# Mathematics and Mathematics with Physics

**Please note that for applicants applying for Maths from autumn 2018 onwards, all Cambridge colleges, including St Catharine’s, will be asking for full A-levels in both Maths and Further Maths. This is important to consider if you are starting A-levels in 2017. As a result, the STEP syllabus will change and St Catharine’s will be asking all applicants to sit STEP 2 and STEP 3. We will, as before, take full account of cases in which applicants have had to self-teach elements of Further Maths A-level, or STEP. For candidates taking the IB, we would require IB Higher Level Mathematics.**

St Catharine’s accepts students for both the mathematics course and the mathematics with physics course. University mathematics is not at all like school mathematics. While it does involve quantities of long calculations, it also involves putting considerable effort into understanding how the ideas behind these long calculations fit together, how they work, how they can be extended and when they will fail. A degree in mathematics can be enormous intellectual fun, providing an opportunity to think about many of the hardest problems facing the modern world and a structure to guide the search for answers.

**Course structure**

To avoid extra paperwork students should apply for the **full four-year course** although the majority will graduate after three years and leave. Students may choose to study either Mathematics or Mathematics with Physics in their first year. At the end of the first year students who take the ‘with physics’ option must choose between the Mathematics Tripos and the Natural Science Tripos. The Mathematics with Physics course may involve a higher workload for those students who make their second year course choice quite late in the year.

The Mathematics with Physics course caters for students who wish to study physics as a part of the Natural Science Tripos in their second year but also to take more mathematics than the first year NST course allows. It also caters for students who are unsure whether they wish to continue as mathematicians or as physicists. These students take three of the four mathematics papers and an additional physics paper in their summer exams. At St Catharine’s they have a Director of Studies in Mathematics, who is responsible for arranging their preparation for the mathematics papers, and a Director of Studies in Physics who organises their physics teaching.

The Mathematics course is designed to provide all mathematics students with a solid foundation in the subject before they choose to specialise in pure, applied or applicable subject areas. It also caters extremely well for students with a main interest in mathematical physics. First-year students reading for the Mathematics Tripos prepare compulsory courses for four mathematics exam papers which everyone sits at the end of their first year. (These exams are not modular and there are no retakes. Except in extenuating circumstances, students who fail their exams are expected to leave the course and the University.) The mathematics first-year courses are intended to create common ground so that the students speak the same mathematical language and have the same level of knowledge in a broad range of topics drawn from algebra, analysis, geometry, number theory, probability, differential equations, dynamics and calculus. The teaching style is only slightly more formal than at school. Students are also encouraged to attend extra courses in the first year, which are examined in the second year. This allows well-prepared students more flexibility in their timetables.

The second year begins with more core material in both pure and applied topics in the first term. These include topology, analysis, Markov chains, quantum theory and linear algebra. The presentation style expected from the students is more formal in character. In the second term, the students can begin to specialise. Most students do not attend all the available courses and revise, for exams, only a small subset of the courses which they study. The exams consist of four compulsory papers at the end of the year. Each three-hour paper consists of a mixture of questions on different topics. There is also an optional computing component which is assessed during the year. The computing marks are added to the exam total at the end of the year. At the end of the second year, a very small number of students may choose to switch to another subject. Popular choices of alternative final year subjects include Part 2 NST Astrophysics or Management Studies.

In the third year of the Mathematics Tripos there is an enormous variety of specialist topics available for study. There are about 38 courses of different lengths and levels of difficulty, so that students can choose a timetable which suits their interests and abilities. The course is examined at the end of the year in four three-hour papers, each of which contains a mixture of different topics. Once again, there is an optional computing component with the marks added to the exam total at the end of the year. This is an opportunity for students to indulge themselves and devote the year to their favourite topics. Most students study fewer than ten of the available topics and take about eight to the exams.

Students who attain a Class 1 or a very high Class 2.1 in their third-year exams may be accepted into a fourth year which is colloquially known as Part 3 of the Mathematical Tripos. Students from Cambridge who take Part 3 obtain both a BA (Bachelor of Arts Honours) degree and a MMath (Master of Mathematics) degree. Part 3 Mathematics is also open as a one-year course to students with Class 1 Honours degree in appropriate subjects from other good universities. Students from outside Cambridge obtain a MASt (Master of Advanced Study) degree. Part 3 is intended as a preparation for research and consists of a vast number of specialist courses from which students may choose their exam topics. The exams consist of individual subject papers at the end of the year. There is an optional assessed essay component. The course has a reputation as one of the most demanding courses, in the world, available at this level.

