# Mathematics

## Introduction to tutors and students

## Why study mathematics?

Wonderful theorems, beautiful proofs, great applications.

David Acheson, *1089 and All That*

Mathematics is the language of science and logic, and the language of argument. Science students are often surprised, and sometimes daunted, by the prevalence of mathematical ideas and techniques which form the basis for scientific theory. The more abstract ideas of pure mathematics may find fewer everyday applications, but their study instills an appreciation of the need for rigorous, careful argument and an awareness of the limitations of an argument or technique. A mathematics degree teaches the skills to see clearly to the heart of difficult technical problems, and provides a “toolbox” of ideas and methods to tackle them.

## Who is this course good for?

- If you enjoy doing mathematics and understanding where maths comes from.
- If you want to solve problems, whether abstract or practical.
- If you enjoy a challenge and want to explore new and fascinating areas of mathematics.

## What's in the course?

As an undergraduate in mathematics, you will firstly cover the basics. At Oxford, the first year course has no options - we aim to teach you core material, covering ideas and techniques fundamental to later years. These topics include linear algebra, groups, differentiation, integration, probability, statistics, geometry, dynamics, optimisation, Fourier series, and multivariable calculus. Some of these you will have met in some form already as part of your studies (for example, integration and differentiation). At university, however, it's not sufficient to know how to integrate and how to differentiate (though this is pretty essential!). We need to know what kind of things can be integrated and what kind of things can't be, what happens when there are discontinuities in the function, and whether there are key properties that functions must have in order to be integrable.

From the second year onwards you can choose certain options as part of your course. In the second year, you take 3 short options from 8 courses, and 5 or 6 long options from a choice of 9 courses. You also continue learning some core material, including differential equations and complex analysis. Short options include graph theory, number theory, projective geometry, and special relativity. Long options include topology, quantum theory, probability, and numerical analysis. There is a range of pure and applied courses on offer, and many people at this stage continue taking a mixture of both.

In the third year, there are around 48 different courses on offer, of which you choose 8, and no compulsory courses. It is at this point that many students begin to specialise. If you choose to continue to the fourth year, there are around 59 courses on offer, of which you again choose 8. You can, in effect, create your own degree tailored to your personal mathematical interests - which may in the end differ wildly from those you started your degree with! Regardless of what courses you end up studying we look forward to exploring the world of mathematics with you.

## Problems to think about

Imagine a knight on a chessboard (8x8 grid). Is it possible, using the knight's move (an L-shape - that is, moving two squares in one direction, then one square perpendicular to that line) to visit all the squares on a chessboard? Does it matter where the knight starts? Is it possible to visit all the squares on a chessboard precisely once? Is this possible for all sizes of grid? What determines whether this is possible or not?

## For more information

For more information about the Mathematics degree, you can look at the departmental prospectus. You can also find more information in the course handbook.

The Oxford Mathematics Alphabet, gives an insight into the world leading research that goes on in the department.

If you'd like to deepen your understanding of the mathematics you're studying as part of your curriculum, we highly recommend Underground Mathematics.

If you're interested in extending your mathematical knowledge, we recommend you look at taking STEP or AEA.

Nrich has some excellent problems on interesting mathematics.

Plus magazine has some excellent articles on interesting mathematics.