### Degree Type:

Doctor of Philosophy### Department:

Department of Mathematics### Programme Duration:

3 years (Standard Entry)

### Modes of Study:

Regular### Entry Requirements:

A good M.Phil or MSc. degree in Mathematics from a recognized university.

### Career Opportunities:

Not Published

## Programme Structure

### Level 800

### First Semester

## MAT 801: General Topology

** This course is about the study of properties of topological spaces. Topological spaces turn up naturally in mathematical analysis, abstract algebra and geometry. A topological space is a structure that allows one to generalize concepts such as convergence, connectedness and continuity. Topics covered include: open and closed sets, neighbourhood, basis, convergence, limit point, completeness, compactness, connectedness, continuity of functions, separation axioms, subspaces, product spaces, and quotient spaces.**

## MAT 803: Functional Analysis I

This course covers major theorems in Functional Analysis that have applications in Harmonic and Fourier, Ordinary and Partial Differential Equations. Topics covered include: Hilbert space as an infinite dimensional generalization of geometric spaces; linear closed subspaces and orthogonality, linear transformations, projections, and spectral theory.

## MAT 805: Ordinary Differential Equations I

This course presents the student with advanced techniques for analysing the behaviour of solutions of ordinary differential equations. Topics include systems of first order linear differential equations, existence and uniqueness of solutions; adjoint systems, linear system associated with a linear homogeneous differential equation of order n, adjoint equation to a linear homogeneous differential equation, Lagrange Identity, linear boundary value problems on a finite interval; homogeneous boundary value problems and Green’s function; non-self-adjoint boundary value problems, self-adjoint eigenvalue problems on a finite interval, the expansion and completeness theorems, oscillation and comparison theorem for second-order linear equations and applications.

## MAT 807: Modern Algebra

This course focuses on traditional algebra topics that have found greatest application in science and engineering as well as in mathematics. Topics include direct product of groups, finite abelian groups, sylow theorem, finite simple groups, polynomial rings, ordered integral domain, extension fields, algebraic extensions, bilinear and quadratic forms, real and complex inner product spaces, the spectral theory and normal operators.

## MAT 809: Advanced Topics in Operations Research

This course serves as an introduction to the field of operations research. It will quip students with scientific approaches to decision-making and mathematical modelling techniques required to design, improve and operate complex systems in the best possible way. Topics include the formulation of linear programming models: goal programming, transportation problem, case study. Further topics are mathematical programming: project planning and control, dynamic programming, integer programming, probabilistic models: application of queuing theory, forecasting and simulation, decision analysis (making hard decisions), and multi-criteria decision making.

## MAT 811: Partial Differential Equations

This course focuses on partial differential equation (PDE) models, which will be developed in the context of modelling heat and mass transport and, in particular, wave phenomena, such as sound and water waves. This course develops students' skills in the formulation, solution, understanding and interpretation of PDE models. Topics include classical theory of partial differential equations together with the modern theory based on functional analysis; systems of partial differential equations, boundary value problems, stability and convergence; applications to the classical problems of mathematical physics.

## MAT 813: Mathematical Epidemiology

This course will examine applications of mathematics in biological contexts including genetics, ecology, physiology, neuroscience and epidemiology. Topics include variants of the MSEIRS epidemic models, disease-free and endemic equilibrium points, determination of the basic reproduction number using the next-generation matrix approach, local stability and global stability analysis of equilibrium points and case studies : HIV/AIDS, TB and Vector-Host Models including Malaria. Further topics are parameter estimation for selected epidemic models, simulation and prediction.

## MAT 815: Computational Linear Algebra

This course is an introduction to numerical Linear Algebra. Topics include: matrix factorizations: QR-factorization, Cholesky factorization , vector and matrix norms: properties of the ‖.‖1, ‖.‖2|| and ‖.‖ norms of vectors in Rn, properties of the ‖.‖1, ‖.‖2|| , ‖.‖ and ‖.‖F norms of an mxn matrix, condition number of a matrix, ill-conditioned systems, the Hilbert matrix, perturbation analysis of linear systems, singular value decomposition (SVD) of an mxn matrix, Moore-Penrose inverse, rank k approximation of a matrix, applications of the SVD to least-squares problems, iterative methods for large sparse linear systems: the Jacobi and Gauss-Seidel methods, the SOR method, applications to the solution of linear systems with banded coefficient matrices, regularization methods for ill-conditioned linear systems, regularization of orders 0, 1 and 2, and the L-curve method for choosing an optimal regularization parameter.

