Contents
 Mathematics at Oxford
 The Mathematics Course
 The Mathematics and Statistics Course
 The Mathematics and Philosophy Course
 The Mathematics and Computer Science Course
 The Mathematical & Theoretical Physics Fourth Year
 Admissions
 Careers
 Information for International Students
 Preparation for the Oxford Mathematics Course
 Recommended Mathematics Reading
 Puzzles
Mathematics at University
Few people who have not studied a mathematics or science degree will have much idea what modern mathematics involves. Most of the arithmetic and geometry seen in schools today was known to the Ancient Greeks; the ideas of calculus and probability you may have met at Alevel were known in the 17th century. And some very neat ideas are to be found there! But mathematicians have not simply been admiring the work of Newton and Fermat for the last three centuries; since then the patterns of mathematics have been found more profoundly and broadly than those early mathematicians could ever have imagined. Looking through any university’s mathematics prospectus you will see course titles that are familiar (e.g. algebra, mechanics) and some that appear thoroughly alien (e.g. Galois Theory, Martingales, Information Theory). These titles give an honest impression of university mathematics: some courses are continuations from school mathematics, whilst others will be thoroughly new.
The clearest change of emphasis is in the need to prove things, especially in pure mathematics. Much mathematics is too abstract or technical to simply rely on intuition, and so it is important that you can write clear and irrefutable arguments, which make plain to you, and others, the soundness of your claims. But pure mathematics is more than an insistence on rigour, arguably involving the most beautiful ideas and theorems in all of mathematics, and including whole new areas, such as topology, untouched at school.
Mathematics, though, would not be the subject it is today if it hadn’t had been for the impact of applied mathematics and statistics. There is much beautiful mathematics to be found here, such as in relativity or in number theory behind the RSA encryption widely used in internet security, or just in the way a wide range of techniques from all reaches of mathematics might be applied to solve a difficult problem.
Courses
Oxford offers threeyear and fouryear degrees in Mathematics, and also in three joint courses:
 Mathematics and Statistics,
 Mathematics and Philosophy,
 Mathematics and Computer Science.
Decisions regarding continuation to the fourth year do not have to be made until the third year. There is also a separate fourth year stream on Mathematics and Theoretical Physics.
Each degree boasts a wide range of options, available from the second year onwards. They will train you to think carefully, critically and creatively about a wide range of mathematical topics, and about arguments generally, with a clear and analytical approach. Details of the courses are given below.
Lectures
Mathematicians from across all the colleges come together for lectures which are arranged by the University. This is usually how students first meet each new topic of mathematics. A lecture is a 50 minute talk, usually given in the Mathematical Institute, with up to 280 other students present. The lecturer discusses the material, gives examples, and make notes at the boards. The lecturer will usually be a member of the Faculty, and may perhaps also be a tutor at one of the colleges. Independent study is also important, because the lectures contain a lot of mathematical content, and it falls to a student to review their lecture notes and consult textbooks, determine which elements are still causing difficulty, and try to work through these. The lecturer will usually set problem sheets or exercises based on the material and these problems will typically form the basis of the tutorials in college.
Smallgroup teaching
Mathematics students at Oxford are both members of the University and one of 29 colleges, and mathematics teaching is shared by these two institutions. Oxford’s collegiate system makes the teaching, and the daytoday routine, a rather different experience from other universities.
Tutorials are an important part of the teaching of mathematics in Oxford. These are hour long lessons organised in college between a tutor, who is usually a senior member of the college, and a small group of students (typically a pair). This form of teaching is very flexible, allowing a tutor time with the specific difficulties of the group and allowing the students opportunities to ask questions. It is particularly helpful for first year mathematicians who naturally begin university from a wide range of educational backgrounds. College tutors follow closely their students’ academic progress, guide them in their studies, discuss subject options and recommend textbooks. Tutorials do not count towards the degree classification awarded; rather, they are a way to learn and grow, ahead of the endofyear examinations.
Each college is more than a hall of residence, it is a society in its own right, and the students in college often become friends during university and beyond. See www.ox.ac.uk/ugcolls and www.maths.ox.ac.uk/r/colleges for links to the colleges’ webpages.
By the third and fourth years the subject options become much more specialized and are taught in intercollegiate classes organized by the departments. These are given by a class tutor (usually a member of Faculty, or a graduate student who has taken and passed teaching qualifications) and a teaching assistant. They range in size – typically there are 812 students – and there is again plenty of chance to ask questions and discuss ideas with the tutors.
