*Students should note that all of the modules below may not be available to them.*

**Undergraduate** students should refer to the relevant section of the UCC Undergraduate Calendar for their programme requirements.

**Postgraduate** students should refer to the relevant section of the UCC Postgraduate Calendar for their programme requirements.

- MA1001 Calculus for Science Part 1
- MA1002 Calculus for Science Part 2
- MA1011 Mathematical Methods for Engineers 1
- MA1012 Mathematical Methods for Engineers II
- MA1055 Mathematics (Honours)
- MA1057 Introduction to Abstract Algebra
- MA1058 Introduction to Linear Algebra
- MA1059 Calculus
- MA1060 Introduction to Analysis
- MA1100 Introductory Mathematics for Business I
- MA2006 Intermediate Calculus for Scientists
- MA2007 Linear Algebra
- MA2013 Mathematics for Engineering
- MA2051 Mathematical Analysis I
- MA2054 Ordinary Differential Equations
- MA2055 Linear Algebra
- MA2061 Game Theory and Linear Algebra
- MA2062 Financial Mathematics
- MA2071 Multivariable Calculus
- MA2200 Introductory Mathematics for Business II
- MA3051 Mathematical Analysis II
- MA3052 Ring and Field Theory
- MA3054 Complex Analysis
- MA3056 Metric Spaces and Topology
- MA3062 Introduction to Modern Algebra
- MA3063 Introduction to Differential Geometry
- MA3301 Multivariable Calculus and Optimisation
- MA3901 Multivariable Calculus and Optimisation
- MA4052 Functional Analysis
- MA4053 Project
- MA4058 Measure Theory and Martingales
- MA4059 Topics in Mathematics
- MA4062 Topics in Modern Algebra
- MA4063 Topics in Differential Geometry
- MA4402 Game Theory and Linear Algebra
- MA4403 Financial Mathematics

MA1001 Calculus for Science Part 1

**Credit Weighting: **5

**Semester(s): **Semester 1.

**No. of Students: **Min 50, Max 600.

**Pre-requisite(s): **None

**Co-requisite(s): **None

**Teaching Method(s): **36 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Mr Martin Quirke, Department of Mathematics.

**Module Objective: **To introduce the fundamental mathematical techniques of science.

**Module Content: **Indices, logarithms, transposition of formulae and basic algebra. Revision of trigonometry and polynomial functions. Basic techniques and applications of differentiation, computational skill in practical examples taken from biology, chemistry, computer science, environmental science, food science and medicine, using real world data.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Solve scientific problems using differential calculus, algebra, trigonometry, polynomials, exponential and logarithm functions.

Find derivatives of elementary functions, and apply to graph sketching, maximizing and physical problems.

Demonstrate knowledge of derivatives as rates of change in scientific contexts.

Demonstrate mastery of algebra of polynomial and rational functions.

**Assessment: **Total Marks 100: Formal Written Examination 70 marks; Continuous Assessment 30 marks (1 x test, 20 marks; 5 computer-based assessments of equal weight, 10 marks).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Winter 2015.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward (in the case of students who have failed continuous assessment, the supplemental examination paper alone will provide a satisfactory assessment of the module learning outcomes worth 100 marks).

MA1002 Calculus for Science Part 2

**Credit Weighting: **5

**Semester(s): **Semester 2.

**No. of Students: **Min 50, Max 600.

**Pre-requisite(s): **None

**Co-requisite(s): **MA1001

**Teaching Method(s): **36 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To introduce the fundamental mathematical techniques of science.

**Module Content: **Basic techniques and applications of integration, linear regression and ordinary differential equations, emphasizing computational skill in practical examples taken from biology, chemistry, computer science, environmental science, food science and medicine, using real world data.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Solve scientific problems using integral calculus.

Evaluate indefinite integrals of elementary functions.

Apply substitution and integration by parts to evaluate indefinite integrals.

Use both definite and indefinite integration in scientific problems.

Find best fitting curves of various kinds.

Solve elementary ordinary differential equations and draw planar phase portraits.

**Assessment: **Total Marks 100: Formal Written Examination 70 marks; Continuous Assessment 30 marks (1 x test, 20 marks; 5 computer-based assessments of equal weight, 10 marks).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Summer 2016.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward (in the case of students who have failed continuous assessment, the supplemental examination paper alone will provide a satisfactory assessment of the module learning outcomes worth 100 marks).

MA1011 Mathematical Methods for Engineers 1

**Credit Weighting: **5

**Semester(s): **Semester 1.

**No. of Students: **Min 50, Max 250.

**Pre-requisite(s): **None

**Co-requisite(s): **None

**Teaching Method(s): **36 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Stephen Wills, Department of Mathematics.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To provide an overview of techniques and applications of univariate calculus.

**Module Content: **Functions, limits, continuity, methods and applications of differentiation and integration. First-order ordinary differential equations.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Solve problems using differentiation, especially extremal problems;

Solve problems using integration, for example the computation of areas;

Apply the techniques they have learned to compute derivatives and integrals;

Approximate integrals numerically;

Solve problems involving first-order differential equations.

**Assessment: **Total Marks 100: Formal Written Examination 60 marks; Continuous Assessment 40 marks (1 in-class test (20 marks), 10 assignments of equal weight (20 marks)).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Winter 2015.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

MA1012 Mathematical Methods for Engineers II

**Credit Weighting: **5

**Semester(s): **Semester 2.

**No. of Students: **Min 50, Max 250.

**Pre-requisite(s): **None

**Co-requisite(s): **MA1011

**Teaching Method(s): **36 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Stephen Wills, Department of Mathematics.

**Lecturer(s): **Staff, Department of Mathematics.

**Module Objective: **To provide an overview of mathematical techniques and applications, including partial differentiation, differential equations, linear algebra and complex functions.

