Module Details |
The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module. |
Title | THE MAGIC OF COMPLEX NUMBERS: COMPLEX DYNAMICS, CHAOS AND THE MANDELBROT SET | ||
Code | MATH345 | ||
Coordinator |
Professor L Rempe Mathematical Sciences L.Rempe@liverpool.ac.uk |
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Year | CATS Level | Semester | CATS Value |
Session 2024-25 | Level 6 FHEQ | Second Semester | 15 |
Aims |
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1. To introduce students to the theory of the iteration of functions of one complex variable, and its fundamental objects; 2. To introduce students to some topics of current and recent research in the field; 3. To study various advanced results from complex analysis, and show how to apply these in a dynamical setting; 4. To illustrate that many results in complex analysis are "magic", in that there is no reason to expect them in a real-variable context, and the implications of this in complex dynamics; 5. To explain how complex-variable methods have been instrumental in questions purely about real-valued one-dimensional dynamical systems, such as the logistic family. 6. To deepen students' appreciations for formal reasoning and proof. |
Learning Outcomes |
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(LO1) To understand the compactification of the complex plane to the Riemann sphere, and be able to use spherical distances and derivatives. |
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(LO2) To be able to use Möbius transformations to transform the Riemann sphere and to normalise complex dynamical systems. |
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(LO3) To be able to state and apply the definitions of Julia and Fatou sets of polynomials, and understand their basic properties. |
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(LO4) To be able to determine whether points with simple orbits, such as certain periodic points, belong to the Julia set or the Fatou set. |
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(LO5) To know how to apply advanced results from complex analysis in a dynamical setting. |
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(LO6) To be able to determine whether certain types of quadratic polynomials belong to the Mandelbrot set or not. |
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(S1) Problem solving/ critical thinking/ creativity analysing facts and situations and applying creative thinking to develop appropriate solutions. |
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(S2) Problem solving skills |
Syllabus |
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Syllabus: |
Recommended Texts |
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Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module. |
Pre-requisites before taking this module (other modules and/or general educational/academic requirements): |
MATH243 COMPLEX FUNCTIONS |
Co-requisite modules: |
Modules for which this module is a pre-requisite: |
Programme(s) (including Year of Study) to which this module is available on a required basis: |
Programme(s) (including Year of Study) to which this module is available on an optional basis: |
Assessment |
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EXAM | Duration | Timing (Semester) |
% of final mark |
Resit/resubmission opportunity |
Penalty for late submission |
Notes |
Final Assessment | 90 | 50 | ||||
CONTINUOUS | Duration | Timing (Semester) |
% of final mark |
Resit/resubmission opportunity |
Penalty for late submission |
Notes |
short quiz 5 Standard UoL penalty applies for late submission. This is not an anonymous assessment. | 0 | 10 | ||||
short quiz 1 Standard UoL penalty applies for late submission. This is not an anonymous assessment. | 0 | 10 | ||||
short quiz 2 Standard UoL penalty applies for late submission. This is not an anonymous assessment. | 0 | 10 | ||||
short quiz 3 Standard UoL penalty applies for late submission. This is not an anonymous assessment. | 0 | 10 | ||||
short quiz 4 Standard UoL penalty applies for late submission. This is not an anonymous assessment. | 0 | 10 |