Mathematics teaching at Cambridge consists of lectures and supervisions. Lectures are run by the University and are attended by all the mathematics students at Cambridge colleges. The lengths of the lecture courses vary but, for example, a course of 24 lectures will be accompanied by 4 problem sets issued by the lecturer. Supervisions are arranged by the colleges and these teaching sessions usually allocate one hour with one supervisor, who is a member of the teaching staff or a graduate student, to each pair of undergraduate students for each problem set. The courses are fast and intense. To get the most out of a supervision, the undergraduate students need to be willing to ask "the dumb question". The supervisions are an essential opportunity for assessment of progress but, more importantly, the best line of defence for undergraduates to get help with difficulties before their exams.

**Faculty website:** http://www.maths.cam.ac.uk/

In a typical year, about eight to ten students come up to St Catharine's to read Mathematics. This is a similar number to many other colleges and is intended to form a year group which is large enough so that the students do not feel isolated but is not overwhelmingly large. The number of Part 3 students can vary considerably but about five is typical, so the total number of undergraduate students reading Mathematics at St Catharine's is therefore usually about thirty. The mathematicians are encouraged to form a friendly and cohesive social group. The college expects the students to demonstrate a willingness to work together and help each other in their studies. There is no such thing as a typical mathematician. The students have a variety of interests, mathematical and recreational and they are encouraged to make the best use of all the intellectual and cultural enrichment opportunities which St Catharine’s has to offer.

At St Catharine's many of the first-year and second-year supervisions are given by Fellows of the college. The remaining supervisions are usually given by Fellows of other colleges who have a long connection with St Catharine’s. St Catharine’s employs at College Teaching Officer who is a teaching specialist for the college. In common with most colleges, the third year supervisions are arranged by groups of colleges who share supervisors teaching specialist topics.

St Catharine’s, in common with other colleges, seeks the strongest young mathematicians. Applicants come from a wide range of social and educational backgrounds. They share a genuine passion for their subject, raw talent, a stubbornness which will allow them to attempt and complete difficult tasks, imagination, a degree of emotional robustness and an ability to work individually or as a part of a team. Applicants who do not find the Mathematics A2 course very easy are unlikely to be able to cope with Cambridge Mathematics.

St Catharine's is looking for students who can cope with the high workload and difficult material in mathematics at Cambridge. Applicants need to demonstrate that they can cope with a workload which is equivalent to three full ‘solid’ (academically demanding) A2s taken within two years. In the absence of special circumstances candidates who have taken three or more years to cover the same workload as is normally covered in two years are unlikely to obtain conditional offers. At many schools pupils take mathematics or languages a year early. Candidates who take an A2 in year 12 but are concurrently studying for their remaining two A2s in solid subjects and take those in the following year will certainly be considered. Many students take A2 mathematics in one year and then take A2 further mathematics the following year instead of studying for the two exams concurrently, which is perfectly acceptable.

Many will have studied Mathematics, Further Mathematics, Physics and/or Chemistry at A2 level, but this is very far from being the only possibility. Applicants who offer an unusual combination of A2s because they already have a career path in mind are not disadvantaged. St Catharine’s will give equal consideration to those taking International Baccalaureate, Scottish Highers or Pre-U qualifications. All mathematics applicants should make every effort to take as much mathematics as possible within their study program before starting their degree course. In common with many Cambridge Colleges, it is assumed that applicants who are able enough to cope with the course in Cambridge are able enough to self-teach most of the material up to the equivalent of A2 Further Mathematics. Applicants are also encouraged to enhance their knowledge by making use of free online help such as, for example, NRICH. Students who may have a choice between mechanics and statistics modules and enjoy both subjects will probably find the mechanics more helpful in their first year in Cambridge, but the choice is quite evenly balanced and no one is disadvantaged should they prefer to take statistical modules.

