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MATH4431: Advanced Probability IV

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Type	Open
Level	4
Credits	20
Availability	Available in 2024/2025
Module Cap
Location	Durham
Department	Mathematical Sciences

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 Combinations 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
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.

More information

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