**Module Content: **Sequences, power series, Taylor expansions. Linear 2nd-order ordinary differential equations. Matrices, solutions of simultaneous linear equations, determinants, eigenvalues and eigenvectors. Introduction to multivariable calculus: partial derivatives, tangent planes. Complex numbers.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Derive and manipulate power series;

Solve simple ordinary differential equations of second order;

Perform basic calculations in linear algebra: solve systems of linear equations; calculate matrix inverses and solve matrix equations; evaluate determinants; compute eigenvalues and eigenvectors of matrices;

Perform basic calculations in differential calculus of functions of several variables: compute partial derivatives; find equations of tangent planes; locate and classify critical points of functions of two variables;

Undertake elementary calculations with complex numbers, interpreting the results geometrically using Argand diagrams.

**Assessment: **Total Marks 100: Formal Written Examination 60 marks; Continuous Assessment 40 marks (1 in-class test (20 marks), 10 assignments of equal weight (20 marks)).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Summer 2016.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

**Credit Weighting: **15

**Semester(s): **Semesters 1 and 2.

**No. of Students: **Min 5, Max 100.

**Pre-requisite(s): **None

**Co-requisite(s): **None

**Teaching Method(s): **120 x 1hr(s) Lectures; 40 x 1hr(s) Workshops.

**Module Co-ordinator: **Dr Thomas Carroll, Department of Mathematics.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To provide an introduction to concepts and techniques of higher mathematics.

**Module Content: **Calculus: differentiation and integration of functions of a single variable, applications, approximation techniques. Analysis: real number system, inequalities, completeness, sequences and series, limits. Foundations: sets, proofs, Boolean algebra, relations & functions. Algebra & Number Theory: symmetry groups, primes, groups, rings. Linear Algebra: vectors, dot products, conics & quadrics, matrices, determinants, linear equations.

**Learning Outcomes: **On successful completion of this module, students should be able to:

recall the basic definitions and theorems of limits, continuity, differentiation and integration;

solve problems using differentiation, integration and related techniques;

recall the completeness axiom for the Real Numbers and apply it to solve problems involving infimum and supremum;

prove and apply basic theorems on sequences and series;

prove theorems in number theory and algebra using induction;

recall algebraic concepts such as relations, functions, binary operations, and axiom systems for algebraic groups, rings and fields;

perform standard matrix-related computations and operations;

determine whether one vector is a linear combination of others, and deduce the dimension of the kernel and image of a matrix, using Gaussian elimination and examining pivots;

prove the equivalence of the dozen invertibility criteria of Strang's nutshell using the main theorems of linear algebra.

**Assessment: **Total Marks 300: Formal Written Examination 130 marks (Paper I (65 marks); Paper II (65 marks)); Continuous Assessment 170 marks (2 x 1.5hr examinations, Paper III (65 marks); Paper IV (55 marks); in class tests (50 marks)).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **2 x 1.5 hr(s) paper(s) to be taken in Summer 2016.

**Requirements for Supplemental Examination: **4 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward (for in-class tests only).

MA1057 Introduction to Abstract Algebra

**Credit Weighting: **5

**Semester(s): **Semester 2.

**No. of Students: **Min 20, Max 200.

**Pre-requisite(s): **None

**Co-requisite(s): **None

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Thomas Carroll, Department of Mathematics.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To provide an overview of concepts of abstract algebra.

**Module Content: **Sets, relations, functions and equivalence relations, elementary theory of groups.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Deal with abstraction in algebra;

Work with algebraic concepts such as relations, functions and binary operations;

Identify generators and subgroups of given groups;

Work with group homomorphisms and isomorphisms;

Find Kernels and Images in specific examples;

Work with group actions, identify orbits and stabilizers.

**Assessment: **Total Marks 100: Formal Written Examination 75 marks; Continuous Assessment 25 marks (2 in-class tests of equal weight).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **None.

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Summer 2016.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

MA1058 Introduction to Linear Algebra

**Credit Weighting: **5

**Semester(s): **Semester 1.

**No. of Students: **Min 20, Max 200.

**Pre-requisite(s): **None

**Co-requisite(s): **None

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To provide an introduction to Linear Algebra

**Module Content: **Linear Algebra: vectors, dot products, conics & quadrics, matrices, determinants, linear equations.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Solve systems of linear equations.

Prove theorems and identities using induction.

Explain the meanings of the words determinant, eigenvalue, eigenvector, invertibility, kernel, image, and spectrum.

Find the eigenvectors and eigenvalues of a square matrix.

Find the determinant and inverse of a 3 x 3 matrix, and solve associated linear equations.

Deduce the solvability of a system of linear equations, without finding the solutions, via Gaussian elimination.

Deduce whether one vector is a linear combination of others, and by the same method deduce the dimension of the kernel and image of any matrix, using Gaussian elimination and examining pivots.

Prove the equivalence of the dozen invertibility criteria of Strang's nutshell using the main theorems of linear algebra.

**Assessment: **Total Marks 100: Formal Written Examination 80 marks; Continuous Assessment 20 marks (1 in-class test).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **None.

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Winter 2015.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

**Credit Weighting: **5

**Semester(s): **Semester 1.

**No. of Students: **Min 20, Max 200.

**Pre-requisite(s): **None

**Co-requisite(s): **None

**Teaching Method(s): **36 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Thomas Carroll, Department of Mathematics.

**Lecturer(s): ** Dr Thomas Carroll, Department of Mathematics.

**Module Objective: **To provide an introduction to concepts and techniques of calculus.

**Module Content: **Differentiation and integration of functions of a single variable, applications, approximation techniques.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Work with the concept of limit;

Prove basic theorems of limits, differentiation and integration;

Solve problems using differentiation, especially extremal problems;

Solve problems using integration, especially the computation of areas and volumes;

Apply the techniques they have learned to compute derivatives and integrals;

Approximate integrals numerically.

