# Courses Under Pure Mathematics University of Ghana UG

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###### PROGRAMME STRUCTURE

## LEVEL 100

**FIRST SEMESTER **

Core

Code | Title | Credits |

UGRC 150 | Critical Thinking and Practical Reasoning | 3 |

MATH 121 | Algebra and Trigonometry | 3 |

MATH 123 | Vectors and Geometry | 3 |

STAT 111 | Introduction to Statistics and Probability I | 3 |

Total | 12 | |

Electives (Select 3-4 credits) | ||

PHYS 105 | Practical Physics I | 1 |

PHYS 143 | Mechanics and Thermal Physics | 3 |

ABCS 101 | Introductory Animal Biology | 3 |

DCIT 101 | Introduction to Computer Science I | 3 |

ECON 101 | Introduction to Economics I | 3 |

* *

** **

**SECOND SEMESTER **

Core

Code | Title | Credits |

URGC 110 | Academic Writing I | 3 |

UGRC 130 | Understanding Human Society | 3 |

MATH 122 | Calculus I | 3 |

MATH 126 | Algebra and geometry | 3 |

STAT 112 | Introduction to Statistics and Probability II | 3 |

Total | 15 | |

Electives (Select 3-4 credits) | ||

PHYS 106 | Practical Physics II | 1 |

PHYS 144 | Electricity and Magnetism | 3 |

BOTN 104 | Growth of Flowering Plants | 3 |

DCIT 104 | Programming Fundamentals | 3 |

ECON 102 | Introduction to Economics II | 3 |

** **

**SINGLE MAJOR IN MATHEMATICS **

**LEVEL 200 **

**FIRST SEMESTER **

Core

Code | Title | Prerequisite- Pass in | Credits |

UGRC 210 | Academic Writing II | 3 | |

MATH 223 | Calculus II | MATH 122 | 3 |

MATH 225 | Vectors and Mechanics | MATH 122 | 3 |

STAT 221 | Introductory Probability I | 3 | |

3-6 credits from one other department from 100 level | 3-6 | ||

Total | 15-18 |

** **

**SECOND SEMESTER **

Core

Code | Title | Prerequisite- Pass in | Credits |

UGRC 220 | Liberal and African Studies | 3 | |

MATH 222 | Vector Mechanics | MATH 225 | 3 |

MATH 224 | Introductory Abstract Algebra | MATH 126 | 3 |

MATH 220 | Introductory Computational Mathematics | MATH 122 | 3 |

STAT 224 | Introductory Probability II | 3 | |

Total | 15 | ||

Electives | |||

3 credits from one other department from 100 level | |||

15-18 |

** **

**LEVEL 300 **

**FIRST SEMESTER **

Core

Code | Title | Prerequisite- Pass in | Credits |

MATH 351 | Linear Algebra | MATH 224 | 3 |

MATH 353 | Analysis I | MATH 223 | 3 |

MATH 355 | Calculus of Several Variables | MATH 223 | 3 |

^{i}MATH 350* | Differential Equations I | MATH 223 | 3 |

Total | 9-12 | ||

Electives (Select 6-9 credits) | |||

MATH 359 | Discrete Mathematics | MATH 224 | 3 |

MATH 361 | Classical Mechanics | MATH 222 | 3 |

MATH 363 | Introductory concepts of financial mathematics | MATH 223/STAT 221 | 3 |

STAT 331 | Probability distributions | STAT 221,224 | 3 |

**SECOND SEMESTER **

Core

Code | Title | Prerequisite- Pass in | Credits |

MATH 354 | Abstract Algebra I | MATH 224 | 3 |

MATH 356 | Analysis II | MATH 223 | 3 |

MATH 372 | Topology | MATH 353 | 3 |

^{i}MATH 350* | Differential Equations I | MATH 223 | 3 |

Total | 9-12 | ||

Electives (Select 6-9 credits) | |||

MATH 366 | Electromagnetic Theory I | MATH 222 | 3 |

MATH 362 | Analytical Mechanics | MATH 222 | 3 |

MATH 358 | Computational Mathematics I | MATH 220 | 3 |

MATH 368 | Introductory number theory | MATH 224 | 3 |

STAT 332 | Multivariate distributions | STAT 331 | 3 |

*Please note MATH 350 may be taken in either First or the Second Semester **LEVEL 400 **