The Invariant Society
The Oxford University Invariant Society is the student mathematical society. Its primary aim is to host weekly popular mathematicallyrelated events, like talks by notable speakers, on a wide variety of topics. Past speakers have included Benoit Mandelbrot, Sir Andrew Wiles, Sir Roger Penrose, Marcus du Sautoy, and Simon Singh. We also host industryrelated events, for those who want to discover the use of mathematics beyond academia. The Invariants also publishes a yearly magazine and runs a weekly puzzle competition. Website: www.invariants.org.uk Facebook: https://www.facebook.com/oxford.invariants
The Mirzakhani Society
The Mirzakhani Society is the society for women and nonbinary students studying Maths at Oxford University. The society is named after Maryam Mirzakhani, the first female mathematician to win a Fields Medal. Every week we host 'Sip & Solve' where we meet and have drinks and snacks between lectures  it's a great way to meet girls doing maths from other years and colleges and work on problem sheets together! We also have events such as panel discussions, a termly lunch with Mathematrix (our sister postgrad society), talks by female mathematicians, and trips such as our annual exchange with female mathematicians at Cambridge. Website: www.mirzakhanisociety.org.uk
LGBTI³
LGBTI³ is a studentled group in the Mathematical Institute which aims to provide a platform and network for LGBTIQ+ people and Allies, undergraduates and graduates, to discuss and explore topics related to life in academia and issues faced by LGBTIQ+ people, in Maths and, more broadly, in STEM. LGBTI³ hosts discussions three times a term accompanied by a free lunch. The events are relaxed and informal, and people may come and go as they please.
The Mathematics Course
Mathematics is the language of science and logic the language of argument. Science students are often surprised, and sometimes daunted, by the prevalence of mathematical ideas and techniques which form the basis for scientific theory. The more abstract ideas of pure mathematics may find fewer everyday applications, but their study instils an appreciation of the need for rigorous, careful argument and an awareness of the limitations of an argument or technique. A mathematics degree teaches the skills to see clearly to the heart of difficult technical problems, and provides a “toolbox” of ideas and methods to tackle them.
The Mathematics degrees can lead to either a BA after three years or an MMath after four years, though you will not be asked to choose between these until your third year. Both courses are highly regarded: the employability of graduates of both degrees is extremely high, and BA graduates can still go on to second degrees, Masters or PhDs. The fouryear course has been accredited by the Institute of Mathematics and its Applications as meeting the educational requirements for the Chartered Mathematician (CMath) designation, and the threeyear course has similarly been accredited to meet the educational requirements when followed by subsequent training and experience in employment equivalent to a taught Masters degree.
The first year consists of compulsory courses that all mathematics students take. A list of current courses is given in the table below. Computational Mathematics involves practical computing classes using MATLAB – a popular piece of mathematical software. The course involves introductory sessions in the first term, and two projects in the second term, which count towards the grade for the year. Otherwise, the year is assessed through endofyear exams, with students taking five papers, each 23 hours in duration.
In the second year students have some choice over the content that they study. There are currently three compulsory courses in the first term that are each contain prerequisite knowledge for many of the later courses, and then secondyear students take five or six long options and three short options.
In the third year still more options become available, giving students the opportunity to specialise, or to take courses in a variety of topics. There are currently courses on a wide range of topics in pure mathematics and applied mathematics, and options such as the philosophy or the history of mathematics, an extended essay, or a structured project. A current list of courses is shown in the table below. Students choose eight units (some "double unit" courses count as two units), with written exams at the end of the year, some of which may be replaced by practicals or projects, depending on the options chosen. At most four of those units can come from the Statistics, Computer Science, or from the Other options, with at most two units from any one of those categories. If you're interested in taking more Statistics options than that, then you might be interested in the joint honours Mathematics and Statistics degree course, which offers students the option to take more than 25% of their thirdyear courses in Statistics.
Currently an upper second over second and third year, as well as an upper second in third year alone, is required to progress to fourth year. The fourth year has even more options (too many to list below!) with a current list at https://coursesarchive.maths.ox.ac.uk/overview/undergraduate#49743.
First Year  Second Year  Third Year 

Core courses
Long options
Short options

Mathematics department options
Statistics options
Computer Science options
Other options

The Mathematics and Statistics Course
The twentieth century saw Statistics grow into a subject in its own right (rather than just a single branch of mathematics), and the applicability of statistical analysis is all the more important in the current information age. The probabilities and statistics associated with a complex system are not to be lightly calculated, or argued from, and the subjects contain many deep results and counterintuitive surprises.
