Unit name | Random Matrix Theory |
---|---|

Unit code | MATH30016 |

Credit points | 10 |

Level of study | H/6 |

Teaching block(s) |
Teaching Block 2C (weeks 13 - 18) |

Unit director | Professor. Snaith |

Open unit status | Not open |

Pre-requisites |
Year 2 Theoretical Physics OR MATH20015 Multivariable Calculus |

Co-requisites |
None |

School/department | School of Mathematics |

Faculty | Faculty of Science |

**Unit Aims**

By the end of the unit you will master some of the most important mathematical techniques used in random matrix theory and have an understanding of how these are relevant in various areas of mathematics, physics, engineering and probability.

**Unit Description**

Random matrices are often used to study the statistical properties of systems whose detailed mathematical description is either not known or too complicated to allow any kind of successful approach. It is a remarkable fact that predictions made using random matrix theory have turned out to be accurate in a wide range of fields: statistical mechanics, quantum chaos, nuclear physics, number theory, combinatorics, wireless telecommunications and structural dynamics, to name only few examples.

Particular emphasis will be given to computing correlations of eigenvalues of ensembles of unitary and Hermitian matrices. Different ensembles have distinct invariance properties, which in the applications are used to model systems whose physical or mathematical behaviour depends only on their symmetries. In some cases the dimension of the matrices will be treated as a large asymptotic parameter. In addition we will develop several techniques to compute certain types of multiple integrals. There will be general discussion of how this relates to current research in various fields of mathematics and physics. The course will appeal to students in applied and pure mathematics as well as in statistics. The theory can be thought of probabilistically and so would be of interest to students focusing on probability, however the background needed is very minimal, so this should not put off students who are not so keen on probability.

**Relation to Other Units**

The material covered provides a useful background for the Level 7 unit Quantum Chaos. Some aspects of this course are related to topics presented in the Level 6 unit Statistical Mechanics.

Learning Objectives

After completing this unit successfully you should be able to:

- Define and comprehend the notions of spectral statistics for various matrix ensembles.
- Compute typical examples of spectral statistics.
- Recognize and compute with a few common matrix ensembles

Transferable Skills

- Clear, logical thinking.
- Problem solving techniques.

The unit will be taught through a combination of

- synchronous online and, if subsequently possible, face-to-face lectures
- asynchronous online materials, including narrated presentations and worked examples
- guided asynchronous independent activities such as problem sheets and/or other exercises
- synchronous weekly group problem/example classes, workshops and/or tutorials
- synchronous weekly group tutorials
- synchronous weekly office hours

80% Timed, open-book examination 20% Coursework

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

If you fail this unit and are required to resit, reassessment is by a written examination in the August/September Resit and Supplementary exam period.

If this unit has a Resource List, you will normally find a link to it in the Blackboard area for the unit. Sometimes there will be a separate link for each weekly topic.

If you are unable to access a list through Blackboard, you can also find it via the Resource Lists homepage. Search for the list by the unit name or code (e.g. MATH30016).

**How much time the unit requires**

Each credit equates to 10 hours of total student input. For example a 20 credit unit will take you 200 hours
of study to complete. Your total learning time is made up of contact time, directed learning tasks,
independent learning and assessment activity.

See the Faculty workload statement relating to this unit for more information.

**Assessment**

The Board of Examiners will consider all cases where students have failed or not completed the assessments required for credit.
The Board considers each student's outcomes across all the units which contribute to each year's programme of study.
If you have self-certificated your absence from an assessment, you will normally be required to complete it the next time it runs
(this is usually in the next assessment period).

The Board of Examiners will take into account any extenuating circumstances and operates
within the Regulations and Code of Practice for Taught Programmes.