**FIRST SEMESTER **

Core

Code | Title | Prerequisite- Pass in | Credits |

^{i}MATH 400 | Project | 3 | |

MATH 441 | Advanced Calculus | MATH 353 or MATH 351 | 3 |

MATH 440* | Abstract Algebra II | MATH 354 | 3 |

MATH 447 | Complex Analysis | MATH 223 | 3 |

Total | 9-12 | ||

Select at least 6 credits | |||

MATH 443 | Differential Geometry | MATH 355 | 3 |

MATH 445 | Introductory Functional Analysis | MATH 356 | 3 |

MATH 449 | Electromagnetic Theory II | MATH 366 | 3 |

MATH 451 | Introduction to Algebraic Field Theory | MATH 354 | 3 |

MATH 453 | Introduction to Quantum Mechanics | MATH 362 | 3 |

MATH 455 | Computational Mathematics II | MATH 358 | 3 |

MATH 457 | Mathematical Biology I | 3 |

Project may be replaced by two elective courses in mathematics

MATH 440 may be taken in either semester

**SECOND SEMESTER **

Core

Code | Title | Prerequisite- Pass in | Credits |

MATH 400 | Project | 3 | |

MATH 442 | Integration Theory and Measure | MATH 356 | 3 |

MATH 440* | Abstract Algebra II | MATH 354 | 3 |

Total | 6-9 | ||

Electives (Select a minimum of 9 credits) | |||

MATH 444 | Calculus on Manifolds | MATH 441 | 3 |

MATH 446 | Module Theory | MATH 440 | 3 |

MATH 448 | Special Relativity | MATH 362 | 3 |

MATH 452 | Introduction to Lie Groups and Lie Algebras | MATH 354 | 3 |

MATH 450 | Differential Equations II | MATH 350 | 3 |

MATH 458 | Mathematical Biology II | MATH 457 | 3 |

MATH 460 | Fourier series and Fourier transforms | MATH 356 | 3 |

**MAJOR **±** MINOR IN MATHEMATICS **

** **

**LEVEL 200 FIRST SEMESTER **

Core

Code | Title | Prerequisite- Pass in | Credits |

UGRC 210 | Academic Writing II | 3 | |

MATH 225 | Vectors and Mechanics | MATH 122 | 3 |

MATH 223 | Calculus II | MATH 122 | 3 |

Total | 9 | ||

Electives (Select a minimum of 3 credits) | |||

MATH 220* | Introductory Computational Mathematics | MATH 122 | 3 |

STAT 221 | Introductory Probability I | 3 |

**SECOND SEMESTER **

Core

Code | Title | Credits | |

UGRC 220 | Liberal and African Studies | 3 | |

MATH 224 | Introductory Abstract Algebra | MATH 126 | 3 |

Total | 6 | ||

Electives (Select a minimum of 3 credits) | |||

MATH 222 | Vector Mechanics | MATH 225 | 3 |

MATH 220* | Introductory Computational Mathematics | MATH 122 | 3 |

STAT 224 | Introductory Probability II | 3 |

Students take 6 credits each semester from their minor department

MATH 220 may be taken in either semester

** **

** **

**LEVEL 300 **

**FIRST SEMESTER **

Core

Code | Title | Prerequisite- Pass in | Credits |

MATH 351 | Linear Algebra | MATH 224 | 3 |

MATH 353 | Analysis I | MATH 223 | 3 |

MATH 355 | Calculus of Several Variables | MATH 223 | 3 |

Total | 9 |

**SECOND SEMESTER **

Core

Code | Title | Prerequisite- Pass in | Credits |

MATH 354 | Abstract Algebra I | MATH 224 | 3 |

MATH 356 | Analysis II | MATH 223 | 3 |

MATH 350 | Differential Equations I | MATH 223 | 3 |

MATH 372 | Topology | MATH 353 | 3 |

Total | 9-12 |

Students take 6 credits each semester from their minor department.

Students may choose to add an elective from the single subject elective list.

MATH 372 Topology may be done in level 400.

