Undergraduate Mathematics 2014-2015

Calculus is the mathematical study of change, and is used in many areas of mathematics, science, and the commercial world. This course covers differentiation, limits, finding maximum and minimum values, and continuity.  There may well be some overlap with school mathematics, but the course is brisk and will go a long way quickly.

This course introduces the concepts of complex numbers, matrices and other basic notions of linear algebra over the real and complex numbers. This provides the necessary mathematical background for further study in mathematics, physics, computing science, chemistry and engineering.

This is a basic maths course which is suitable for students who hold the equivalent of Standard Grade maths. The aim is to help students to develop their confidence and accuracy in a range of topics, starting with basic skills and progressing to more advanced topics. There are weekly tests with good performance leading to the possibility of gaining an exemption from the final exam. Learning is mainly done in the students’ own time. This course is not suitable for students who have recently achieved an A or a B in Higher Maths or equivalent.

This course studies elementary planar geometry, such as lines, triangles and circles.  It will introduce students to the basic ideas of definition, theorem and proof, and will provide essential training in rigorous problem solving. The course is strongly recommended for all students aiming at a degree in mathematics.

This course is a continuation of the material covered in MA1007 Introductory Mathematics 1. It is suitable for students who have taken MA1007 and is a good course for students who use some maths in their degree but do not have a strong background in the subject. Topics covered include differentiation, integration, complex numbers, vectors and matrices among others. Students are continually assessed through weekly tests, with the possibility of gaining an exemption from the final exam if the results from the weekly tests are good enough. Learning is mainly done in the students’ own time.

The aim of the course is to provide an introduction to Integral Calculus and the theory of sequences and series, to discuss their applications to the theory of functions, and to give an introduction to the theory of functions of several variables. 

This provides the necessary mathematical background for further study in mathematics, physics, computing science, chemistry and engineering.

Combinatorics is the branch of mathematics concerned with counting. This includes counting structures of a given kind (enumerative combinatorics), deciding when certain criteria can be met, finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization), and applying algebraic techniques to combinatorial problems (algebraic combinatorics). The course is recommended to students of mathematics and computing science.

Set theory was introduced by Cantor in 1872, who was attempting to understand the concept of "infinity" which defied the mathematical world since the Greeks. Set Theory is fundamental to modern mathematics - any mathematical theory must be formulated within the framework of set theory, or else it is deemed invalid. It is the alphabet of mathematics.

In this course we will study naive set theory. Fundamental object such as the natural numbers and the real numbers will be constructed. Structures such as partial orders and functions will be studied. And of course, we will explore infinite sets.

The importance of introducing the study of sets into mathematics was recognised  in the late 19th century by Georg Cantor, who was the first to discover the existence of the “infinity of infinities”. Following that and some further remarkable achievements, “Set Theory" became a vibrant subject, and is regarded as the very foundation of mathematics. The course will start with an introduction to the language of set theory, and some of the basics of the subject. Having set the foundations, the course will then proceed with its main bulk consisting of an introduction to a variety of algebraic topics.

Analysis provides the rigourous, foundational underpinnings of calculus. It is centred around the notion of limits: convergence within the real numbers. Related ideas, such as infinite sums (a.k.a. series), continuity, differentiation, and Riemann integration, are also visited in this course.

Care is needed to properly use the delicate formal concept of limits. At the same time, limits are often intuitive, and we aim to reconcile this intuition with correct mathematical reasoning. The emphasis throughout this course is on rigourous mathematical proofs, valid reasoning, and the avoidance of fallacious arguments.

Probability theory is concerned with the analysis of random phenomena by providing an abstract mathematical framework to study them within the language of set theory. This is done by the concepts of "probability spaces" and "random variables". The theory began in the 16th century in attempts to analyze games of chance; In 1812 Pierre Simon Laplace wrote: "It is remarkable that a science which began with the consideration of games of chance should have become the most important object of human knowledge."

The course is recommended to anyone interested in the foundations and applications of mathematics.

Linear algebra is the study of vector spaces and linear maps between them.

It provides foundations for almost all branches of mathematics and sciences in general. For example, special relativity and quantum mechanics are formulated within the framework of linear algebra.

This is a course in multivariable calculus.  As the name suggests, it generalises familiar concepts from calculus (such as limits, derivatives, integrals and differential equations) to situations with many variables.

In addition to lectures and tutorials, there will be practical training through several computer sessions. Recommended to mathematicians and physicists.