**Assessment: **Total Marks 100: Formal Written Examination 80 marks; Continuous Assessment 20 marks (1 in-class test).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Winter 2015.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward (in the case of students who have failed continuous assessment, the supplemental examination paper alone will provide a satisfactory assessment of the module learning outcomes).

MA1060 Introduction to Analysis

**Credit Weighting: **5

**Semester(s): **Semester 2.

**No. of Students: **Min 20, Max 200.

**Pre-requisite(s): **None

**Co-requisite(s): **MA1059

**Teaching Method(s): **36 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Thomas Carroll, Department of Mathematics.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To provide an introduction to concepts and techniques of mathematical analysis.

**Module Content: **Real number system, complex numbers, inequalities, completeness, sequences and series.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Recall the basic properties of rational numbers, real numbers and complex numbers;

Recall the completeness axiom and apply it to solve problems involving infimum and supremum;

Recall the basic notions of metric spaces;

Recall and apply the basic definitions and theorems of sequences;

Recall and apply the basic definitions and theorems of infinite series.

**Assessment: **Total Marks 100: Formal Written Examination 80 marks; Continuous Assessment 20 marks (1 in-class test).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Summer 2016.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward (in the case of students who have failed continuous assessment, the supplemental examination paper alone will provide a satisfactory assessment of the module learning outcomes).

MA1100 Introductory Mathematics for Business I

**Credit Weighting: **5

**Semester(s): **Semester 2.

**No. of Students: **Min 50, Max 400.

**Pre-requisite(s): **None

**Co-requisite(s): **None

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Workshops.

**Module Co-ordinator: **Dr Martin Kilian, Department of Mathematics.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To provide an introduction to fundamental quantitative techniques for business.

**Module Content: **Exponential, logarithmic and polynomial functions. Series, annuities and introduction to mathematics of finance. Elementary calculus with business applications.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Determine present and future values of single payment and annuity financial transactions;

Compute mortgage payments;

Interpret graphs of functions;

Compute equilibrium point for supply and demand functions;

Determine price elasticities of supply and demand functions.

**Assessment: **Total Marks 100: Formal Written Examination 80 marks; Continuous Assessment 20 marks (1 test (20 marks)).

**Compulsory Elements: **Formal Written Examination and In-Class Tests.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Summer 2016.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

MA2006 Intermediate Calculus for Scientists

**Credit Weighting: **5

**Semester(s): **Semester 2.

**No. of Students: **Min 5, Max 50.

**Pre-requisite(s): **MA1001 and MA1002, or MA1059

**Co-requisite(s): **None

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To study techniques of multivariable calculus.

**Module Content: **Partial differentiation. Double integrals and line integrals.

**Learning Outcomes: **On successful completion of this module, students should be able to:

sketch the graphs of functions of two variables;

compute partial derivatives, mixed partial derivatives and higher-order partial derivatives of functions of several variables using various methods including the chain rules;

find the equations of tangent planes and normal lines to surfaces which are the graphs of functions of several variables;

compute directional derivatives and maximum/minimum rates of change in terms of the gradient vector;

solve simple unconstrained and constrained maxima/minima problems, using tests to determine the nature of local extrema of functions of two variables;

set up line integrals arising in physical and geometrical problems concerned with curvature, mass, and work done by forces, and determine their value;

calculate Riemann sums of functions of two or more variables, and the values of double and triple integrals of continuous functions by means of iterated integrals and Stokes' theorem of computation.

**Assessment: **Total Marks 100: Formal Written Examination 75 marks; Continuous Assessment 25 marks (1 in-class test).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **None.

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Summer 2016.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

**Credit Weighting: **5

**Semester(s): **Semester 1.

**No. of Students: **Min 5, Max 50.

**Pre-requisite(s): **MA1001 and MA1002, or MA1059

**Co-requisite(s): **None

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Martin Kilian, Department of Mathematics.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To provide an introduction to techniques of linear algebra.

**Module Content: **Linear algebra. Matrices. Determinants. Linear Equations.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Manipulate matrices and use them to solve systems of simultaneous equations via Gaussian Elimination, Gauss-Jordan Elimination and the Inverse Matrix Method;

Calculate determinants of square matrices using minors and cofactors;

Use Cramer's Rule to solve systems of simultaneous equations;

Perform calculations involving dot products, norms, the Cauchy-Schwarz inequality and the triangle inequality;

Identify and provide examples of subspaces of a vector space;

Manipulate linear combinations of vectors, and determine linear (in)dependence, span, null space, bases and dimension of a vector space;

Use the Gram-Schmidt orthogonalization process to construct an orthogonal basis of a vector space V from any basis of V;

Form the characteristic polynomial of a matrix, calculate eigenvalues and eigenvectors of a matrix, and form eigenspaces.

**Assessment: **Total Marks 100: Formal Written Examination 75 marks; Continuous Assessment 25 marks (1 test).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **None.

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Winter 2015.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

MA2013 Mathematics for Engineering

**Credit Weighting: **5

**Semester(s): **Semester 1.

**No. of Students: **Min 30, Max 200.

**Pre-requisite(s): **MA1008

**Co-requisite(s): **None

**Teaching Method(s): **36 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Stephen Wills, Department of Mathematics.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To provide an introduction to the concepts and techniques of multivariable integral calculus and complex function theory.

**Module Content: **Multivariable integral calculus: line integrals, multiple integrals, Green's theorem, curl and

divergence, surface integrals, Gauss's divergence theorem.

