Program Overview

Mathematics and Computer Science

Overview

Mathematics is the universal language of science while computer science is the study of the hardware and algorithms that are used in modern computer systems. Since many of the early pioneers of computer science, for instance Alan Turing, were mathematicians it is not surprising that these two subjects are closely related. This is a three-year joint degree programme, in conjunction with the School of Electronics, Electrical Engineering and Computer Science, that combines the study of the two subjects at each level.

Course Structure

• Stage 1: Students must take six compulsory modules.
• Stage 2: Students must take two compulsory modules plus another four optional modules approved by an advisor of studies.
• Stage 3: Students must take modules totally 120 units as approved by an advisor of studies.

Modules

Core Modules

• Mathematical Methods 1 (30 credits)
  • Review of A-level calculus: elementary functions and their graphs, domains and ranges, trigonometric functions, derivatives and differentials, integration.
  • Maclaurin expansion.
  • Complex numbers and Euler’s formula.
  • Differential equations (DE); first-order DE: variable separable, linear; second-order linear DE with constant coefficients: homogeneous and inhomogeneous.
  • Vectors in 3D, definitions and notation, operations on vectors, scalar and vector products, triple products, 2x2 and 3x3 determinants, applications to geometry, equations of a plane and straight line.
  • Rotations and linear transformations in 2D, 2x2 and 3x3 matrices, eigenvectors and eigenvalues.
  • Newtonian mechanics: kinematics, plane polar coordinates, projectile motion, Newton’s laws, momentum, types of forces, simple pendulum, oscillations (harmonic, forced, damped), planetary motion (universal law of gravity, angular momentum, conic sections, Kepler’s problem).
  • Curves in 3D (length, curvature, torsion).
  • Functions of several variables, derivatives in 2D and 3D, Taylor expansion, total differential, gradient (nabla operator), stationary points for a function of two variables.
  • Vector functions; div, grad and curl operators and vector operator identities.
  • Line integrals, double integrals, Green's theorem.
  • Surfaces (parametric form, 2nd-degree surfaces).
  • Curvilinear coordinates, spherical and cylindrical coordinates, orthogonal curvilinear coordinates, Lame coefficients.
  • Volume and surface integrals, Gauss's theorem, Stokes's theorem.
  • Operators div, grad, curl and Laplacian in orthogonal curvilinear coordinates.
• Algorithmic Thinking (10 credits)
  • Basic programming skills (e.g. in Python); introduction of software to present mathematical contents (e.g. LaTex) and to solve mathematical problems (e.g. Mathematica, R or packages like numpy and matplotlib); basic understanding of the complexity of algorithms (Big Oh notation).
• Mathematical Reasoning (10 credits)
  • The notion of mathematical statements and elementary logic.
  • Mathematical symbols and notation.
  • The language of sets.
  • The concept of mathematical proof, and typical examples.
  • Communicating mathematics to others.
• Introduction to Algebra and Analysis (30 credits)
  • Elementary logic and set theory, number systems (including integers, rationals, reals and complex numbers), bounds, supremums and infimums, basic combinatorics, functions.
  • Sequences of real numbers, the notion of convergence of a sequence, completeness, the Bolzano-Weierstrass theorem, limits of series of non- negative reals and convergence tests.
  • Analytical definition of continuity, limits of functions and derivatives in terms of a limit of a function.
  • Properties of continuous and differentiable functions.
  • L'Hopital's rule, Rolle's theorem, mean-value theorem.
  • Matrices and systems of simultaneous linear equations, vector spaces, linear dependence, basis, dimension.
• Procedural Programming (20 credits)
  • This module introduces the fundamentals of procedural programming.
  • Using a problem-solving approach, real-world examples are explored to promote code literacy and good algorithm design.
  • Students are introduced to the representation and management of primitive data, structures for program control and refinement techniques, which guide the development process from problem specification to code solution.
• Object Oriented Programming (20 credits)
  • This module introduces the fundamentals of object-oriented programming.
  • Real-world problems and exemplar code solutions are examined to encourage effective data modelling, code reuse and good algorithm design.
  • Fundamental OO programming concepts including abstraction, encapsulation, inheritance and polymorphism are practically reviewed through case studies, with an emphasis on testing and the use of code repositories for better management of code version control.
• Linear Algebra (20 credits)
  • Recap and extend to fields such as C, the notions of abstract vector spaces and subspaces, linear independence, basis, dimension.
  • Linear transformations, image, kernel and dimension formula.
  • Matrix representation of linear maps, eigenvalues and eigenvectors of matrices.
  • Matrix inversion, definition and computation of determinants, relation to area/volume.
  • Change of basis, diagonalization, similarity transformations.
  • Inner product spaces, orthogonality, Cauchy-Schwarz inequality.
  • Special matrices (symmetric, hermitian, orthogonal, unitary, normal) and their properties.
  • Basic computer application of linear algebra techniques.
• Data Structures and Algorithms (20 credits)
  • Data structures: Stacks, Lists, Queues, Trees, Hash tables, Graphs, Sets and Maps
  • Algorithms: Searching, Sorting, Recursion (with trees, graphs, hash tables etc.)
  • Asymptotic analysis of algorithms
  • Programming languages representation and implementation