Minor students choose any two courses each semester

** **

**LEVEL 400 **

**FIRST SEMESTER **

Core

Code | Title | Prerequisite- Pass in | Credits |

^{i}MATH 400 | Project | 3 | |

MATH 441 | Advanced Calculus | MATH 351 or MATH 353 | 3 |

MATH 447 | Complex Analysis | MATH 223 | 3 |

Total | 9 | ||

Electives (Select 6-9 credits) | |||

MATH 440 | Abstract Algebra II | MATH 354 | 3 |

MATH 443 | Differential Geometry | MATH 355 | 3 |

MATH 451 | Introduction to Algebraic Field Theory | MATH 354 | 3 |

MATH 453 | Introduction to Quantum Mechanics | MATH 362 | 3 |

MATH 455 | Computational Mathematics II | MATH 358 | 3 |

MATH 445 | Introductory Functional Analysis | MATH 353 | 3 |

MATH 457 | Mathematical Biology I | 3 | |

MATH 449 | Electromagnetic theory II | MATH 366 | 3 |

**SECOND SEMESTER **

Core

Code | Title | Prerequisite- Pass in | Credits |

^{i}MATH 400 | Project | 3 | |

MATH 442 | Integration Theory and Measure | MATH 356 | 3 |

Total | 3 | ||

Electives (Select 12 credits) | |||

MATH 444 | Calculus on Manifolds | MATH 441 | 3 |

MATH 446 | Module Theory | MATH 440 | 3 |

MATH 448 | Special Relativity | MATH 362 | 3 |

MATH 452 | Introduction to Lie Groups and Lie Algebras | MATH 354 | 3 |

MATH 450 | Differential Equations II | MATH 350 | 3 |

MATH 458 | Mathematical Biology II | MATH 457 | 3 |

MATH 460 | Fourier series and Fourier transforms | MATH 356 | 3 |

*Project may be replaced by two elective courses in mathematics **COMBINED MAJOR **

**LEVEL 200 **

**FIRST SEMESTER **

Core

Code | Title | Prerequisite- Pass in | Credits |

UGRC 210 | Academic Writing II | 3 | |

MATH 225 | Vectors and Mechanics | MATH 122 | 3 |

MATH 223 | Calculus II | MATH 122 | 3 |

Total | 9 |

**SECOND SEMESTER **

Core

Code | Title | Prerequisite- Pass in | Credits |

UGRC 220 | Liberal and African Studies | 3 | |

MATH 224 | Introductory Abstract Algebra | MATH 126 | 3 |

Total | 6 | ||

Electives (Select a minimum of 3 credits) | |||

MATH 222 | Vector Mechanics | MATH 225 | 3 |

MATH 220 | Introductory Computational Mathematics | MATH 122 | 3 |

Students take 6-9 credits each semester from their other department

** **

**LEVEL 300 **

**FIRST SEMESTER **

Core

Code | Title | Prerequisite- Pass in | Credits |

MATH 351 | Linear Algebra | MATH 224 | 3 |

MATH 353 | Analysis I | MATH 223 | 3 |

MATH 355 | Calculus of Several Variables | MATH 223 | 3 |

Total | 9 |

**SECOND SEMESTER **

Core

Code | Title | Prerequisite- Pass in | Credits |

MATH 354 | Abstract Algebra I | MATH 224 | 3 |

MATH 356 | Analysis II | MATH 223 | 3 |

MATH 350 | Differential Equations I | MATH 223 | 3 |

Total | 9 |

Students take 9 credits each semester from their other department.

** **

**LEVEL 400 **

**FIRST SEMESTER **

Core

Code | Title | Prerequisite- Pass in | Credits |

MATH 441 | Advanced Calculus | MATH 351 or MATH 353 | 3 |

MATH 447 | Complex Analysis | MATH 223 | 3 |

Total | 6 | ||

Electives (Select 3-6 credits) | |||

MATH 440 | Abstract Algebra II | MATH 354 | 3 |

MATH 443 | Differential Geometry | MATH 355 | 3 |

MATH 451 | Introduction to Algebraic Field Theory | MATH 354 | 3 |

MATH 453 | Introduction to Quantum Mechanics | MATH 362 | 3 |

MATH 455 | Computational Mathematics II | MATH 358 | 3 |

MATH 445 | Introductory Functional Analysis | MATH 353 | 3 |

MATH 449 | Electromagnetic theory II | MATH 366 | 3 |

MATH 457 | Mathematical Biology I | 3 |

** **

**SECOND SEMESTER **

Core

Code | Title | Prerequisite- Pass in | Credits |

MATH 442 | Integration Theory and Measure | MATH 356 | 3 |

Electives (Select 6 credits) | |||

MATH 372 | Topology | MATH 353 | 3 |

MATH 444 | Calculus on Manifolds | MATH 441 | 3 |

MATH 446 | Module Theory | MATH 440 | 3 |

MATH 448 | Special Relativity | MATH 362 | 3 |

MATH 452 | Introduction to Lie Groups and Lie Algebras | MATH 354 | 3 |

MATH 450 | Differential Equations II | MATH 350 | 3 |

MATH 458 | Mathematical Biology II | MATH 457 | 3 |

MATH 460 | Fourier series and Fourier transforms | MATH 356 | 3 |

Students take 9 credits each semester from their other department.