Functions of a complex variable: elementary complex mappings, contour integrals, residue calculus. Fourier series, separation of variables techniques for solution of PDEs.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Visualise and work with surfaces and regions in 3-dimensional space;

Evaluate volume, surface, and line integrals in 2 and higher dimensions, and apply them using Green's theorem and Gauss' theorem;

Describe the mapping properties of basic functions of a complex variable (powers, fractional, exponential);

Construct an analytic function from its real part using the Cauchy-Riemann equations;

Compute residues and apply the Cauchy residue theorem to evaluate certain integrals;

Expand a function as a Fourier Series;

Use the technique of separation of variables to solve PDEs.

**Assessment: **Total Marks 100: Formal Written Examination 60 marks; Continuous Assessment 40 marks (2 in-class tests of equal weight).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Winter 2015.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

MA2051 Mathematical Analysis I

**Credit Weighting: **5

**Semester(s): **Semester 1.

**No. of Students: **Min 5, Max 75.

**Pre-requisite(s): **MA1059 and MA1060; or MA1055

**Co-requisite(s): **None

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Stephen Wills, Department of Mathematics.

**Lecturer(s): ** Dr Thomas Carroll, Department of Mathematics.

**Module Objective: **To present elementary classical analysis in a concrete setting, emphasizing specific techniques important to classical analysis and its applications.

**Module Content: **Topology of Euclidean space; continuous and differentiable mappings; Riemann integration

**Learning Outcomes: **On successful completion of this module, students should be able to:

Formulate basic definitions and results and prove basic results concerning the topology of Euclidean spaces;

Recall the basic definitions and theorems of limits, continuity and differentiation and their proofs.

Formulate and prove the Minimum-Maximum Theorem, the Intermediate Value Theorem, the Uniform Continuity Theorem, and some of their applications;

Give criteria that ensure functions are Riemann-Stieltjes integrable, and establish basic properties of such integrals.

**Assessment: **Total Marks 100: Formal Written Examination 80 marks; Continuous Assessment 20 marks (3 assignments of equal weight).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Winter 2015.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

MA2054 Ordinary Differential Equations

**Credit Weighting: **5

**Semester(s): **Semester 2.

**No. of Students: **Min 5, Max 75.

**Pre-requisite(s): **MA1058, MA1059 and MA1060; or MA1055

**Co-requisite(s): **MA2051

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To provide an introduction to the theory of ordinary differential equations.

**Module Content: **Uniform convergence. Power series. Existence and uniqueness theory. Picard's theorem. Method of undetermined coefficients, reduction of order, variation of parameters. Theory of linear differential equations.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Formulate definitions, basic results and their proofs in relation to uniform convergence of sequences and series of functions, the corresponding Cauchy criteria, and some of their applications;

Formulate and prove basic results about absolute and uniform convergence of power series and determine the radius of convergence.

Determine existence and uniqueness of solutions of an ordinary differential equation;

Solve ordinary differential equations by elementary methods such as separation of variables, undetermined coefficients, or variation of parameters;

Apply reduction of order to differential equations;

Solve constant-coefficient linear ordinary differential equations of all orders;

Solve inhomogeneous linear ordinary differential equations, either by Laplace transform or by other elementary methods;

Apply Picard iteration to approximate solutions of ordinary differential equations.

**Assessment: **Total Marks 100: Formal Written Examination 80 marks; Continuous Assessment 20 marks (1 in-class test).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Summer 2016.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

**Credit Weighting: **5

**Semester(s): **Semester 1.

**No. of Students: **Min 5, Max 75.

**Pre-requisite(s): **MA1058, MA1059 and MA1060; or MA1055

**Co-requisite(s): **None

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Dr Claus Michael Koestler, School of Mathematical Sciences.

**Module Objective: **To provide an introduction to the concepts of the theory of linear algebra.

**Module Content: **Linear equations and matrices; vector spaces; determinants; linear transformations and eigenvalues; norms and inner products; linear operators and normal forms.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Verify the linearity of mappings on real and complex vector spaces,

and the linear independence of sets of vectors;

Evaluate bases, transition matrices and matrices representing linear transformations;

Compute eigenvalues and eigenvectors of linear operators;

Construct orthonormal bases for vector spaces;

Verify properties of projection mappings, adjoint mappings, self-adjoint operators and isometries.

**Assessment: **Total Marks 100: Formal Written Examination 75 marks; Continuous Assessment 25 marks (2 in-class tests of equal weight).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Winter 2015.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

MA2061 Game Theory and Linear Algebra

**Credit Weighting: **5

**Semester(s): **Semester 2.

**No. of Students: **Max 10.

**Pre-requisite(s): **MA1055

**Co-requisite(s): **None

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To provide an introduction to business-related applications of linear algebra.

**Module Content: **Applications of linear analysis to game theory, Markov chains, linear programming.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Solve linear optimization problems subject to linear inequalities;

Use the simplex method;

Find optimal strategies and calculate expected values of games;

Employ Markov matrices in business models.

**Assessment: **Total Marks 100: Formal Written Examination 80 marks; Continuous Assessment 20 marks (1 class test).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Summer 2016.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

**Credit Weighting: **5

**Semester(s): **Semester 1.

**No. of Students: **Max 10.

**Pre-requisite(s): **MA1055

**Co-requisite(s): **None

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To develop facility with quantitative techniques for finance and investment.

**Module Content: **An introduction to the theory of options, the time value of money, rate of return of an investment cash-flow sequence and the arbitrage theorem.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Calculate probabilities and expectations of events and random variables associated to finite probability spaces and to standard variants of Brownian motion, using conditioning and independence techniques;

Carry out calculations based on present-value analysis and arbitrage arguments;

Calculate the price of European call and put options using the multiperiod model;

Derive and apply the Black-Scholes formula for option pricing;

Estimate volatility of shares from price history data.

