###### Mathematics

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## The Degree

## Teaching

Teaching takes place in an atmosphere of outstanding research across a broad spectrum of topics. This may sound a little forbidding but our students, past and present, tell us that we are friendly and informal and that they really appreciate the personal attention the staff are still able to give them. They also find the staff’s commitment to their subject both infectious and stimulating.

As would be expected from one of the top departments of Mathematics in the country, our degree programmes are quite deliberately demanding and academic in nature - this is why we require high grades for entry. We aim to give students a good grounding in basic advanced mathematics in the early part of the degree so allowing a wide choice from over thirty options in the third and fourth years of their degrees.

Each degree programme offered contains a significant amount of rigorous and abstract pure mathematics which is studied from the very beginning. University mathematics is very different from school mathematics and it takes time to adjust to the ideas of rigour, proof and abstraction. Ultimately the rewards are enormous not only for the understanding of mathematics but for the multiple applications in many other areas such as astronomy, biology, chemistry, computing, finance, physics and statistics, as well as giving students transferable skills which are greatly in demand.

## Computing

Students are not expected to have any experience of computers on arrival at UCL. Of course, the vast majority will have some knowledge. All students will take a comprehensive course in Python programming in their first year as part of the Mathematical Methods 2 module. Once mastered, this coding knowledge may subsequently be developed in the optional Computational Methods, in year two, and a further course, Numerical Methods, in year three. There are also courses in Mathematica, a sophisticated programme which can manipulate symbols, in other words, can ‘do’ mathematics. Clusters of PCs are provided throughout UCL and these can be used for word-processing, email and access to the internet. Most Halls of Residence also have computer terminals, many students’ rooms are networked and the departmental student common room has wireless access.

## Statistics

Courses in statistics are taught by the Department of Statistical Science and not the Department of Mathematics. There are no compulsory statistics courses in our degree programmes except in the BSc and MSci programmes in Mathematics and Statistical Science (MASS). However, in the second half of all our degrees, courses in probability and statistics are available in both the Department of Statistical Science and in our own department.

## Problem-Solving

Mathematics is all about solving problems. With this in mind the department organises a weekly, two-hour informal session devoted to problem-solving. This is open to all students. The students involved in these sessions form the basis of the team which the department sends to the International Mathematical Competition for University Students, which over the last 22 years has had participants from 150 different universities from 40 countries.

## Organisation

There are two teaching terms, the autumn term (12 weeks) and the winter/spring term (11 weeks), with the summer term (7 weeks) being reserved for revision and examinations. Each of the teaching terms in the Mathematics Department has a reading week in the middle. The degree programmes are organised on a course-unit system, in which students take a number of individual courses, each assigned a course-unit (CU) value depending on the amount of work involved. UCL has extended this system to assign each course a European Credit Transfer System (ECTS) value. ECTS allows students to gain recognition for academic achievement at participating institutions across Europe, which can assist UCL students who wish to pursue educational or career opportunities throughout Europe. Each year a student completes courses to the value of 4.0 CU, equivalent to 60 ECTS credits. To obtain a BSc degree a student must pass 11 units (165 ECTS), that is 22 half-unit courses, although most students will actually take and pass 12 units (180 ECTS). Students also have to pass a certain number of units to proceed to each succeeding year. Classification for Honours is based on a weighting by year of 1:3:5 for the BSc degree and 1:3:5:5 for the MSci degree based on the results in all examinations.

At the beginning of January there is a mid-sessional examination for first-year students in each of the four courses taken in the first term. This gives students an opportunity to test their progress and is an encouragement to work hard! The results of these examinations do not count towards the final degree assessment.

Each half-unit course consists of three one-hour lectures each week often with a further hour taken up with a problems class. In the first two years of the degree a weekly problem sheet will be set for each course. Students’ answers to these will be marked and returned to them. The marks from these exercises contribute a small percentage to the total marks in the examinations taken at the end of the year. This system also applies to some third-year courses. In the first one-and-a-half years there will be an additional one-hour problem class/workshop every week for each course, which provides help with the problem sheets.

First-year students also receive at least two tutorials a week, in groups of five or six students. In these tutorials, students are encouraged to ask questions about what they have been learning. (If they do not have any questions, they may be asked to explain some point, to ensure they do really understand the subject!) Students have to attend 18 hours or so of formal tuition a week consisting of lectures, problem classes and small-group tutorials. This is just the tip of the iceberg. Students must expect to spend many more hours each week understanding their notes, reading books and doing the problem sheets. A mathematics degree is definitely not an easy option!

In addition to the small-group tutorials, all courses have ‘office hours’ where lecturers make themselves formally available to students. The department also operates an ‘open-door’ policy enabling students to see staff at mutually convenient times. Some courses involve project work which may include elements of presentational skills. Students are encouraged to participate in the UCL student tutoring scheme, where students spend half a day each week helping in local schools.

Each student is assigned a Personal Tutor to whom they can go
if they have any particular problems. In addition, the college runs a system of student mentors for first-year
students as part of its Transition Programme.

## The Structure of the Degree Programmes

The Mathematics Department offers three types of degree programme, each of which can be studied as a BSc over three years or as an MSci over four years, giving a choice of fourteen different degrees.

