Undergraduate Programme and Module Handbook 2024-2025

# Module MATH4431: Advanced Probability IV

## Department: Mathematical Sciences

### MATH4431: Advanced Probability IV

Type | Open | Level | 4 | Credits | 20 | Availability | Available in 2024/2025 | Module Cap | Location | Durham |
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#### Prerequisites

- EITHER: [Complex Analysis II (MATH2011) AND Stochastic Processes III (MATH3251)] OR [Complex Analysis II (MATH2011) AND Probability II (MATH2647)] OR [Markov Chains (MATH2707) AND Analysis III (MATH3011)]

#### Corequisites

- None

#### Excluded Combination of Modules

- None

#### Aims

- To explore in depth fundamental probabilistic systems in both discrete and continuous settings, building on earlier probability courses, and to introduce one or more specialist topics adjacent to contemporary research.

#### Content

- The following content will run every year (Michaelmas term)
- Coin tossing and trajectories of random walks
- Classical limit theorems
- Order statistics
- Some non-classical limits
- Elements of Brownian motion
- One or two of the following topics will be announced to run each year (Epiphany term):
- Random graphs and probabilistic combinatorics
- Random walks in space
- Geometric probability
- Random matrix theory
- Probability and phase transition
- Conformally invariant probability
- Interacting particle systems
- Random permutations
- Random tilings

#### Learning Outcomes

Subject-specific Knowledge:

- By the end of the module students will: be able to solve seen and unseen problems on the given topics.
- Have a knowledge and understanding of this subject demonstrated through an ability to analyse the behaviour of the probabilistic systems explored in the course.
- Reproduce theoretical mathematics concerning probabilistic systems at a level appropriate to Level 4, including key definitions and theorems.

Subject-specific Skills:

- Students will have enhanced mathematical skills in probabilistic intuition.

Key Skills:

- Students will have highly specialised skills in the following areas: problem solving, abstract reasoning, modelling, computation.

#### Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

- Lecturing demonstrates the development of mathematical ideas into a coherent body of material, and how the theory is applied to practical examples.
- Four homework assignments provide formative assessment and feedback to guide students in the correct development of their knowledge and skills in preparation for the summative assessment.
- The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

#### Teaching Methods and Learning Hours

Activity | Number | Frequency | Duration | Total/Hours | |
---|---|---|---|---|---|

Lectures | 42 | 2 per week in Michaelmas and Epiphany; 2 in Easter | 1 hour | 42 | |

Problems Classes | 8 | Fortnightly in Michaelmas and Epiphany | 1 hour | 8 | |

Preparation and Reading | 150 | ||||

Total | 200 |

#### Summative Assessment

Component: Examination | Component Weighting: 100% | ||
---|---|---|---|

Element | Length / duration | Element Weighting | Resit Opportunity |

End of year written examination | 3 hours | 100% |

#### Formative Assessment:

Eight assignments to be submitted.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University