Problems in Mathematics

This 3/4th year module runs every year.

Credit Value: 30 Credits

Aims

This course aims to allow students 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.

Teaching and Assessment

This module is entirely coursework based; split into 40% for problem sheets and 60% for essays. Over the course of the year there will be several evenings of lectures. Each evening will concentrate on one topic (the choice of topics will vary each year). At the end of each lecture evening, students will be given a problem sheet to complete. This will consist of several short compulsory questions to be submitted within 4 weeks of the lecture.

Each problem sheet will count 10% towards the final mark for the module, and students will complete four problem sheets.

In addition, at the end of each lecture students will be given a short list of suggestions for essays with each topic.

Over the course of the year students must choose any two of these questions to complete; each essay should be roughly 2,500 words and no more than 4,000 words.

Optionally a student may, with permission, choose to write ONE essay of the two on a mathematical subject of their own choosing. If a student wishes to do this, he or she must obtain the permission of a member of School to supervise this project, and submit an abstract which must be approved by the essay supervisor before the end of the Spring Term.

Each essay counts 30% giving a total of 60%.

Syllabus

A selection of typical topics is given below but will vary from year to year to keep current. Each topic would be the subject of one evening of lectures.

1. The Riemann Hypothesis: development, background; historical and recent attempts to prove it;
2. 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;
3. Unsolved problems in Number Theory (such as the twin primes conjecture, the Goldbach conjecture, the infinity (or otherwise) of perfect numbers and Mersenne primes.
4. Laying the foundations of Mathematics: attempts to axiomatise from Euclid to Bourbaki.
5. The 4-colour theorem; early attempts at proofs and its eventual computer-based proof.
6. Recreational Mathematics (SuDoku grids, latin squares, magic squares, analysis of Tower of Hanoi, what makes Rubik’s cube so challenging).
7. Computability; the P vs NP problem; implications of a possible solution.

Learning Outcomes

Subject Specific

Knowledge and understanding of a range of results in mathematics and/or statistics.

• In particular: Understanding of some of the most famous problems that have shaped mathematics.

Appreciation of the need for proof in mathematics, and the ability to follow and construct mathematical arguments.

• In particular: Increased knowledge of the way mathematics is done by working mathematicians.

Appreciation of the power of generalization and abstraction in the development of mathematical theories.

A deeper knowledge of some particular areas of mathematics and/or statistics.

Intellectual

Ability to comprehend conceptual and abstract material.

Develop a logical and systematic approach to problem solving.

Practical

Ability to use a wide range of software packages including word processing and spreadsheets.

• In particular: The preparation of typed essays, requiring the use of word processing tools including equation editors.

Problem-solving skills, including the ability to assess problems logically and approach them analytically.

Highly developed quantitative skills.

Ability to transfer knowledge and expertise from one context to another.

•  In particular: Understanding of the way that many different strands of thought and areas of mathematics can be used to work on complex problems in mathematics.

Personal and Social

Ability to learn independently using a variety of media.

Ability to work independently with patience and persistence.

Time-management skills and organizational skills.

General IT skills, including word processing and spreadsheets.

Good communication skills, including the ability to write coherently.

Ability to complete a sustained and substantial task.

Ability to complete work in a limited time period.

Awareness of the development of important ideas in mathematics and the impact of the work of individual mathematicians.

• Awareness that mathematics is a living subject with many problems still unsolved and new research continuing all the time.
• Appreciation of the beauty of some of the most famous theorems of mathematics.

Increased awareness of the context in which mathematics is done.

Recommended Texts

• John Fauvel and Jeremy Gray, The History of Mathematics, Palgrave Macmillan (1985)
• Howard Eves and Carroll V Newson, Foundations and Fundamental Concepts of Mathematics, Dover (1997)
• Keith Devlin, The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time, Granta Books (2005)
• John Derbyshire, Unknown Quantity: A Real and Imaginary History of Algebra, Plume (2007)