**Assessment: **Total Marks 100: Formal Written Examination 80 marks; Continuous Assessment 20 marks (3 assignments of equal weight).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Winter 2015.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continous Assessment is carried forward.

**Credit Weighting: **5

**Semester(s): **Semester 2.

**No. of Students: **Min 5, Max 75.

**Pre-requisite(s): **None

**Co-requisite(s): **MA2051, MA2055

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To provide a foundation in multivariable calculus.

**Module Content: **Calculus of several variables, including continuity, differentiability and constrained and unconstrained optimisation. Line, surface and volume integrals.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Use the definitions to verify continuity and differentiability for simple functions of two (or more) variables;

Compute partial derivatives, mixed partial derivatives and higher-order partial derivatives using various methods including the chain rules;

Find the equations of tangent planes and normal lines to surfaces that are the graphs of functions of several variables;

Compute directional derivatives and maximum/minimum rates of change in terms of the gradient vector;

Solve simple unconstrained and constrained maxima/minima problems, using tests to determine the nature of local extrema of functions of two variables;

Set up and calculate line integrals including examples arising in physical and geometrical problems involving curvature, mass, and work done by forces;

Calculate double and triple integrals of continuous functions, defined over closed, bounded regions, by means of iterated integrals and the fundamental theorem of computation.

**Assessment: **Total Marks 100: Formal Written Examination 75 marks; Continuous Assessment 25 marks (5 x 0.5hr tests of equal weight).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **None.

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Summer 2016.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

MA2200 Introductory Mathematics for Business II

**Credit Weighting: **5

**Semester(s): **Semester 1.

**No. of Students: **Min 30, Max 150.

**Pre-requisite(s): **MA1100

**Co-requisite(s): **None

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Martin Kilian, Department of Mathematics.

**Lecturer(s): ** Dr Andrei Mustata, Department of Mathematics.

**Module Objective: **To complete the introduction to fundamental mathematical techniques for business.

**Module Content: **Least squares approximation and curve fitting. Matrices and simultaneous equations. Further methods and applications of calculus.

**Learning Outcomes: **On successful completion of this module, students should be able to:

apply integration techniques to business situations such as computing consumer and producer surplus;

solve elementary first order differential equations, including differential equations describing exponential growth/decay, with emphasis on applying these techniques to calculating revenue, cost, profits, long-term growth predictions etc.;

formulate and solve a linear programming model from information given in word and table form;

perform basic matrix calculations, including addition/subtraction, multiplication, transpose, inverses, determinants;

solve simultaneous equations using Gaussian Elimination, Gauss-Jordan Elimination, Cramer's Rule and the Inverse Matrix Method and use matrices to perform Input/Output analysis;

form and solve difference equations, given information in word or numerical form and apply this knowledge to income/growth models;

find curves of best fit to given sets of data and use the least squares approximation method to predict future outcomes.

**Assessment: **Total Marks 100: Formal Written Examination 80 marks; Continuous Assessment 20 marks (2 in-class tests of equal weight).

**Compulsory Elements: **Formal Written Examination; In-Class Tests.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Winter 2015.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

MA3051 Mathematical Analysis II

**Credit Weighting: **5

**Semester(s): **Semester 2.

**No. of Students: **Min 5, Max 75.

**Pre-requisite(s): **MA2051, MA2054, MA2055

**Co-requisite(s): **None

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Dr Stephen Wills, Department of Mathematics.

**Module Objective: **To present elementary linear analysis in a concrete setting, emphasizing specific techniques important to analysis and its applications.

**Module Content: **Normed linear spaces, Banach spaces, bounded linear operators; Hilbert spaces, convexity, orthogonal expansions and their applications, Riesz representation theorem for functionals.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Verify the axioms of normed vector spaces, Banach spaces and Hilbert spaces in specific examples, applying relevant tests for completeness or the existence of an inner product;

Use completeness arguments to produce existence proofs for operators with desired properties;

Compute orthogonal expansions with respect to an orthonormal basis of a Hilbert space, and apply such expansions to solve problems;

Prove the Riesz Representation Theorem for bounded linear functionals on Hilbert spaces.

**Assessment: **Total Marks 100: Formal Written Examination 75 marks; Continuous Assessment 25 marks (2 assignments (5 marks each), 1 in-class test (15 marks)).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Summer 2016.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

**Credit Weighting: **5

**Semester(s): **Semester 1.

**No. of Students: **Min 5, Max 75.

**Pre-requisite(s): **MA1057 or MA1055

**Co-requisite(s): **None

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To provide an introduction to concepts of Ring and Field Theory with applications.

**Module Content: **Divisibility, irreducibles and primes in Z and Q[X], rings and ideals and unique factorization, construction of fields, Galois theory, straight edge and compass constructions.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Accurately manipulate computations in a given ring or field, and in ring quotients by appropriate ideals;

Solve arithmetic problems like finding the largest common divisor in a Euclidean ring;

Master basic techniques for investigating when a polynomial over rational numbers is irreducible;

Solve problems in an integral domain via the induced problem in its field of fractions;

Recognize when a ring is principal ideal domain or unique factorization domain;

Solve problems in field extensions of a field F by translating them into linear problems over F;

Compute the Galois group of a given extension.

**Assessment: **Total Marks 100: Formal Written Examination 75 marks; Continuous Assessment 25 marks (3 x assignments of equal weight).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Winter 2015.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

**Credit Weighting: **5

**Semester(s): **Semester 1.

**No. of Students: **Min 5, Max 75.

**Pre-requisite(s): **MA2051 or equivalent

**Co-requisite(s): **None

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To provide an introduction to the theory of functions of a complex variable.