Course Descriptions

**LEVEL 100 **

**MATH 121: Algebra and Trigonometry**

This course is a precalculus course aiming to develop the students ability to think logically, use sound mathematical reasoning and understand the geometry in algebra. It includes more advanced levels of topics addressed in secondary school such as: logic and proof, set theory. The concept and properties of functions such as surjectivity, injectivity, odd and even functions. General properties of the associated graph of a function.Trigonometric functions, their inverses, their graphs, circular measure and trigonometric identities.

**MATH 123: Vectors and Geometry **

Vectors may be used very neatly to prove several theorems of geometry. This course is about applying vector operations and the methods of mathematical proof (of MATH 121) to geometric problems. The areas of study include: vector operations with geometric examples; components of a vector and the scalar product of vectors. Coordinate geometry in the plane including normal vector to a line, angle between intersecting lines, reflection in a line, angle bisectors and the equation of a circle, the tangent and the normal at a point.

** **

**MATH 122: Calculus I **

This is a first course in calculus introducing the elementary idea of limit, continuity and derivative of a function. Rules of differentiation. Applications of differentiation. Derivative of the elementary and transcendental functions. Some methods of integration. Improper integrals. Applications of integration. Formation of differential equations and solution of first order differential equations both separable variable type and using an integrating factor.

**MATH 126: Algebra and Geometry **

This is a course which highlights the interplay of algebra and geometry. It includes topics such as: polar coordinates; conic sections. Complex numbers, Argand diagram, DeMoivre’s theorem, roots of unity. Algebra of matrices and determinants, linear transformations. Transformations of the complex plane. Sketching polar curves and some coordinate geometry in 3 dimensions. Vector product and triple products.

**MATH 101: General Mathematics (Non-Mathematics students)**

The aim of this course is to equip students with sufficient elementary algebra and calculus to allow them to solve elementary problems in the biological and physical world. Topics from secondary school are revised and in some cases extended. The main focus is to provide sufficient precalculus and trigonometry to allow students to apply calculus to problem solving.

**LEVEL200**

**MATH 223: Calculus II-Prerequisite pass in MATH 122 **

The first and the second derivatives of functions of a single variable and their applications. Integration as a sum; definite and indefinite integrals; improper integrals. The logarithmic and exponential functions, the hyperbolic functions and their inverses. Techniques of integration including integration by parts, recurrence relations among integrals, applications of integral calculus to curves: arc length, area of surface of revolution. Ordinary differential equations: first order and second order linear equations with constants coefficients. Applications of first order differential equations.

**MATH 225: Vectors and Mechanics **

This is a first course in the applications of differentiation and integration of vector functions of a scalar variable. Kinematics of a single particle in motion, displacement, velocity and acceleration. Relative motion. Concept of a force, line of action of a force, Newton’s laws of motion. Motion in a straight line, motion in a plane, projectiles, circular motion. Work, energy, power. Impulse and linear momentum. Moment of a force and couple, conditions for equilibrium of rigid bodies.

**MATH 222: Vector Mechanics **

Vector functions of a scalar variable; further differentiation and integration; Serret-Frenet formulae; differential equations of a vector function. Motion of a particle; Kinematics, Newton’s laws; concept of a force; work, energy and power; impulse and momentum, conservation laws of energy and linear momentum. Rectilinear motion, motion in a plane. The two-body problem, variable mass.

**MATH 224: Introductory Abstract Algebra -Prerequisite pass in MATH 126 **an axiomatic presentation of mathematics. Among the topics to be discussed are notions of relations on sets, equivalence relations and equivalence classes as well as the concept of partial ordering. The system of real numbers and their properties will be discussed. The principle of induction will be reviewed. An introduction to number theory will be given as numbers are the most familiar mathematical objects. The course seeks also to introduce axiomatically defined systems such as groups, rings and fields, and vector spaces.

