Teaching

Professor Mauro Mobilia
Applied Mathematics

Teaching qualification: University of Leeds Teaching Award Professional Standard 2 and Fellow of The Higher Education Academy (FHEA)

MATH0390: Foundation Pure and Applied Mathematics (Oct-Dec 2019)

General information on MATH0390: here and on Minerva/VLE

Syllabus:
1. Linear and quadratic relations: Equations for straight lines; intersection of two lines. Linear relations to model socio-economic scenarios: supply and demand; equilibrium price. Quadratic relations and revenue optimisation.
2. Indices and powers: Index laws; examples and applications.
3. Exponential and logarithm functions: Definition and graphs of exponential functions; the number e and its exponential; definition and graphs of logarithm functions. Logarithm laws and relationship between exponential and logarithm functions; applications.
4. Percentages, geometric series, and applications to interest & savings: Percentages. Simple and compound interest; discounting. Geometric series and application to regular savings.
5. Differentiation and applications: Differentiation from first principles and slope (gradient) of a curve. Tangents and normals to a curve. Maxima and minima of a function; stationary and turning points; distinguishing maxima and minima; application to profit maximisation.
6. Integration and applications: Integration as the reverse of differentiation, indefinite integral. Integration as a summation; definite integral and area under the graph; application to consumer surplus.

Objectives of the courses: Darwin’s natural selection paradigm is a cornerstone of modern evolutionary biology and ecology. Darwinian ideas have applications in social and behavioural science, and have also inspired research in the mathematical and physical sciences. In the last decades, mathematical analysis and theoretical modelling have led to tremendous progress in the quantitative understanding of evolutionary phenomena. Yet, many questions of paramount importance, like the “origin of cooperative behaviour” or “what determines biodiversity”, are subjects of intense research and their investigation requires advanced mathematical and computational tools. The students of the co-taught MATH3567/MATH5567M modules will be exposed to fundamental ideas of evolutionary modelling. These will be introduced through influential models and paradigmatic examples that will be analysed by a combination of methods drawn from the theory of nonlinear dynamics and stochastic processes. The students of these modules will thus be introduced to some areas of applied mathematics that currently give rise to exciting new developments and prominent challenges in mathematical biology and in evolutionary dynamics.

Provisional schedule:
1. Introduction to evolutionary modelling; 2. Modelling with difference equations; 3. Modelling with ordinary differential equations; 4. Introduction to Mendelian genetics; 5. Introduction to game theory; 6. Evolutionary games; 7. Random processes: Discrete-time Markov chains; 8. Random processes: Continuous-time Markov chains 9. Evolutionary game theory in finite populations; 10. Diffusion theory (Fokker-Planck); 11. Application of diffusion processes to population genetics.