The Mathematics and Statistics degrees (a three year BA or a four year MMath) teach the same rigour and analysis, and many of the mathematical ideas, as the Mathematics degrees and further provide the chance to specialize in probability and statistics, including some courses only available to students on the Mathematics and Statistics degrees. The course has been accredited by the Royal Statistical Society.
The first year of the joint degree is identical to the first year of the Mathematics degree, but the second year has some differences from the Mathematics degree. The courses in Probability and Statistics are compulsory for Mathematics and Statistics students (rather than optional), and Mathematics and Statistics students have access to an additional long option on Simulation & Statistical Programming. This course introduces Monte Carlo methods, which are important for modern statistical inference, and students are taught to program in R, a programming language widely used in statistics. Overall, Mathematics and Statistics students take the same number of long and short options as students on the Mathematics degree course, choosing from the options in the table above.
In the third year students take eight units of courses. The course in Applied and Computational Statistics is compulsory (and counts as two units) and at least two courses must be taken from the following;
 Applied Probability (covering probabilistic modelling with continuoustime Markov chains, and applications such as queuing theory)
 Foundations of Statistical Inference (understanding how data can be interpreted in the context of a statistical model)
 Statistical Machine Learning (covering a number of different machine learning methods and how to choose an appropriate method)
 Statistical Lifetime Models (estimating eventtime distributions, which appear in many social and medical data contexts)
Mathematics and Statistics students can also choose courses from the Mathematics department options, from the list above.
Currently an upper second over second and third year, as well as an upper second in third year alone, is required to progress to fourth year. The fourth year students are taught jointly alongside the fourthyear Mathematics students, and take eight to ten units from the schedule of courses, which must include a dissertation on a statistics project, and two units chosen from statistics or probability courses.
The Mathematics and Philosophy Course
This course brings together two of the most fundamental and widely applicable of intellectual skills. Mathematical knowledge, and the ability to use it, is the most important means of tackling quantifiable problems, while philosophical training encourages the crucial abilities to analyse issues, question received assumptions and articulate the results clearly. Logic, and the philosophy of mathematics, provide natural bridges between the two subjects.
In the first year, Mathematics and Philosophy students take (broadly speaking) the pure mathematics half of the topics from the Mathematics degree (Introduction to University Mathematics, Introduction to Complex Numbers, Linear Algebra (I, II), Groups and Group Actions, Analysis (I, II, III), Introductory Calculus, and Probability), as well as Philosophy courses on the following topics:
 Introduction to Logic
 General Philosophy
 Elements of Deductive Logic
Mathematics and Philosophy students also read Frege's Foundations of Arithmetic in the first year.
Across the second and third years of study, there are currently compulsory courses in each discipline. In the second year, students currently choose two secondyear Mathematics options shown in the table below.
In the third year, students choose options from Mathematics and from Philosophy, with certain rules on the number of options to be taken from each. There are many options currently available; these are shown in the table below.
Mathematics courses  Philosophy courses 
Core courses
Secondyear options
Thirdyear options
Computer Science options

Core courses
Options

There are mathematics examinations at the end of the second year, in total 7½ hours. At the end of the third year, there are six three hour papers (or equivalent), with at least two in mathematics and at least three in philosophy.
Currently an upper second over second and third year, as well as an upper second in third year alone, is required to progress to fourth year. The fourth year of the course allows you the opportunity to specialize entirely in mathematics, in philosophy or to retain a mixture. There are examinations at the end of the year with the option of replacing some of these papers with a philosophy thesis or a mathematics dissertation.
Informal descriptions of the philosophy courses can be found at https://www.philosophy.ox.ac.uk/coursedescriptionsfirstpublicexamina... (for first year courses) and at https://www.philosophy.ox.ac.uk/coursedescriptionsfinals (for later options).
Recommended Philosophy Reading
Prior study of philosophy is in no way a prerequisite for this degree. It is clearly sensible, though, to find out more about the subject first. Here are some recommendations for philosophy and logic reading, to complement the earlier list of mathematical texts. Selected reading from one or more, or similar texts, will help you get a flavour of the degree.
 Simon Blackburn, Think
 One or more of the shorter dialogues of Plato such as Protagoras, Meno or Phaedo. (Each widely available in English translation.)
 Bertrand Russell, The Problems of Philosophy
 Bertrand Russell, Introduction to Mathematical Philosophy
 Jonathan Glover, Causing Death and Saving Lives
 A.J. Ayer, The Central Questions in Philosophy
 Martin Hollis, Invitation to Philosophy
 A.W. Moore, The Infinite
 Thomas Nagel, What Does It All Mean?