**MATH220: Introductory Programming for Computational Mathematics**

This course is in two parts. The first part is an introduction to programming using the python programming language. This part of the course begins with the basics of python. Vectorization, and visualization in python are also treated. The second part is an introduction to solving mathematical problems numerically. These problems include finding the roots of nonlinear equations, solving large systems of linear equations and fitting polynomials to data. By the end of this course, students will be able to use python to solve basic mathematical problems.

**LEVEL 300 **

**MATH 350: Differential Equations I-Prerequisite MATH 223 **

Differential equations can be studied analytically, numerically and qualitatively. The focus of this course is to find solutions to differential equations using analytic techniques. Differential forms of 2 and 3 variables. Exactness and integrability conditions. Existence and uniqueness of solution. Second order differential equations with variable coefficients. Reduction of order, variation of parameters. Series solutions. Ordinary and regular singular points. Orthogonal sets of functions. Partial differential equations.

**MATH 351: Linear Algebra-Prerequisite MATH 224 **

We will develop a core of material called linear algebra by introducing certain definitions, creating procedures for determining properties and proving theorems. Although the student will be doing some computations, the goal in most problems is not merely to get the ³ how to get the answer and then interpret the result. Topics to be discussed include: spanning sets; subspaces, solution spaces. Bases. Linear maps and their matrices. Inverse maps. Range space, rank and kernel. Eigenvalues and eigenvectors. Diagonalization of a linear operator. Change of basis. Diagonalizing matrices. Diagonalization theorem. Bases of eigenvectors. Symmetric maps, matrices and quadratic forms.

**MATH 353: Analysis I-Prerequisite MATH 223**

This is the first rigorous analysis course. Topics to be discussed include: normed vector spaces, limits and continuity of maps between normed vector spaces. Students will be expected to produce proofs to justify their claims. We study the algebra of continuous functions. Bounded sets of real numbers. Limit of a sequence. Subsequences. Series with positive terms. Convergence tests. Absolute convergence. Alternating series. Cauchy sequences and complete spaces.

**MATH 354: Abstract Algebra I-Prerequisite MATH 224 **

The primary aim of Math 354 is to study groups and their properties. We shall develop the foundations of group theory and study some notable groups like cyclic groups, permutation groups, finite Abelian groups and their characterization. Other ideas include: subgroups, cyclic groups.The Stabilizer-Orbit theorem. Lagrange’s theorem. Classifying groups. Structural properties of a group. Cayley’s theorem. Generating sets. Direct products. Finite abelian groups. Cosets and the proof of Lagrange’s theorem. Proof of the Stabilizer-Orbit theorem.

**MATH 355: Calculus of Several Variables-Prerequisite MATH 223 **

The major goal for this course is to understand and apply the concepts of differentiation and integration to functions of several variables. Functions of several variables and partial derivative. Directional derivative, gradient. Local extema, constrained extrema. Lagrange multipliers. The gradient, divergence and curl operators. Line, surface and volume integrals. Green’s theorem, divergence theorem, Stokes’ theorem.

**MATH 356: Analysis II-Prerequisite MATH 223 **

This is a continuation of MATH 353. We now consider vector spaces of functions and discuss convergence of sequences of functions; pointwise and uniform convergence. Other topics discussed include; power series, the contraction mapping theorem and applications. We examine the definition of the Riemann integral and conditions for integrability. We give a proof of the fundamental theorem of calculus and other major basic results involved in its proof. We finish with some point set topology in R.

**MATH 358: Computational Mathematics I-Prerequisite MATH 220**

This course is a sequel to MATH 220. In this course, we continue the solution of linear systems by treating matrices with special structures. We also continue with data fitting using polynomials. Several high order methods for discretizing the derivative and definite integral are also treated. The course ends with approximations of eigenvalues for large matrices. We explain the concept of the dominant eigenvalue and its eigenvector. We also look at simultaneous approximation of eigenvalues.

**MATH 359: Discrete Mathematics-Prerequisite MATH 224 **

This course is a study of discrete rather than continuous mathematical structures. Topics include: asymptotic analysis and analysis of algorithms, recurrence relations and equations, Counting techniques (examples include: Inclusion-exclusion and pigeon-hole principles and applications, Multinomial theorem, generating functions), Elementary number theory and cryptography, Discrete probability theory and graph theory; including planarity, Euler circuits, shortest-path algorithm. Network flows. Modelling computation:languages and grammars, models, finite state machines, Turing machines.