**Module Content: **Bilinear mappings, complex differentiable functions, power series, complex contour integrals, Cauchy's theorem and integral formula, Taylor's theorem, zeros of analytic functions and Rouche's theorem, maximum modulus principle, singularities and Laurent series, poles and residues, residue calculus.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Analyse the mapping properties of fundamental functions (bilinear, exponential, trigonometric) of a complex variable;

Define the derivative of a complex function and derive and apply the Cauchy-Riemann equations;

Prove and apply Cauchy's integral formula, and Taylor's theorem on the power series expansion of analytic functions;

Derive Liouville's theorem on entire functions and establish the fundamental theorem of algebra;

Compute residues and apply Cauchy's theorem to the evaluation of integrals and the summation of series;

Apply Rouche's theorem on the zeros of analytic functions.

**Assessment: **Total Marks 100: Formal Written Examination 80 marks; Continuous Assessment 20 marks (1 in-class test).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Winter 2015.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

MA3056 Metric Spaces and Topology

**Credit Weighting: **5

**Semester(s): **Semester 1.

**No. of Students: **Min 5, Max 75.

**Pre-requisite(s): **MA2051

**Co-requisite(s): **None

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Martin Kilian, Department of Mathematics.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To provide an introduction to the theory and concepts of metric and topological spaces.

**Module Content: **Topological spaces, compactness, connectedness, product spaces, continuity, countability, homotopy.

**Learning Outcomes: **On successful completion of this module, students should be able to:

State the basic concepts of topological and metric spaces.

Construct examples and counterexamples of topological spaces with certain properties.

Perform set theoretic computations.

Formulate the basic properties of continuous functions.

Apply the theory of various notions of convergence.

Determine if a set is simply-connected.

**Assessment: **Total Marks 100: Formal Written Examination 75 marks; Continuous Assessment 25 marks (2 tests of equal weight).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Winter 2015.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

MA3062 Introduction to Modern Algebra

**Credit Weighting: **5

**Semester(s): **Semester 2.

**No. of Students: **Min 5, Max 75.

**Pre-requisite(s): **MA1057 and MA1058, or MA1055; MA2055

**Co-requisite(s): **None

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Professor Bernard Hanzon, School of Mathematical Sciences.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To provide an introduction to major concepts in Modern Algebra with applications.

**Module Content: **Basic algebraic concepts and techniques such as rings and modules, computational algebra techniques and optimization, group representations, fields and field extensions, Galois theory.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Identify generators for given ideals of polynomials;

Apply division algorithms for multivariable polynomials;

Construct free resolutions of modules;

Define the homology group of a complex;

Complete character tables for given examples of groups;

State the main theorems of Galois theory;

Identify the ideal associated to an algebraic variety;

Identify properties of an algebraic variety such as irreducibility, dimension, degree, genus.

**Assessment: **Total Marks 100: Formal Written Examination 75 marks; Continuous Assessment 25 marks (2 in-class tests of equal weight).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Summer 2016.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

MA3063 Introduction to Differential Geometry

**Credit Weighting: **5

**Semester(s): **Semester 2.

**No. of Students: **Min 5, Max 75.

**Pre-requisite(s): **MA2071

**Co-requisite(s): **MA3051, MA3056

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Martin Kilian, Department of Mathematics.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To provide an introduction to concepts of differential geometry.

**Module Content: **Basic differential geometry concepts and techniques such as differential manifolds and submanifolds, tangent and normal bundles, Riemannian metric, curvatures, geodesics, with emphasis on the cases of curves and surfaces; differentiable maps, geometric aspects of the theory of differential equations, symplectic geometry, homogeneous spaces.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Explain and manipulate the concepts of differential manifolds, tangent bundle and cotangent bundles, tensor fields, differential forms, differentiable maps, symplectic forms;

Use the implicit function theorem to pass between parametric and level set descriptions of given manifolds. Use immersions and submersions to describe given manifolds;

Pass between parametric and level set descriptions of given manifolds;

Use Riemannian metrics on given manifolds to calculate volumes, the Levi-Civita connection, curvatures, geodesics, with emphasis on surfaces;

Calculate the induced metric on a submanifold, its second fundamental form, and explain the Gauss equation with particular emphasis on surfaces;

Calculate the normal and geodesic curvatures of a curve on a surface, and decide whether the curve is a geodesic;

Calculate critical point indices of a differentiable map on a given manifold, and use this to describe its topological properties;

Use the concepts of manifold, Lie groups and algebras in the study of differential equations;

Work with Fuchsian groups and their applications to hyperbolic geometry.

**Assessment: **Total Marks 100: Formal Written Examination 75 marks; Continuous Assessment 25 marks (2 tests of equal weight).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Summer 2016.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

MA3301 Multivariable Calculus and Optimisation

**Credit Weighting: **5

**Semester(s): **Semester 1.

**No. of Students: **Min 20, Max 100.

**Pre-requisite(s): **MA2200

**Co-requisite(s): **None

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Martin Kilian, Department of Mathematics.

**Lecturer(s): ** Dr Spyridon Dendrinos, School of Mathematical Sciences.

**Module Objective: **To develop and understanding of techniques required for dealing with multi-parameter quantitative problems in business and management.

**Module Content: **Multivariable functions, partial derivatives, optimisation problems and Language multipliers. Growth rates and applications.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Calculate first and second order partial derivatives of functions in two or three variables and apply this to problems in relation to economic concepts such as partial elasticity, production and utility;

Solve unconstrained optimisation problems for functions in two variables and apply this to optimisation problems in economics;

Solve constrained optimisation problems for functions in two variables and apply this to optimisation problems in economics;

Apply the Lagrange Multiplier Method to solve constrained optimisation problems in two variables and apply this to optimisation problems in economics;

Demonstrate understanding of basic mathematical rules for exponential and logarithmic functions and apply these rules to solve problems in economics in relation to relative growth and decay;

Solve elementary differential equations by the method of separation of variables and apply this to solve problems in economics.