 P.F. Strawson, Introduction to Logical Theory
 Earl Conee and Theodore Sider, Riddles of Existence
 Øystein Linnebo, Philosophy of Mathematics
The Mathematics and Computer Science Course
Mathematics is a fundamental intellectual tool in computing, but computing is increasingly also a tool in mathematical problem solving. This course concentrates on areas where mathematics and computing are most relevant to each other, emphasizing the bridges between theory and practice. It offers opportunities for potential computer scientists both to develop a deeper understanding of the mathematical foundations of the subject and to acquire a familiarity with the mathematics of application areas where computers can solve otherwise intractable problems. It also gives mathematicians access to both a practical understanding of the use of computers, and a deeper understanding of the limits to the use of computers in their own subject. This training leads to a greater flexibility of approach and a better handling of new ideas in one of the fastest changing of all degree subjects.
The first year currently consists of lectures on the following topics:
 Introduction to University Mathematics
 Introduction to Complex Numbers
 Linear Algebra (I, II)
 Groups and Group Actions
 Analysis (I, II, III)
 Probability
 Continuous Mathematics
 Functional Programming
 Design and Analysis of Algorithms
 Imperative Programming (I, II, III)
Most of the computer science topics have associated practicals which must be passed in order to progress.
At the beginning of the second year students take two of the core mathematics courses; Linear Algebra, and Metric Spaces and Complex Analysis, and take two core Computer Science courses on Models of Computation and Algorithms. In the second year, students will also typically choose two of the long options from the Mathematics course (excluding Differential Equations), or exchange a long option for two or three short options (chosen from Number Theory, Group Theory, Projective Geometry, Introduction to Manifolds, or Graph Theory). Students also undertake a group design practical, and take additional Computer Science options in either their second or third years. A current list of these options can be found at www.cs.ox.ac.uk/undergradcourses. Students sit exams at the end of the second year, and some of the computer science courses include practicals which must be passed in order to progress.
In the third year there is a still wider range of options available – students take ten courses in all, at least two of which must be in Mathematics and at least four of which must be in Computer Science. See www.cs.ox.ac.uk/teaching/mcs/PartB/ for further information on all the options currently available. In the fourth year students may choose to specialize entirely in mathematics or computer science or to retain a mixture. Every student must take either a Mathematics Dissertation (and six further units) or a Computer Science Project (and five further units); each unit is assessed by examination or equivalent.
The Mathematical & Theoretical Physics 4th Year
This course is a fourthyear Masterslevel course, which unites these two classic disciplines. Theoretical physics utilizes many mathematical techniques, and there are many elegant mathematical proofs to be found in string theory, quantum field theory, and other realms of study usually considered to be applied mathematics.
As this is a fourthyear course, you cannot apply for it as a prospective undergraduate. Instead students who are in their third year of Mathematics, Physics, or Physics and Philosophy degrees can apply to transfer onto this course. As with our other joint degrees, in this course you may choose to be highly specialised or gain a broad knowledge of the discipline.
The course covers the areas of Quantum Field Theory, Particle Physics and String Theory, Theoretical Condensed Matter Physics, Theoretical Astrophysics, Plasma Physics and Physics of Continuous Media, and Mathematical Foundations of Theoretical Physics. Students currently choose at least 10 options (16 hour lecture courses) from a sizeable list of courses. Students may also choose a maximum of 3 options from the Mathematics degree or the Physics degree.
For more information please see https://mmathphys.physics.ox.ac.uk.
Admissions
The following applies to prospective students for the Mathematics degree, or for any of the joint degrees, who are considering applying in October 2021 for entry in 2022 or 2023. Much like applying for any other UK university, applications to Oxford are made through UCAS, though the deadline is earlier, on October 15th. Your application may include a preference for one college, or may be an “open” application in which case a college is assigned to you.
Mathematics Admissions Test
The Mathematics Admissions Test (MAT) is sat by candidates in their schools, colleges or at a test centre. The test, which lasts 2½ hours, will be in the same format as in 200720 and these past tests, and two further specimen tests, are available with solutions at www.maths.ox.ac.uk/r/mat.
The date of the test is Wednesday 3rd November 2021.
All applicants attempt the first ten multiplechoice questions, and then four from six longer questions depending on their proposed degree. Instructions are in the test on which questions to complete. No aids, calculators, formula booklets or dictionaries are allowed. A syllabus for the test is available at the above website; it roughly corresponds to material from the first year of Alevel Mathematics, though the questions are devised to test for a deeper understanding of, and imagination with, the syllabus’ methods and material.