**MATH 361: Classical Mechanics -Prerequisite MATH 222**

The methods of classical mechanics have evolved into a broad theory of dynamical systems and therefore there are many applications outside of Physics; for example to biological systems. Topics to be discussed will include 1-dimensional dynamics: damped and forced oscillations. Motion in a plane: projectiles, circular motion, use of polar coordinates and intrinsic coordinates. Two-body problems, variable mass. Motion under a central, non-inertial frame. Dynamics of a system of particles.

**MATH 362: Analytical Mechanics- Prerequisite MATH 222 **

In this course the student is introduced to a collection of closely related alternative formulations of classical ** mechanics. It ** provides a detailed introduction to the key analytical techniques of classical mechanics. Topics discussed include rigid body motion, rotation about a fixed axis. General motion in a plane, rigid bodies in contact, impulse. General motion of a rigid body. Euler-Lagrange equations of motion.

**MATH 363: Introductory Concepts in Financial Mathematics-Prerequisite **

**MATH 223, STAT 221 **

This course introduces the basic methods applied in financial mathematics. We will discuss probability functions, stochastic processes, random walks and martingales; Ito’s lemma and stochastic calculus. Students will understand the stochastic differential equations for a geometric Brownian motion process. We will study mean reverting models such as the Ornstein- Uhlenbeck process, as well as stochastic volatility models such as the Heston Model. Stochastic models for stock pricing are also discussed; we study a binomial option pricing model, the Black-Scholes model and the capital asset pricing model.

** **

**MATH366: Electromagnetic Theory I-Prerequisite MATH 222**

This course develops the mathematical foundations for the application of the electromagnetic model to various problems. Mathematics discussed includes scalar and vector fields, grad, div and curl operators. Orthogonal curvilinear coordinates. Electrostatics: charge, Coulomb’s law, the electric field and electrostatic potential, Gauss’s law, Laplace’s and Poisson’s equations. Conductors in the electrostatic field. Potential theory.

**MATH 368: Introductory Number Theory-Prerequisite MATH 224 **

This course builds on the elementary number theory introduced in MATH 224 Topics include: the Fundamental theorem of Arithmetic, Proof and Application: GCD, LCM. Asymptotic notations, Congruences, Residue systems and Euler Phi-function, linear congruence, Chinese Remainder theorem, Theorems of Euler, Fermat and Wilson. Arithmetic functions and Dirichlet mu inversion formula, averages of arithmetic functions. Quadratic residues and quadratic reciprocity law, the Jacobi symbol. Prime Number distribution.

**MATH 372: Topology **±**Prerequisite MATH 353 **

This is a first course in point set topology. Students will be introduced to topological spaces and be able to identify open and closed sets with respect to the given topology. Other aspects to be discussed are basis for a topological space. Separation and countability properties. Limit points. Connectedness. Subspace topology. Homeomorphism. Continuity. Metrizability. Continuity via convergent sequences. Compactness.

**MATH 440: Abstract Algebra II-Prerequisite MATH 354 **

This is a second course in group theory. Topics covered will include: finite groups, Sylow theorems and simple groups. Composition series. We state and prove the Zassenhaus Lemma, the Schreier theorem and the Jordan-Hölder theorem. Direct and semi-direct products. Abelian groups, torsion, torsion-free and mixed abelian groups. Finitely generated group and subgroups. p-groups, nilpotent groups and solvable groups.

**MATH 441: Advanced Calculus-Prerequisite MATH 351 or MATH 353**

Here we think of differentiation as a process of approximating the function *f *near* a*, by a linear map. This linear map is called the Frèchet derivative of *f *at *a*. The main aim of this course is to understand two of the most important theorems for modern analysis: the Inverse Map Theorem and the Implicit Function Theorem. Other ideas include: linear and affine maps between normed vector spaces. Limits, continuity, tangency of maps and the derivative as a linear map. Component-wise differentiation, partial derivatives, the Jacobian as the matrix of the linear map. Generalized mean value theorem.