**Assessment: **Total Marks 100: Formal Written Examination 80 marks; Continuous Assessment 20 marks (1 in-class test).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Winter 2015.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

MA3901 Multivariable Calculus and Optimisation

**Credit Weighting: **5

**Semester(s): **Semester 1.

**No. of Students: **Min 20, Max 100.

**Pre-requisite(s): **MA2200

**Co-requisite(s): **None

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Professor Bernard Hanzon, School of Mathematical Sciences.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To develop advanced calculus techniques and then apply them to quantitative problems in business and management.

**Module Content: **Multivariable functions, partial derivatives, optimisation problems and Lagrange multipliers.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Calculate first and second order partial derivatives of functions in two or three variables and apply this to problems in relation to economic concepts such as partial elasticity, production and utility;

Solve unconstrained optimisation problems for functions in two variables and apply this to optimisation problems in economics;

Solve constrained optimisation problems for functions in two variables and apply this to optimisation problems in economics;

Apply the Lagrange Multiplier Method to solve constrained optimisation problems in two variables and apply this to optimisation problems in economics;

Demonstrate understanding of basic mathematical rules for exponential and logarithmic functions and apply these rules to solve problems in economics in relation to relative growth and decay;

Solve elementary differential equations by the method of separation of variables and apply this to solve problems in economics.

**Assessment: **Total Marks 100: Continuous Assessment 100 marks (2 x in-class Examinations (1 x 20 marks, 1 x 80 marks)).

**Compulsory Elements: **Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **No Formal Written Examination.

**Requirements for Supplemental Examination: **Marks in passed element(s) of Continuous Assessment are carried forward, Failed element(s) of Continuous Assessment must be repeated (A departmental test will take place during the period set aside by the University for supplemental examinations; students should consult the Department in early August for date).

**Credit Weighting: **5

**Semester(s): **Semester 1.

**No. of Students: **Min 5, Max 50.

**Pre-requisite(s): **MA3051

**Co-requisite(s): **None

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Dr Claus Michael Koestler, School of Mathematical Sciences.

**Module Objective: **To provide a grounding in modern functional analysis.

**Module Content: **A study of the interaction of topological and algebraic structures, of fundamental importance in contemporary mathematics and its applications. Normed linear spaces, Banach spaces, bounded linear operators, adjoint operators, spectrum.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Verify the axioms of normed vector spaces, Banach spaces and Hilbert spaces in specific examples, applying relevant tests for completeness or the existence of an inner product;

Compute the norms and spectra of (certain classes of) operators;

Use completeness arguments to produce existence proofs for operators and linear functionals with desired properties;

Establish well-known isometric isomorphisms between sequence spaces and their dual spaces.

**Assessment: **Total Marks 100: Formal Written Examination 75 marks; Continuous Assessment 25 marks (5 assignments of equal weight).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Winter 2015.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

**Credit Weighting: **5

**Semester(s): **Semesters 1 and 2.

**No. of Students: **Min 1, Max 25.

**Pre-requisite(s): **None

**Co-requisite(s): **None

**Teaching Method(s): **12 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Stephen Wills, Department of Mathematics.

**Lecturer(s): ** Dr Andrei Mustata, Department of Mathematics.

**Module Objective: **To develop skills of mathematical investigation and report writing.

**Module Content: **Project on an assigned mathematical topic.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Write a logical and coherent account of a mathematical topic;

Construct proofs of mathematical results;

Assimilate and explain details of selected known theorems and theories;

Carry out independent library research;

Give a clear and accurate oral presentation of project results.

**Assessment: **Total Marks 100: Continuous Assessment 100 marks (1 x Project Report).

**Compulsory Elements: **Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Where work is submitted up to and including 7 days late, 10% of the total marks available shall be deducted from the mark achieved. Where work is submitted up to and including 14 days late, 20% of the total marks available shall be deducted from the mark achieved. Work submitted 15 days late or more shall be assigned a mark of zero.

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **No Formal Written Examination.

**Requirements for Supplemental Examination: **Marks in passed element(s) of Continuous Assessment are carried forward, Failed element(s) of Continuous Assessment must be repeated (resubmision of revised project report and presentation of findings as prescribed by the Department).

MA4058 Measure Theory and Martingales

**Credit Weighting: **5

**Semester(s): **Semester 1.

**No. of Students: **Min 5, Max 50.

**Pre-requisite(s): **MA3051

**Co-requisite(s): **None

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Dr Spyridon Dendrinos, School of Mathematical Sciences.

**Module Objective: **To provide an overview of the theory of measurable sets, integration and martingales

**Module Content: **Measurable spaces and functions; measures and integrals; integrable functions; conditional expectation; martingales; convergence theorems.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Identify and formulate the basic concepts and theorems of sigma algebras, measures and abstract measure spaces;

Discuss the completion and construction of measures including the basics of Caratheodory's Extension Procedure;

Synthesise techniques that have been developed in the course to solve particular problems;

Explain the basic concepts and main theorems of Lebesgue and Lesbesgue-Stieltjes integration including the main convergence theorems;

Solve problems involving Lebesgue and Lesbesgue-Stieltjes integration;

Explain the basic concepts and theorems of conditional expectation and martingales;

Evaluate basic martingales.