The distribution of the test will be administered by Cambridge Assessment Admissions Testing (CAAT) and all applicants must separately register with them by the application deadline (15th October), through their school or college or through a test centre. Schools can register with CAAT to become test centres, but please note that the deadline to become a new centre is 30 September; see www.ox.ac.uk/tests.
All applicants are expected to take the Mathematics Admissions Test and must notify the college and University as soon as possible in the event of any potential difficulties or schedule clashes.
Interviews
Applicants will be shortlisted for interview on the basis of their test marks and UCAS form, with around 3 applicants per place being shortlisted. In December 2021, all interviews will be held online (as in 2020). The advice in this section is valid for online interviews, and further information will be added as soon as this becomes available, including guidance on the technology we will use.
Typically, interviews last 2030 minutes with two interviewers, and you may have more than one at a particular college. Applicants for the joint degrees with Philosophy and with Computer Science should expect at least one interview on each discipline. In an interview, you may be asked to look at problems of a type that you have never seen before. We want to see how you tackle new ideas and methods and how you respond to helpful prompts, rather than simply find out what you have been taught. Interviews are academic in nature, essentially imitating tutorials, this being how much of Oxford’s teaching is done; feel free to ask questions, do say if unsure of something, and expect hints.
Offers
Around 10% of all applicants are currently made offers. After the interviews, if your application is unsuccessful with your first college, another may make you an offer; around 2530% of offers made are not with the applicant’s first college. Applicants are informed of the decision by midJanuary.
In Mathematics, Mathematics and Statistics, Mathematics and Philosophy, the standard offer for students taking Alevels is A*A*A with A* in Mathematics and in Further Mathematics, and A in any third subject. An exception is made for Alevel candidates at schools or colleges that cannot teach Further Mathematics, in which case the standard offer is either A*AAa with A* in Mathematics and a in ASlevel Further Mathematics or A*AA with A* in Mathematics, as appropriate. In Mathematics and Computer Science, the standard Alevel offer is A*AA with A*A between Maths and Further Maths if taken, and otherwise with the A* in Maths.
Note that Alevel Philosophy is not required for Mathematics and Philosophy. An essaybased subject may be helpful, but is not essential.
For those taking Advanced Highers, the standard offer is AAB/AA, depending on the number of Advanced Highers that you are able to take, and including A in Mathematics.
The standard offer for students taking the International Baccalaureate is 39 with 766 at HL including 7 in HL Maths. Information on typical offers involving other qualifications can be found at www.ox.ac.uk/intquals.
Admissions FAQs
Q: Which college is the best for Mathematics?
A: None of them; the colleges all teach the same degree, and their students attend the same lectures, are set the same problem sheets for tutorials, and take the same exams. Colleges differ much more in their size, age, location than they do in their teaching of mathematics. Note that not all colleges take students in the joint degrees see www.maths.ox.ac.uk/r/colleges for details. Other than that, there is no right or wrong way to choose a college. You could review college websites or prospectuses. If you can't choose or don't want to choose, you are welcome to make an open application, in which case a college will be assigned to you at random (and you will then be treated exactly as if you applied to that college).
Q: Do I need Further Mathematics?
A: Further Mathematics Alevel is highly recommended. For Mathematics, Mathematics and Statistics, or Mathematics and Philosophy, if your school or college offers Further Mathematics Alevel, then we would usually expect you to be studying it. We recognize that it is not available to some students; in which case we can make a reduced offer based on the mathematics that you are being taught. Further Mathematics helps with the transition to university, but our courses are taught from first principles, and the tutorials are very helpful in making sure that we support all students from all their educational backgrounds.
Q: I'm doing four A levels, do I get a harder offer? Do I get an easier offer?
A: No, the same standard offer applies to you (for Mathematics, that would be A* in Maths, A* in Further Maths, A in any third subject so one or the other of your third and fourth subjects).
Q: Are GCSE grades important? What are the minimum GCSE requirements?
A: We have no minimum GCSE requirements. We do look at your GCSE grades (if you have some) as one measure of your overall academic achievement. This is contextualised with cohort data about typical GCSE results of the school where you achieved your GCSEs, where we have that data. All applicants take the MAT, which is more relevant than GCSE grades because it is entirely on material relevant to your intended course of study.
Q: How do I prepare for the Mathematics Admissions Test (MAT)?
A: Revise mathematics that you've learned in school, and look at the MAT syllabus (www.maths.ox.ac.uk/r/mat). Try some of the past papers available on www.maths.ox.ac.uk/r/mat, and familiarise yourself with the format and style of the test. Try other mathematics problems (not necessarily MAT), to practice getting stuck and unstuck on mathematics problems.