### Second Semester

## MAT 804: Functional Analysis II

This course covers major theorems in Functional Analysis that have applications in Harmonic and Fourier, Ordinary and Partial Differential Equations. Topics covered include: linear spaces, semi-norms, norm, locally convex spaces, linear functional, Hahn-Banach theorem, factor spaces, product spaces conjugate spaces, linear operators, and adjoints.

## MAT 806: Ordinary Differential Equations II

This course presents the student with advanced techniques for analysing the behaviour of solutions of ordinary differential equations. Topics include linear systems with isolated singularities, linearisation of systems of differential equations, asymptotic behaviour of non-linear systems: stability, perturbation of systems having a periodic solution, perturbation theory of two-dimensional real autonomous systems.

## MAT 808: Boundary Condition Functions

This course introduces students to the construction of Green’s functions for boundary value problems. Topics include boundary condition functions for self-adjoint and non-self-adjoint boundary value problems, construction of Green’s functions in terms of boundary condition functions, aymptotic behaviour of boundary condition functions and Green’s functions, and singular self-adjoint boundary value problem.

## MAT 810: Complex Analysis

This course provides advanced topics in complex analysis such as conformal mappings, and physical applications of conformal mapping. Further topics include analytic continuation and Riemann surfaces, Rouches’s theorem and principle of the argument, mapping properties of analytic functions, inverse function theory, maximum modulus theory, infinite products and gamma function, Sterling’s formula and Bessel’s functions, homotopy and homology theory, and analytic properties of holomorphic functions.

## MAT 812: Image Processing

This course introduces the basic theories and methodologies of digital image processing. The topics include manipulating images in MATLAB/OCTAVE, images as Arrays of Numbers, digital image, compression the singular value decomposion, the image de-blurring problem: a mathematical model of the blurring process. Further topics include de-blurring using a general linear model, obtaining the point spread function (PSF). De-bluring images using TSVD method, total variation method, and the Tikhonov regularization method, general image reconstruction as an inverse problem.

## MAT 814: Optimization

This course introduces the principal algorithms for linear, network, discrete, nonlinear, dynamic optimization and optimal control. Emphasis is on methodology and the underlying mathematical structures. Topics include unconstrained optimization: optimality conditions, Newton's method, quasi-Newton's methods, Steepest Descent Method, Conjugate-Gradient methods, Line Search methods, Trust Region Methods, Derivative-Free Methods, constrained Optimization: optimality conditions for (a) linear equality constraints, (b) linear inequality constraints, (c) nonlinear constraints, feasible-point methods, sequential quadratic programming (SQP), reduced-gradient method, penalty and barrier methods.

## MAT 816: Optimal Control

This course investigates how dynamical systems should be controlled in the best possible way. Topics include: OCP with bounded and unbounded controls. Bang-Bang controls, Singular controls. OCPs with linear and nonlinear dynamical systems. OCPs for systems with fixed or free terminal times. OCPs for systems with equality and inequality constraints on functions of state and control variables. Numerical Methods for OCPS: Control parametrization method, State Discretization methods, Lenhart's Forward-Backward Sweep method. Application to the conrol of dynamical systems, including the control of infectious diseases.

## MAT802: Measure and Integration

This course covers advanced topics in abstract measure theory and Lebesgue integration. Topics covered include: measurable sets and functions, measure spaces, Lebesgue integral, monotone convergence theorem, Fatou’s lemma, Lebesgue dominated convergence theorem, Vitali’s theorem, decomposition of measures, Caratheordory and Hahn extension theorem, spaces, Riesz representation theorem, and product measures.