- The single-subject degree where only mathematics is studied in all its many different forms.
- Degrees where mathematics is studied as the major subject with another minor subject x – (Mathematics with x). The major subject will form roughly 75% of the degree and the minor 25%. The minor subject will replace the compulsory applied mathematics courses in the first one and a half years of the degree. A student following any of these degree programmes will be considered to be a member of the Mathematics Department and will be fully involved in all its activities.
- Degrees where mathematics is studied on an equal basis with another subject x – (Mathematics and x). A student following such a degree programme will be considered to be a member of the Mathematics Department and the other department, and so will be looked after by staff in both departments and will be able to participate in a double set of social events. Students should consult the appropriate department about courses available there.

The degree programmes in the Mathematics Department all lead to either a BSc or MSci Honours degree.

## BSc or MSci?

Each degree programme can be taken either as a three-year BSc or as a four-year MSci (Master in Science). The MSci degree is an undergraduate degree programme as distinct from the MSc (Master of Science) which is a one-year graduate degree programme. The first two years of the BSc and MSci are identical. Whilst BSc students have a wide choice in their third year, MSci students are required to take a selection of designated courses in their third year and do a major project in their fourth year. This project will involve a substantial piece of written work and a presentation. The MSci programmes are intended for those students who really want to understand advanced mathematics in all its beauty and who may wish to become trained mathematicians, possibly going on to do academic research in mathematics or into employment where mathematics is directly involved. MSci degrees attract the normal mandatory financial assistance, give a broad-based training in mathematics and are well suited to the increasingly interdisciplinary nature of modern mathematics. As four-year degrees they are more easily recognised in Europe. The three-year BSc degrees are ideal for students who wish to obtain transferable skills such as numeracy, problem-solving and logical thinking, which are required for a large number of careers where mathematics may not be directly involved. Students are advised to apply for the MSci degree in the first instance, as it is always possible to transfer to the BSc at any time. Similarly the department reserves the right after two or three years to transfer students from the MSci degree to the BSc, if they are not making sufficiently good progress.

## The Core Courses

In the first year, all mathematics degree programmes are based on the principle of ensuring that all students gain a good grounding in basic advanced mathematics – ‘the mathematics that every mathematics graduate should know!’ Likewise for our combined degree programmes. Below we describe the core mathematics courses given in the first one-and-a-half years and which ensure students possess the necessary knowledge to embark upon the very large number of options in the third and fourth years. Every student taking one of our degrees will have to take the majority of these courses. For this reason we describe them in rather more detail.

## First Year

Since this year is the one of most immediate concern to prospective students, a fuller description of what it will be like is given below.

#### Pure Mathematics

The work in pure mathematics falls into two distinct parts.

(i) Algebra 1 and 2

The first-year work in this subject divides naturally into two parts. The first part deals with structures, introducing the ideas of sets and functions, matrices and linear equations. Linear algebra develops the ideas of abstraction, rigour and proof, and has applications to nearly every branch of mathematics as well as to economics and the biological and physical sciences.

The second part deals with the subject of groups - a concept of remarkable power and beauty, which has applications to all branches of mathematics, and, in particular, to the study of symmetry and elementary particle theory in physics. It also continues with the study of linear algebra.

(ii) Analysis 1 and 2

Starting with a few basic properties of the real numbers, the whole of school calculus is rigorously developed. The course helps to instil the vital disciplines of proof, logical argument and rigour as well as being beautiful in its own right. This work is designed to give a solid foundation to the study of calculus, and includes topics such as limits, sequences, series, functions, continuity, differentiation and integration. It is one thing to be able to use a mathematical tool, but it is also important to know when the procedure has a firm logical basis so that it can be used with confidence.

#### Applied Mathematics and Mathematical Methods

(i) Applied Mathematics 1 and 2

One of the reasons why mathematics is such an important subject is that it is possible to use it to provide answers to questions raised in other areas of study. One of the most successful applications of mathematics is to mechanical and dynamical systems, and this forms the central theme of this introductory course in applied mathematics. Many students will have done some work in this area already, if they have taken A level or AS level Mechanics/Applied Mathematics or Physics. However, the course is designed so that no previous knowledge of the subject matter is assumed other than that required for GCSE Physics or equivalent. However, students who have not done any A level Mechanics/Applied Mathematics are strongly advised to do some preliminary reading before starting their degree.

The courses use vector methods and the solution of differential
equations to study such topics as the three-dimensional motion
of a particle
under a variety of force fields, oscillations, Newtonian mechanics
and central orbits.

#### (ii) Mathematical Methods 1 and 2

Much in these courses is a continuation of the sort of pure mathematics studied at A level. It is aimed at extending the range of mathematical tools at students’ disposal and at providing them with instruction in their use. These courses revise parts of A level Pure and Further Mathematics and include such important techniques as the solution of differential equations, functions of several variables, multiple integrals, vector calculus, partial differential equations and numerical methods. The course also includes some topics on elementary probability and a comprehensive course in Python programming.

## Second Year

In the first term the algebra course (Algebra 3) concentrates on linear algebra where topics such as quadratic forms, diagonalization of matrices and other canonical forms, and the theory of determinants are considered. The analysis course (Analysis 3) continues with the study of functions of a complex variable – a very elegant and useful subject. The mathematical methods course (Mathematical Methods 3) studies the solution and properties of certain partial differential equations which arise, in particular, in theoretical physics. The applied mathematics course continues with the study of fluid mechanics in which the department has particular expertise. Students will complete a group project after the examinations in the third term, providing an "apprenticeship in research"

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