| Unit name | Combinatorics |
|---|---|
| Unit code | MATH30030 |
| Credit points | 20 |
| Level of study | H/6 |
| Teaching block(s) |
Teaching Block 1 (weeks 1 - 12) |
| Unit director | Dr. Ellis |
| Open unit status | Not open |
| Units you must take before you take this one (pre-requisite units) |
MATH10010 Introduction to Proofs and Group Theory, MATH10011 Analysis, MATH10013 Probability and Statistics and MATH10015 Linear Algebra |
| Units you must take alongside this one (co-requisite units) |
None |
| Units you may not take alongside this one |
None |
| School/department | School of Mathematics |
| Faculty | Faculty of Science |
Why is this unit important?
Combinatorics is the study of discrete structures, which are ubiquitous in our everyday lives. While combinatorics has important practical applications (for example to networking, optimisation, and statistical physics), problems of a combinatorial nature also arise in many areas of pure mathematics such as algebra, probability, topology and geometry.
How does this unit fit into your programme of study?
This unit serves as an introduction to combinatorics, developing fundamental aspects of a diverse range of topics in discrete mathematics such as extremal graph theory, Ramsey theory and the probabilistic method. The unit aims to develop and improve students’ problem-solving and theorem-proving skills, building on those acquired in first-year courses. Moreover, it seeks to enhance students’ appreciation of the interconnectedness between combinatorics and different areas of mathematics, by introducing probabilistic, analytic and algebraic techniques.
An overview of content.
The course will start with introducing the basics of graph theory: definitions, concepts and basic results, concerning trees, cycles, connectivity, matchings, planarity. Hall’s Matching Theorem and its variants and strengthenings will be discussed in depth. It will then cover the fundamentals of Extremal Graph Theory, including Turan’s theorem and the Erdos-Stone theorem. After a short section on graph colourings, we will introduce the fundamentals of Ramsey theory. While doing so we will introduce (and further explore) the ‘probabilistic method’. The course will end with a section on algebraic graph theory, together with some elegant applications.
How will students, personally, be different as a result of this unit?
Students who successfully complete the unit will have received a thorough grounding in the fundamentals of combinatorics / discrete mathematics, and will have learned techniques and ways of thinking that have broad applicability in mathematics, science and technology. Their problem-solving, analytical and creative skills will have been greatly enhanced by studying this unit.
Learning Outcomes
Students who successfully complete the unit will:
The unit will be taught through a combination of:
Note that the problem-sheets and exercises are part of the formative assessment mentioned above. The tutorials will be interactive, aimed at developing students’ problem-solving skills in real time.
Tasks which help you learn and prepare for summative tasks (formative).
There will be regular homework exercises set each week, to enable students to fully understand and master the material and to develop their problem-solving skills.
Tasks which count towards your unit mark (summative).
The summative assessment for this unit will consist of a coursework task (10%) and a timed examination (90%).
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. MATH30030).
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 University 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. For appropriate assessments, if you have self-certificated your absence, you will normally be required to complete it the next time it runs (for assessments at the end of TB1 and TB2 this is usually in the next re-assessment period).
The Board of Examiners will take into account any exceptional circumstances and operates
within the Regulations and Code of Practice for Taught Programmes.
