Problems in Mathematics

• Credit value: 30 credits at Level 6
• Assessment: four problem sheets consisting of several short compulsory questions to be submitted within four weeks of the lecture (10% each) and two 2500-4000-word essays (30% each)

This course will allow you to engage with some of the important problems which have shaped mathematics. Problems will be put in their historical context and will be used to illustrate the development of different areas of mathematics. You will have the opportunity to tackle more open-ended work and make links between the many branches of mathematics that have been studied on the degree programme.

Indicative module content

• The Riemann Hypothesis: development, background; historical and recent attempts to prove it
• Euler's legacy: 300 years since his birth we look at the ways in which mathematics has been, and is still, affected by his work
• Unsolved problems in number theory (such as the twin primes conjecture, the Goldbach conjecture, the infinity (or otherwise) of perfect numbers and Mersenne primes
• Laying the foundations of mathematics: attempts to axiomatise from Euclid to Bourbaki
• The four-colour theorem; early attempts at proofs and its eventual computer-based proof
• Recreational mathematics (SuDoku grids, latin squares, magic squares, analysis of Tower of Hanoi, what makes Rubik’s cube so challenging)
• Computability; the P vs NP problem; implications of a possible solution

By the end of this module, you will have:

• knowledge and understanding of a range of results in mathematics and/or statistics, in particularsome of the most famous problems that have shaped mathematics
• appreciation of the need for proof in mathematics
• the ability to follow and construct mathematical arguments, in particular, the way mathematics is done by working mathematicians
• appreciation of the power of generalisation and abstraction in the development of mathematical theories
• a deeper knowledge of some particular areas of mathematics and/or statistics.