Many schools no longer offer Further Mathematics at A2 level and some mathematics applicants to St Catharine's have only AS Further Mathematics or even single subject Mathematics at A2. Those who cannot obtain teaching to A2 Further Mathematics are often self-taught in one or more of the Further Mathematics modules. Some applicants for mathematics will be asked to self-teach to AS Further Mathematics level or A2 Further Mathematics level as a part of their offer. It is important for applicants to inform their potential colleges and universities if they are having to self-teach particular subject areas.

Conditional offers can vary quite widely and depend on supporting paperwork, interviews and background support available to applicants. There is not really a typical offer. The college will only make conditional offers to candidates it considers capable of attaining at least a good Class 2.1 in their degree course in Cambridge. Sometimes, in a year with exceptional pressure on places, the college may pool a candidate who in another year would have given a conditional offer. Pooled candidates are referred to other Cambridge colleges who may have some spare capacity and be able to give them a conditional offer. Therefore any candidate who fails to secure a conditional offer should only assume that, **on the day**, there were candidates who seemed stronger. They should **not** assume that they are not potentially good mathematicians.

Good individual module scores are important in Mathematics and Further Mathematics A2, with most of the successful applicants having module scores above 90%. As a very rough guide, St Catharine’s might ask for A*A*A at A2 often including an A* in Further Mathematics or the equivalent and the remaining grades in solid subjects

The pre-university qualification offers are often around D2, D2 and D2 but possibly with a D1 in some cases.

For Scottish Highers the offers are often A1, A1, A1.

For the International Baccalaureate the offers are around 7, 7, 7 at IB higher level with one being in mathematics.

Almost all candidates are asked to sit two STEP papers. Those taking the equivalent of the full double mathematics at A2 will usually be asked for STEP 2 and STEP 3 papers, often at both at grade 1. Students with the equivalent of AS Further Mathematics, Scottish Highers or International Baccalaureate will often be asked for STEP 1 and STEP 2 either both at grade 1 or sometimes at grades 1 and S in some or in any order. Some candidates are given conditional offers which require them to do some extra coursework in order to sit their STEP papers. Most students have only informal help with STEP at school. STEP past papers, solutions and a guide to STEP-style problems called 'Advanced problems in mathematics' are all available from OCR publications, Mill Wharf, Mill St, Birmingham B6 4BU. Dr S Siklos also has a book on how to do STEP questions. See also http://www.damtp.cam.ac.uk/user/stcs/STEP.html

St Catharine’s places a high importance on the role of the mathematics course as a preparation for later life. As part of their development of transferable skills, the mathematics students at St Catharine's are encouraged to work together and support each other in their studies so that students with a weaker background can expect a high level of support from their director of studies, supervisors and peers. All students are expected to demonstrate a willingness to be supportive. The students are also encouraged to consider Spring Week placements after their first year, when that is appropriate, and also longer work placements at the end of their second year. The college encourages mathematics students to take opportunities where available within their course or within the university community to develop computing, presentation and report writing skills. Those who arrive with some language skills are encouraged to maintain and develop them. Many of the mathematicians go on to further training after graduation, but many choose to go into the employment market where there is a huge variety of options from which to choose career paths. Mathematics graduates career choices include academic careers as mathematicians, teachers, engineers, consultants, bankers, traders. Some have moved across disciplines and employed their skills in other subject areas. Employers are eager to take on mathematics graduates because they see the degree as a guarantee of an ability reason clearly and to master difficult material quickly. It is a degree course which opens doors rather than closing them. The college takes the need to prepare for a future career very seriously.

As a college, Catz has a strong reputation for being friendly and sociable. Everyone is encouraging and supportive, especially maths students, where everyone has their own strengths and weaknesses. Each year group has around 8 mathematicians, so you get to know everyone really well. In addition, the college has its own maths society (the Hyperbolics) which organises various social events, so you get to know people from different years, including post-grads. Last year, we had pub trips, bowling and a garden party, as well as the inaugural Hyperbolics football match!

Unfortunately, it's quite easy to become a bit lazy at Catz, especially if you're a mathematician, since everything is so convenient! Lectures are less than a 5-minute walk away (or a 3-minute sprint!), and the majority of first year supervisions take place in college, so you never have to travel far. This also means you see your supervisors around and get to know them quite well - they're all really friendly as well. We've also got a really great library should you need any books, and the librarians are happy to order in anything extra.

Sam Burr and Matt Temple (second-year Mathematicians)