Q: How do I prepare for interview?
A: Interviews are academic in nature. Revise past material you have already met. In an interview a tutor will typically discuss problems involving new mathematical ideas. You could practice talking to a school teacher or a friend about a favourite area of mathematics or go through a past MAT question with them. The tutor will be interested to see how you respond as the problem is adapted and new ideas introduced, and in how well you can solve problems and express your arguments. You can work on this by practicing solving problems (see MAT advice above).
Q: I haven't studied philosophy Alevel; can I still apply to the Maths and Philosophy degree?
A: Yes, absolutely, though you should be able to demonstrate an interest in philosophy (e.g. through further reading, suggested above).
More information
You can find out more at one of our open days. There are two Departmental Open Days on 24 April and 01 May 2021, and three University Open Days on 30 July, 01 July, and 17 September 2021, when colleges are also open. These include talks on Mathematics and the joint degrees, live broadcasts, and online questionandanswer sessions for you to talk to tutors and current students. See www.maths.ox.ac.uk/opendays for the Departmental Open Days, and see www.ox.ac.uk/opendays for full details of the University Open Days.
You can contact us with any enquiries about admissions relating to Mathematics or its Joint Degrees by email: undergraduate.admissions [at] maths.ox.ac.uk.
The Mathematics Department welcomes applications from disabled students and is committed to making reasonable adjustments so that disabled students can participate fully in our courses. You can find out more about the accessibility of our building at: www.accessguide.ox.ac.uk/andrewwilesbuilding. We encourage prospective disabled students to contact the Department’s Administrator (departmentaladministrator [at] maths.ox.ac.uk) at their earliest convenience, to discuss particular needs and the ways in which we could accommodate these needs. See also: https://edu.admin.ox.ac.uk/disabilitysupport  the University Disability Office’s website which includes FAQs and further information.
Careers
Demand for mathematics graduates has always been strong, but has been growing rapidly with the increased use of highly technical mathematical models and the growing prevalence of computers.
Over 30% of our graduates continue on a course of further study, ranging from a research degree in mathematics to a postgraduate course in teacher training. Mathematics at Oxford has many very active research groups, ranging from Geometry, Group Theory, Topology, and Number Theory to the applied research groups of the Centre of Mathematical Biology, the Oxford Centre for Industrial and Applied Mathematics, Numerical Analysis and Stochastic Analysis. You can find out more at: www.maths.ox.ac.uk/studyhere/postgraduatestudy.
There are no clearly defined career routes after a mathematics degree, unlike Law or Medicine. However, a degree in mathematics gives you excellent quantitative skills, which are applicable in a wide range of careers. Our graduates have gone into careers as consultants, analysts and a variety of financial roles. Additionally many of our graduates go into academicrelated positions, such as research roles in companies, the intelligence services or the civil service.
Not only are there many career options after graduating, the average starting salary for our mathematics graduates, six months after finishing their degree, was £33,000, according to data from the Destination of Leavers from Higher Education (DLHE) survey.
In order to support your future career, the University runs a Careers Service which offers free advice and services such as internship programmes at companies across the world, advice sessions from alumni, and tailored careers advice. This service is available to you for life, so we can support you whenever you need us. See www.careers.ox.ac.uk for more information.
“Studying maths at Oxford gave me the analytical and reasoning skills I use in my job as a Public Health Intelligence Officer, as well as teaching me a great deal about communicating difficult mathematical/statistical concepts and how to translate public health questions (e.g. "Does this service work?") into questions that can be answered well by data  and translating the answers back out again."
"Oxford has given me the opportunities to get where I am today through two main areas in my personal development: academia, as the drive and discipline required to complete a degree at Oxford have to come from yourself; and the interpersonal skills developed through sport, student politics, and relaxing in the bar with very bright and interesting people."
Preparation for the Oxford Mathematics Course
Whilst some courses, early in the degree, have a firstprinciples approach and assume very little mathematical knowledge, other areas would prove rather difficult without certain ideas and techniques being familiar. The following is a list of topics, largely in pure mathematics, most of which we would expect you to have studied before starting the course (but many students will have a few gaps, especially those who have not taken Alevel Further Mathematics or equivalent):
 Polynomials and basic properties of the roots of polynomial equations.
 Partial fractions.
 Simultaneous equations.
 Inequalities and their manipulation.
 Basic properties of triangles and circles.
 Equations of the parabola, ellipse and hyperbola.
 Elementary properties of lines and planes in three dimensions.