**MATH 442: Integration Theory and Measure **±**Prerequisite MATH 356 **

Algebra of sets, measurable sets and functions, measures and their construction (in particular Lemma, Monotone Convergence Theorem, Lebesgue Dominated Convergence Theorem. Lebesgue spaces, elementary inequalities, modes of convergence. Product measures and Generalisation of the Riemann (R) integral (eg Kurzweil-Henstock (KH) integral). Lebesgue (L) integral. Relationship between the KH-integrable, L-integrable and Rintegrable functions.

**MATH 443: Differential Geometry-Prerequisite MATH 355 **

The modern approach to differential geometry uses the language of manifolds. This provides a theory and a variable free notation which frees us from always having to consider the coordinate system. We want to be able to deal with the elements of calculus both invariantly (i.e. independently of the local coordinates) and intrinsically (i.e. independently of the way a geometric object is embedded in Euclidean space). But to appreciate the great contribution to differential geometry made by the theory of manifolds, we will first study classical differential geometry and then a little of the modern approach.

**MATH 444: Calculus on Manifolds-Prerequisite MATH 441 **

This course aims to provide an introduction to Differentiable Manifolds and the tools for performing calculus on these objects; tangent vectors and differential forms. We will see how concepts like the derivative in R^{n} is extended to a smooth *n*-dimensional manifold. Topics include: manifolds and submanifolds, differentiability of maps between manifolds, the tangent space, the tangent bundle and the tangent functor. Vector bundles. The exterior algebra, the notion of a differentiable form on a manifold, singular n-chains and integration of a form over a chain. Partition of unity. Application to Stokes’ theorem.

**MATH 445: Introductory Functional Analysis-Prerequisite MATH 356**

Finite dimensional normed vector spaces. Equivalent norms. Banach spaces. Infinitedimensional normed vector spaces-Hamel and Schauder bases; separability. Compact linear operators on a Banach space. Complementary subspaces and the open-mapping theorem. Closed Graph theorem. Hilbert spaces, their special subspaces and the dual space. The completion of a normed vector space. Reflexive Banach spaces.

**MATH 446: Module Theory-Prerequisite MATH 440 **

In this course we shall study the mathematical objects called modules. The use of modules was pioneered by one of the most prominent mathematicians, Emmy Noether (a German), who led the way in demonstrating the power and elegance of this structure. Topics include: modules, submodules, homomorphism of modules. Quotient modules, free (finitely generated) modules. Exact sequences of modules. Direct sum and product of modules. Chain conditions, Noetherian and Artinian modules. Projective and injective modules. Tensor product, categories and functors. Hom and duality of modules.

**MATH 447: Complex Analysis-Prerequisite MATH 223 **

The objective of this course is to introduce students to complex numbers and functions of a complex variable. We introduce the notions of differentiability, analyticity and integrability for a function defined on the complex plane. We also look at ways in which one can integrate complex-valued functions. Elementary topology of the complex plane. Complex functions and mappings. The derivative and harmonic functions. Integrals. Maximum modulus, CauchyGorsat and Cauchy theorems. Applications of the theorems. Taylor and Laurent series, zeros and poles of a complex function. Residue theorem and consequences. Conformal mapping, analytic continuation.

**MATH 448: Special Relativity- Prerequisite MATH 362 **

By employing the mathematics of sets, mappings and relations we aim to develop an ability to think relativistically. Topics include: Galilean relativity, postulates of special relativity; Lorentz transformations. Lorentz-Fitzgerald contraction, time dilation. 4-vectors, relativistic mechanics, kinematics and force, conservation laws; decay of particles; collision problems, covariant formulation of electrodynamics.

**MATH 449: Electromagnetic Theory II-Prerequisite MATH 366 **

This is a second course in the development of the mathematical foundations for the application of the electromagnetic model to various problems. Magnetostatics: steady currents, heating affect and magnetic field, magnetic vector potential, magnetic properties of matter, dipoles, induced magnetism, permanent magnetism. Time-varying fields: electromagnetic induction. ‘LIIHUHQWLDOIRUPRI)DUDGD\¶VODZHQHUJ\LQHOHFWURPDJQHWLFILHOGV0D[ZHOO¶VHTXDWLRQVDQG their consequences Poynting vector; electromagnetic potentials formation of electrodynamics.

**MATH 450: Differential Equations II-Prerequisite MATH 350 **

This course introduces undergraduate students to the qualitative theory of Ordinary Differential Equations. We will use the Picard-Lindelöf Theorem to analyze whether an ODE or a system of ODEs has a solution and the behavior of the solution as the parameter is varied (bifurcation). We will especially consider autonomous linear and nonlinear systems and investigate the stability of the solutions that result. We will also introduce the concept of a Lyapunov function. Other topics might include partial differential equations, the method of characteristics and classification.