**Assessment: **Total Marks 100: Formal Written Examination 80 marks; Continuous Assessment 20 marks (1 in-class test).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Winter 2015.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

**Credit Weighting: **5

**Semester(s): **Semester 2.

**No. of Students: **Min 6, Max 50.

**Pre-requisite(s): **MA3051, MA3062, MA3063, MA3054

**Co-requisite(s): **None

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Professor Bernard Hanzon, School of Mathematical Sciences.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To further develop an understanding of, and the skills required for, advanced mathematics.

**Module Content: **The module will include topics from: algebra, analysis and geometry.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Solve basic problems in an area of current research in mathematics;

Explain the most important recent results of algebra, analysis, geometry or topology;

Rigorously prove theorems using geometric intuition, algebraic symbol manipulation, and analytic estimates;

Use theorems from algebra to give insight in the areas of number theory or complex differential geometry;

Use the concept of manifold in the study of differential equations.

**Assessment: **Total Marks 100: Formal Written Examination 75 marks; Continuous Assessment 25 marks (3 assignments of equal weight).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Summer 2016.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

MA4062 Topics in Modern Algebra

**Credit Weighting: **5

**Semester(s): **Semester 2.

**No. of Students: **Min 5, Max 75.

**Pre-requisite(s): **MA1057 and MA1058, or MA1055; MA2055

**Co-requisite(s): **None

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Professor Bernard Hanzon, School of Mathematical Sciences.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To further develop major concepts in Modern Algebra with applications.

**Module Content: **A range of algebraic concepts and techniques chosen from rings and modules, computational algebra techniques and optimization, group representations, fields and field extensions, Galois theory, Lie groups and algebra, homology, algebraic varieties.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Compute Groebner bases and apply elimination theory techniques;

Calculate homology groups of complexes;

Perform irreducible decompositions for given representations by finite groups and some classical groups;

Compute the Galois group of given field extensions;

Explain the connection between an affine algebraic variety and its ring/field of functions and use these to understand basic properties of given varieties.

Apply homology techniques to calculate simple invariants of algebraic varieties.

**Assessment: **Total Marks 100: Formal Written Examination 75 marks; Continuous Assessment 25 marks (2 in-class tests of equal weight).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Summer 2016.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

MA4063 Topics in Differential Geometry

**Credit Weighting: **5

**Semester(s): **Semester 2.

**No. of Students: **Min 5, Max 75.

**Pre-requisite(s): **MA2071, MA3051

**Co-requisite(s): **None

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Martin Kilian, Department of Mathematics.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To further develop an understanding of differential geometry.

**Module Content: **Topics to be chosen from the differential geometry of manifolds and submanifolds, tangent and normal bundles, Riemann metric, connections, curvatures, geodesics, with particular emphasis on the cases of curves and surfaces; differential forms, differentiable maps, geometric aspects of the theory of differential equations, Lie groups, symplectic geometry, homogeneous spaces, elliptic and hyperbolic geometries.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Explain and manipulate the concepts of differential manifolds, tangent bundle and cotangent bundles, tensor fields, differential forms, differentiable maps, symplectic forms.

Use the implicit function theorem to pass between parametric and level set descriptions of given manifolds. Use immersions and submersions to describe given manifolds.

Use Riemannian metrics on given manifolds to calculate volumes, the Levi-Civita connection, curvatures, geodesics, with emphasis on surfaces.

Calculate the induced metric on a submanifold, its second fundamental form, and explain the Gauss equation with particular emphasis on surfaces.

Calculate the normal and geodesic curvatures of a curve on a surface, and decide whether the curve is a geodesic.

Calculate critical point indices of a differentiable map on a given manifold, and use this to describe its topological properties.

Use the concepts of manifold, Lie groups and algebras in the study of differential equations.

Work with Fuchsian groups and their applications to hyperbolic geometry.

**Assessment: **Total Marks 100: Formal Written Examination 75 marks; Continuous Assessment 25 marks (2 tests of equal weight).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Summer 2016.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

MA4402 Game Theory and Linear Algebra

**Credit Weighting: **5

**Semester(s): **Semester 2.

**No. of Students: **Min 15, Max 100.

**Pre-requisite(s): **MA2200

**Co-requisite(s): **None

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To provide an introduction to business related applications of linear algebra.

**Module Content: **Applications of linear analysis to game theory, Markov chains, linear programming.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Solve linear optimization problems subject to linear inequalities;

Use the simplex method;

Find optimal strategies and calculate expected values of games;

Employ Markov matrices in business models.

**Assessment: **Total Marks 100: Formal Written Examination 80 marks; Continuous Assessment 20 marks (1 class test).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Summer 2016.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continuous Assessment is carried forward.

**Credit Weighting: **5

**Semester(s): **Semester 1.

**No. of Students: **Min 15, Max 100.

**Pre-requisite(s): **MA3301, ST1023 or equivalent

**Co-requisite(s): **None

**Teaching Method(s): **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To develop facility with quantitative techniques for finance and investment.

**Module Content: **An introduction to the theory of options, the time value of money, rate of return of an investment cash-flow sequence and the arbitrage theorem.

**Learning Outcomes: **On successful completion of this module, students should be able to:

Calculate probabilities and expectations of events and random variables associated to finite probability spaces and to standard variants of Brownian motion, using conditioning and independence techniques;

Carry out calculations based on present-value analysis and arbitrage arguments;

Calculate the price of European call and put options using the multiperiod model;

Derive and apply the Black-Scholes formula for option pricing;

Estimate volatility of shares from price history data.

**Assessment: **Total Marks 100: Formal Written Examination 80 marks; Continuous Assessment 20 marks (3 assignments of equal weight).

**Compulsory Elements: **Formal Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**Formal Written Examination: **1 x 1.5 hr(s) paper(s) to be taken in Winter 2015.

**Requirements for Supplemental Examination: **1 x 1.5 hr(s) paper(s) to be taken in Autumn 2016. The mark for Continous Assessment is carried forward.