 Arithmetic and geometric progressions.
 Product, quotient and chain rules of differentiation.
 Solving simple differential equations.
 Integration by parts.
 Recognition of the shape of a plane curve from its equation, maxima and minima, tangents and normals.
 Binomial Theorem, combinations.
 Taylor series, the binomial series for noninteger exponent.
 Matrices and determinants.
 Induction.
 Complex numbers – their algebra and geometry.
 Exponential and trigonometric expansions and Euler’s relation between them.
 Standard integration techniques and spotting substitutions.
 Secondorder differential equations with constant coefficients
As Alevel syllabuses contain varying amounts of mechanics, probability and statistics, very little prior knowledge is assumed here. You may find the early parts of some courses repeat material from your Alevel whilst other topics may be almost completely new to you. Typically though, even the “old” material will be repackaged and presented with a different emphasis to school mathematics.
After Alevel results come out in the summer, tutors usually write to students with preparatory exercises on topics like the ones above. Similar practice problems are online at www.maths.ox.ac.uk/r/practice. There is also a study guide for incoming Oxford mathematicians www.maths.ox.ac.uk/r/studyguide and a set of bridging material www.maths.ox.ac.uk/studyhere/undergraduatestudy/bridginggap.
Puzzle 1
The LotkaVolterra equations show the interaction of a population of predators (F, for foxes) and a population of prey (R for rabbits). For positive α,β,γ,δ the interactions are given by:
$$ \frac{\mathrm{d}F}{\mathrm{d}t}=\alpha R F  \beta F$$
$$\frac{\mathrm{d}R}{\mathrm{d}t}=\gamma R  \delta R F $$
 What do you think $\alpha$, $\beta$, $\gamma$, and $\delta$ represent?
 The population is at equilibrium if $\frac{\mathrm{d}F}{\mathrm{d}t}=0$ and $\frac{\mathrm{d}R}{\mathrm{d}t}$. Find and interpret all possible population equilibrium states in terms of $\alpha$, $\beta$, $\gamma$, and $\delta$.
 By considering the signs of $\frac{\mathrm{d}F}{\mathrm{d}t}=0$ and $\frac{\mathrm{d}R}{\mathrm{d}t}$ near each equilibrium state, describe what happens if the populations are close to their equilibrium values.
Puzzle 2
Fibonacci numbers are defined by the recurrence relation $F_n=F_{n1}+F_{n2}$ with $F_1=1$ and $F_2=1$. Zeckendorf’s Theorem states that every positive integer can be represented by the sum of nonconsecutive Fibonacci numbers.
 Write 400 as the sum of nonconsecutive Fibonacci numbers.
 Prove Zeckendorf’s Theorem.
 The positive integers $a_1\leq a_2 \leq a_3 \leq \dots$ have the property that, even when any one of the $a_n$ is removed from the list, every positive integer can be written as a sum of some of the remaining $a_n$ without repeat. Show that $a_n\leq F_n$ for $n\geq 1$.
Information for International Students
We welcome applications from international students – currently around 25% of our undergraduate students are from outside the EU and around 15% are from within the EU. Colleges and the department provide you with many opportunities to socialise with a wide range of people. There are over 150 University registered clubs and societies dedicated to a wide range of activities and cultures  for example the German Society, the Chinese Society, and the Italian Society all host a variety of events, including film viewings, talks, and food tasting. You can find a complete list at: www.ox.ac.uk/students/life/clubs/list. Many colleges offer storage for international students over the short vacations, and some may also offer accommodation throughout the year.
The application process for international students is the same as for UK students – applicants need to apply via UCAS by the 15th October and also, separately, register to sit the Mathematics Admissions Test (MAT). International schools and colleges can become test centres, so you can sit the MAT at your school – centres need to register by the end of September. Alternatively you can sit the MAT at a local open centre  see www.admissionstesting.org/findacentre/
All interviews in December 2021 will be held online (as in 2020).
If you are made an offer it may be conditional on you achieving particular grades in your qualifications. We accept a wide range of international qualifications – you can find a complete list at www.ox.ac.uk/intquals. If you have not been taught in English for the last three years your offer may also be conditional on you satisfying English Language requirements. See www.ox.ac.uk/englang for more information.
There are a number of scholarships available to international undergraduates – most have closing dates in midFebruary after you have received your offer. You can search for scholarships at www.ox.ac.uk/admissions/undergraduate/feesandfunding/feesfundingandscholarships/search.
You can find more information about being an international student at Oxford at www.ox.ac.uk/int.
