THIS MODULE DOES NOT RUN IN 2013/14!

Term 2, 10 weeks, oral examination at end of term

Module Leader: Stefan Grosskinsky (Mathematics and Complexity)

link to Online Course Materials

Assessment information 2013:

Week	Assessment	Issued	Deadline	how assessed	%credit
2	homework assignments (calculations)	TBA	TBA	written script	12.5
7	homework assignments (calculations + simulations)	TBA	TBA	written script	20
10	homework assignments (calculations + simulations)			written script	17.5
10	Oral Examination	TBA		Oral examination	50

 

 

Taken by students from:

Code	Degree Title	Year of study	core or option	credits
P-F3P4(5)	Complexity Science MSc (+PhD)	1	option	18
P-G1P8(9)	Complexity Science MSc (+PhD)	1	option	18
P-G3G1	Maths and Stats MSc (+PhD)	1	option	18
P-F3P6(7)	MSc in Complex Systems Science	1 or 2	option	18

 

Context: This is part of of the Complexity DTC taught programme.

 

Module Aims:

This module covers the mathematical description and analysis of complex systems with stochastic time evolution.

 

Syllabus:

1. Short introduction to basic theory
   Markov processes, graphical construction, semigroups and generators, stationary distributions and reversibility, conservation laws, symmetries, absorbing states
2. Population models
   branching processes, Moran model, Wright-Fisher diffusion and duality with Kingman’s coalescent, fixation times, diffusion limits
3. Epidemic models
   Contact process, survival and extinction, mean field rate equations, critical values, general remarks on the DP universality class
4. Interacting random walks
   exclusion processes, stationary currents and conservation laws, hydrodynamic limits, dynamic phase transition

Theoretical techniques (introduced along in lectures):
scaling limits and Fokker-Planck approximations, mean-field rate equations, generating functions, duality;

Computational techniques (covered in classes):

• how to simulate discrete and continuous-time models: random sequential update and other update rules, sampling rates and jump chains, construction with Poisson processes and rejection
• how to measure: stationary averages, ergodic theorem, equilibration times
• maybe also: classical Monte Carlo with heat-bath and Metropolis algorithm
• implementation of 2 simulations with measurements and plots (homework, basic codes will be provided in C to be adapted)

 

 

Illustrative Bibliography:

• Gardiner: Handbook of Stochastic Methods (Springer).
• Grimmett, Stirzaker: Probability and Random Processes (Oxford).
• Grimmett: Probability on Graphs (CUP). (available online here)

 

Teaching:

Lectures per week	3 hours
Classwork sessions per week	1 hour
Module duration	10 weeks
Total contact hours	40
Private study and group working	140