Optional Modules

• Group Theory (20 credits)
  • Definition and examples of groups and their properties
  • Countability of a group and index
  • Lagrange’s theorem
  • Normal subgroups and quotient groups
  • Group homomorphisms and isomorphism theorems
  • Structure of finite abelian groups
  • Cayley’s theorem
  • Sylow’s theorem
  • Composition series and solvable groups
• Professional and Transferrable Skills (20 credits)
  • This module will prepare students for employment by developing an awareness of the business environment and the issues involved in successful career management combined with the development of key transferrable skills such as problem solving, communication and team working.
  • Students will build their professional practice and ability to critically self-reflect to improve their performance.
• Metric Spaces (20 credits)
  • Definition and examples of metric spaces (including function spaces)
  • Open sets, closed sets, closure points, sequential convergence, density, separability
  • Continuous mappings between metric spaces
  • Completeness
• Mathematical Methods 2 (20 credits)
  • Functions of a complex variable: limit in the complex plane, continuity, complex differentiability, analytic functions, Cauchy-Riemann equations, Cauchy’s theorem, Cauchy’s integral formula, Taylor and Laurent series, residues, Cauchy residue theorem, evaluation of integrals using the residue theorem.
  • Series solutions to differential equations: Frobenius method.
  • Fourier series and Fourier transform.
  • Basis set expansion.
  • Introduction to partial differential equations.
  • Separation of variables.
  • Wave equation, diffusion equation and Laplace’s equation.
• Analysis (20 credits)
  • Cauchy sequences, especially their characterisation of convergence.
  • Infinite series: further convergence tests (limit comparison, integral test), absolute convergence and conditional convergence, the effects of bracketing and rearrangement, the Cauchy product, key facts about power series (longer proofs omitted).
  • Uniform continuity: the two-sequence lemma, preservation of Cauchyness (and the partial converse on bounded domains), equivalence with continuity on closed bounded domains, a gluing lemma, the bounded derivative test.
  • Mean value theorems including that of Cauchy, proof of l'Hôpital's rule, Taylor's theorem with remainder.
  • Riemann integration: definition and study of the main properties, including the fundamental theorem of calculus.
• Theory of Computation (20 credits)
  • Automata and Formal Languages
  • Computability Theory (Turing Machines etc) and Decidability Theory (Halting Problem, etc)
  • Complexity Theory
• Introduction to Artificial Intelligence and Machine Learning (20 credits)
  • Concepts of artificial intelligence and machine learning.
  • Fundamentals of supervised and unsupervised learning
  • Fundamentals of experimental settings and hypothesis evaluation
  • The concept of feature selection
  • Evaluation in machine learning
    • Type I and Type II errors
    • Confusion matrices
    • ROC and CMC curves
    • Cross validation
  • Linear and non-linear function fitting
    • Linear Regression
    • Kernels
  • Classification models:
    • Nearest Neighbour
    • Naïve Bayes
    • Decision Trees
  • Clustering models:
    • k-Means
    • hierarchical clustering
    • Anomaly detection
• Employability for Mathematics (0 credits)
  • Introduction to placement for mathematics and physics students, CV building, international options, interview skills, assessment centres, placement approval, health and safety and wellbeing.
  • Workshops on CV building and interview skills.
  • The module is delivered in-house with the support of the QUB Careers Service and external experts.
• Classical Mechanics (20 credits)
  • Introduction to calculus of variations.
  • Recap of Newtonian mechanics.
  • Generalised coordinates.
  • Lagrangian.
  • Least action principle.
  • Conservation laws (energy, momentum, angular momentum), symmetries and Noether’s theorem.
  • Examples of integrable systems.
  • D’Alembert’s principle.
  • Motion in a central field.
  • Scattering.
  • Small oscillations and normal modes.
  • Rigid body motion.
  • Legendre transformation.
  • Canonical momentum.
  • Hamiltonian.
  • Hamilton’s equations.
  • Liouville’s theorem.
  • Canonical transformations.
  • Poisson brackets.

Entry Requirements

• A level requirements: A (Mathematics) AB OR A* (Mathematics) BB
• Irish leaving certificate requirements: H2H3H3H3H3H3 including Higher Level grade H2 in Mathematics
• Access Course: Successful completion of Access Course with a minimum of 80% in each Level 3 module. Must be relevant Access Course with substantial Mathematics modules (eg: Mathematics and Computing).
• International Baccalaureate Diploma: 34 points overall including 6 (Mathematics) 6 5 at Higher Level.
• Graduate: A minimum of a 2:2 Honours Degree, provided any subject requirement is also met.

Careers

• Introduction: Studying for a Mathematics and Computer Science degree at Queen’s will assist students in developing the core skills and employment- related experiences that are valued by employers, professional organisations and academic institutions.
• Employment after the Course: The School has links with over 500 IT companies both here and abroad. We benefit from the fact that there are more software companies located in N Ireland than any other part of the UK, outside of London.
• Prizes and Awards: Top performing students are eligible for a number of prizes within the School.

Fees and Funding

• Tuition Fees:
  • Northern Ireland (NI): £4,855
  • Republic of Ireland (ROI): £4,855
  • England, Scotland or Wales (GB): £9,535
  • EU Other: £22,400
  • International: £22,400
• Additional course costs: All essential software will be provided by the University, for use on University facilities, however for some software, students may choose to buy a version for home use.

How to Apply

• Application for admission to full-time undergraduate and sandwich courses at the University should normally be made through the Universities and Colleges Admissions Service (UCAS).
• When to Apply: UCAS will start processing applications for entry in autumn 2026 from early September 2025.
• The advisory closing date for the receipt of applications for entry in 2026 is Wednesday 14 January 2026 (18:00).