**MATH 451: Introduction to Algebraic Field Theory-Prerequisite MATH 354 **

The famous problems of squaring the circle, doubling the cube and trisecting an angle captured the imagination of both professional and amateur mathematicians for over two thousand years. Despite the enormous effort and ingenious attempts by these men and women, the problems would not yield to purely geometrical methods. It was only the development of abstract algebra in the nineteenth century which enabled mathematicians to arrive at the surprising conclusion that these problems are impossible. Topics include: algebraic numbers. Extending fields. Towers of fields. Irreducible polynomials. Constructible numbers and fields. 7UDQVFHQGHQFHRIʌDQG*e.* Residue rings and fields.

**MATH 452: Introduction to Lie Groups and Lie Algebras- Prerequisite MATH 354 **This course will cover the basic theory of Lie groups and Lie algebras. Topics may include: topological groups and Haar measure, vector fields and groups of linear transformations. The exponential map. Linear groups and their Lie algebras. Structure of semi-simple Lie algebras, Cartan and Iwasawa decompositions. Manifolds, homogeneous spaces and Lie groups. Integration and representations.

**MATH 453: Introduction to Quantum Mechanics-Prerequisite MATH 362**

Principle of least action, Hamilton’s equation, Poisson brackets. Liouville’s equation. Canonical transformations. Symmetry and conservation laws. Postulates of quantum mechanics, the wave formalism. Dynamical variables. The Schrödinger equation in one-dimension; free particles in a box, single step and square well potentials. Orbital angular momentum. The 3-dimensional Schrödinger equation; motion in a central force field, the 3-d square well potential, the hydrogenic atom. Heisenberg’s equation of motion, harmonic oscillator and angular momentum.

**MATH 455: Computational Mathematics II-Prerequisite MATH 358**

This course looks at methods of discretizing and solving differential equations. It begins with the solution of initial value problems for ordinary differential equations. We start with the Euler methods and systematically develop high order solutions for solving problems. The course then develops the concept of finite differences to solve boundary value problems. In addition, we look at the problem of discretizing partial differential equations in space and time both implicitly and explicitly.

**MATH 457: Mathematical Biology I **

In this course we focus on 3 types of biological phenomena to be modelled, namely single species population dynamics, interacting species and molecular dynamics. In single species population dynamics we will use difference equations, graphical analysis, fixed points and linear stability analysis. First order systems of ordinary differential equations: logistic equation, steady states, linearisation, and stability. We will examine applications to harvesting and fisheries. For interacting species we examine systems of difference equations (hostparasitoid systems) and systems of ordinary differential equations (predator-prey and competition models). Finally, we will consider biochemical kinetics, Michaelis-Menten kinetics and metabolic pathways, activation and inhibition.

**MATH 458: Mathematical Biology II- Prerequisite MATH 457 **

The detail of this course may be informed by the student choice(s) of project topic and could include: (i) modelling of biological systems using partial differential equations. Derivation of conservation equations. Different models for movement (e.g. diffusion, convection, directed movement). Connection between diffusion and probability. (ii) Linear reaction-diffusion equations. Fundamental solution for linear diffusion equations. Speed of a wave of invasion. Non-linear reaction-diffusion equations. Travelling wave solutions for monostable equations (e.g. Fisher equation). Travelling wave solutions for bistable equations. (iii) Systems of reaction-diffusion equations. Travelling wave solutions for systems of reaction-diffusion equations. Pattern formations in systems of reaction-diffusion equations. Pattern formations in chemotaxis equations. (iv) Mathematical modelling of infection diseases (SIR). Derivation of a simple SIR model. Travelling wave solutions for the simple SIR model. Generalisation of the simple SIR model. Stochastic SIR model.

**MATH 460: Fourier Series and Fourier Transforms-Prerequisite MATH 356 **

The objective of this course is to introduce the theory of Fourier series and Fourier transforms on the real line. Topics include: convolutions, summability kernels, convergence of Césaro means. Mean-square convergence, pointwise convergence. Fourier transform on the real line, inversion formula, Plancherel formula, Weierstrass approximation theorem. Applications to partial differential equations, Poisson summation formula. The Heisenberg uncertainty principle.