) and provide an opportunity to discuss your research with other researchers in your field. To become an academic students will typically study to undergraduate and Master's level, before doing a PhD in a mathematical area. Around 30% of our undergraduates go onto further study (including Masters and PhD programmes).

The majority of mathematics students don't go on to become academics. You can find out more in our careers section.

To find out more about the research going on in the Mathematical Institute why not take a look at the Oxford Mathematics Alphabet? Each letter is a short post written by one of our academicson a topic that they're researching and comes with links to further reading and a poster for you to print out. You might also be interested in some of our case studies detailing the latest research in the Mathematical Institute. The videos on the right give an introduction to some of the topics studied here.

## Why study Mathematics?

**You like thinking about things.**Mathematics has a beauty and a power that few other subjects can touch. As a mathematician you can answer questions that have puzzled people for hundreds of years.

**You want to make a difference.**Medical researchers, economic policy makers, and technology firms all want highly numerate graduates. If you want to stop the next major disease outbreak, become a mathematician.

**You enjoy working things out.**As technology advances more and more data is being generated, creating new and interesting problems for future mathematicians to solve.

## 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.

## The future of maths

Mathematics wouldn't be the subject it is today if it hadn’t had been for the impact of applied mathematics and statistics. There is much beautiful mathematics to be found here, such as in relativity or in number theory behind the RSA encryption widely used in internet security, or just in the way a wide range of techniques from all reaches of mathematics might be applied to solve a difficult problem. Also with ever faster computers, mathematicians can now model highly complex systems such as the human heart, can explain why spotted animals have striped tails, can treat non-deterministic systems like the stock market or Brownian motion. The high technical demands of these models and the prevalence of computers in everyday life are making mathematicians ever more employable after university.

As a mathematician (academic or not) you have the ability to help answer some of the most pressing issues facing humanity today.

## Next:

Teaching at Oxford is different from most other universities, with more small-group teaching. Find out more on the next page.