Recommended Mathematics Reading
A selection of mathematical texts is given below; some are technical books aimed at bridging the gap between Alevel and university mathematics, which will help you fill in those gaps over the summer; others aim to popularize mathematical ideas, the history of a topic or theorem, or are biographies of great mathematicians, which may give you a flavour for how mathematics is discovered and the variety of topics studied at university. Of course, you aren’t expected to buy or read all, or any, of them, and the list is far from comprehensive.
Bridging Material
 Alcock, Lara. How to Study for a Mathematics Degree (2012)
 Allenby, Reg. Numbers and Proofs (1997)
 Earl, Richard. Towards Higher Mathematics: A Companion (2017)
 Houston, Kevin. How to Think Like a Mathematician (2009)
 Liebeck, Martin. A Concise Introduction to Pure Mathematics (2000)
 Neale, Vicky. Why Study Mathematics? (2020)
Popular Mathematics
 Acheson, David. 1089 and All That (2002), The Calculus Story (2017), The Wonder Book of Geometry (2020)
 Bellos, Alex. Alex’s Adventures in Numberland (2010)
 Clegg, Brian. A Brief History of Infinity (2003)
 Courant, Robbins and Stewart, Ian. What is Mathematics? (1996)
 Devlin, Keith. Mathematics: The New Golden Age (1998), The Millennium Problems (2004), The Unfinished Game (2008)
 Dudley, Underwood. Is Mathematics Inevitable? A Miscellany (2008)
 Elwes, Richard. MATHS 1001 (2010), Maths in 100 Key Breakthroughs (2013)
 Gardiner, Martin. The Colossal Book of Mathematics (2001)
 Goriely, Alain. Applied Mathematics: A Very Short Introduction (2018)
 Gowers, Tim. Mathematics: A Very Short Introduction (2002)
 Hofstadter, Douglas. Gödel, Escher, Bach: an Eternal Golden Braid (1979)
 Körner, T. W.. The Pleasures of Counting (1996)
 Neale, Vicky. Closing the Gap: the quest to understand prime numbers (2017)
 Odifreddi, Piergiorgio. The Mathematical Century: The 30 Greatest Problems of the Last 100 Years (2004)
 Piper, Fred & Murphy, Sean. Cryptography: A Very Short Introduction (2002)
 Polya, George. How to Solve It (1945)
 Sewell, Michael (ed.). Mathematics Masterclasses: Stretching the Imagination (1997)
 Singh, Simon. The Code Book (2000), Fermat’s Last Theorem (1998)
 Stewart, Ian. Letters to a Young Mathematician (2006), 17 Equations That Changed The World (2012)
History and Biography
 Burton, David. The History of Mathematics (2007)
 Derbyshire, John. Unknown Quantity – A Real and Imaginary History of Algebra (2006)
 Goldstein, Rebecca. Incompleteness – The Proof and Paradox of Kurt Gödel (2005)
 Gray, Jeremy. Hilbert’s Challenge (2000)
 Hellman, Hal. Great Feuds in Mathematics (2006)
 Hodgkin, Luke. A History of Mathematics – From Mesopotamia to Modernity (2005)
 Hodges, Andrew. Alan Turing: The Enigma (1992)
 Stedall, Jacqueline. The History of Mathematics: A Very Short Introduction (2012)
 Pesic, Peter. Abel’s Proof (2004)
 Reid, Constance. Julia: A Life in Mathematics (1996)
 Stillwell, John. Mathematics and Its History (2002)
Puzzle 3
Imagine a traveler on the integers between 0 and 5 inclusive. At each time step the traveler either moves up to the next integer or down to the previous integer with equal probability. However, if the traveler reaches 0 or 5, then they stop moving.
 If the traveler starts at 2, what is the probability that they are at 0 after five steps? What is the probability that the traveler is at 5?
 Now consider an infinite number line, without the stopping condition at 0 or 5. What does the distribution of possible locations after N steps look like? How does this vary with N?
Puzzle 4
I have a flock of chickens. For each pair of chickens $C_1$ and $C_2$, either $C_1$ pecks $C_2$ or $C_2$ pecks $C_1$. If $C_1$ pecks $C_2$ and $C_2$ pecks $C_3$, then I say that $C_1$ threatens $C_3$. A king chicken is one which either pecks or threatens every other chicken.
 Consider the chicken that pecks the most other chickens (if there’s a tie, consider one of them). Prove that it’s a king chicken.
 Prove that if a chicken is pecked at all, then one of the chickens that pecks it must be a king chicken.
 Could there be exactly two king chickens in my flock?