## Course Content » 3rd Year Modules

Choose a module from the table below:

 MAT311 MAT312 MAT321 MAT322

###### MAT311 Real Analysis & Group Theory

30 credits

Pre-requisites: [MAT211 and MAT221] OR [MAM211 and MAM221 and MAM231 and MAM241 (pre-2010)]
Bounded subsets of real numbers.
Suprema, Supremum property, Existence of irrational roots.
Inner product and normed spaces.
Open sets, Nested Cells Theorem, Cluster Point.
Bolzano-Weierstrass Theorem (for sets),Compactness.
Heine-Borel Theorem, Connectedness.
Sequences, Convergence and Subsequences, Monotone Convergence Theorem, Bolzano-Weierstrass Theorem for Sequences, Cauchy Sequences, Global Continuity Theorem.
Preservation of compactness and connectedness.
Uniform continuity, Fixed point theorems.
Operations, Groups & subgroups, Normal subgroups and factor groups, Cyclic groups.
Homomorphisms & fundamental homomorphism theorem, Permutation groups, Cayley's theorem.

Main Outcomes

On completing this module students will be able to

Explain the notions of open subset, boundedness and finiteness, connectedness and compactness in Euclidean space.
Explain the notions of continuity and uniform continuity of functions between subspaces of Euclidean space, and in relation to sequences, compactness and connectedness.
Use their knowledge of the basic theory of groups to provide examples and counter-examples of various concepts.
Carry out proofs of mathematical statements.
Use new knowledge in unfamiliar but similar situations.
Relevant People ###### MAT312 Environmental Modeling

30 credits

Pre-requisites: [MAT211 and MAT221] OR [MAM211 and MAM221 and MAM231 and MAM241 (pre-2010)]
Population Growth Models: Single Species Models (Exponential and Logistic Growth Models. Applications and Case Studies).
Phase Plane Analysis of Dynamical Systems: Equilibrium Solutions and Stability; Direction Fields, Phase Portraits. Null-clines. Stability of Linear and Almost Linear systems.
Interacting Species Models: Predator-Prey Models, Competing Species Models, Co-operating Species Models. Multi-Species Models, Food Chains.
Mathematical Models of Epidemics: The SIR, SIRS, SEIR, SIS and SI models for infectious diseases. The Basic Reproduction Number R0. Analysis of disease-free equilibrium and endemic equilibrium. Applications to Models for Malaria, Measles, TB and HIV transmission.
Environmental Modeling.
Dynamic optimization: discrete case, Euler's equation in variational calculus,Hamiltonian method.
Natural resources: Fishing, Forestry, Mining.
Groundwater flow: Darcy's law, Laplace's equation, Dupuit's theory, pumping from a well.
Diffusion: in 1,2,3 dimensions, instantaneous and continuous sources, reflecting boundaries.
Numerical methods and basics of PDEs for the topics mentioned above.

Main Outcomes

On completing this module students will be able to

Use their knowledge to develop mathematical models for single species as well as interacting species.
Use their knowledge to determine conditions for mutual co-existence, or extinction of interacting species.context.
Use Green's Theorem to evaluate line integrals.
Use their knowledge to formulate mathematical models for the transmission of certain diseases.
Use their knowledge to determine threshold values disease-free and endemic equilibria.
Use their knowledge to suggest possible ways of combating the spread of certain diseases.
Use their knowledge of dynamic optimization, discrete and continuous, analytic and numerical solutions in problem solving.
Use their knowledge to model situations such as natural resource exploitation, contaminant transport (groundwater flow and diffusion) and other environmental problems.
Relevant People ###### MAT321 Complex Analysis & Ring Theory

30 credits

Pre-requisites: [MAT211 and MAT221] OR [MAM211 and MAM221 and MAM231 and MAM241 (pre-2010)]
Complex numbers and properties.
Elementary complex functions and properties.
Analytic functions.
Contour integral and properties.
Taylor and Laurent series.
Residue theory.
Rings, subrings, integral domains, fields, field of quotients.
Quotient rings, prime ideals, maximal ideals.
Polynomial rings, factorization, irreducibility tests.
Field extensions.
Finite field construction.
Constructibility by ruler and compass.

Main Outcomes

On completing this module students will be able to

Use properties of complex numbers and complex functions in problem solving.
Use the properties of analytic functions to solve problems.
Use Taylor and Laurent series to solve problems.
Use Residue theory to solve problems.
Generalize concepts from Groups to Rings.
Use different methods to test for irreducibility of polynomials.
Use the theory to construct finite fields.
Use their knowledge of field extensions to prove the impossibility of certain geometric constructions using ruler and compass.
Relevant People ###### MAT322 Financial Modeling

30 credits

Pre-requisites: [MAT211 and MAT221] OR [MAM211 and MAM221 and MAM231 and MAM241 (pre-2010)]
Elementary probability theory.
Normal random variables.
Geometric Brownian motion.
Present value analysis.
Pricing contract via arbitrage, The arbitrage theorem.
Introduction to derivatives.
The Black-Scholes formula.
Vanilla options.
Multi-period binomial model method.
Option valuations by expected utility.
Exotic options.
Portfolio optimization.
Autoregressive models and mean reversion.
Simulations-random walk, Monte Carlo methods.

Main Outcomes

On completing this module students will be able to

Implement the basic tools of the geometric Brownian motion of stock prices.
Use their knowledge of elementary cases of pricing via arbitrage of options to solve problems.
Use their knowledge of vanilla and exotic options and their valuation to solve problems.
Use the Black-Scholes formula to solve problems.
Use their knowledge of Monte Carlo simulation to solve problems.
Use their knowledge to pose a problem numerically.
Use different techniques to discretise partial differential equations and associated boundary conditions.
Implement different techniques to solve finite difference problems.
Use different methods for convergence and stability analyses.
Use MAPLE/ MATLAB in problem solving.
Relevant People 