Welcome to the Mathematical Institute!

Congratulations on receiving a place to study Mathematics, Mathematics and Computer Science, Mathematics and Philosophy, or Mathematics and Statistics at the University of Oxford! We look forward to welcoming you in October. 

Your college will be in touch before term starts, but we hope this page will be a useful resource for preparing you for your studies. You may also want to look at the Information for Prelims Students section of the website, as this is specifically for first years and gives key dates, information on finding your way around, and information about exams.

We would encourage you not only to read about mathematics over the summer but also to do some mathematics. Feel free to do any of the practice problems online that seem helpful - these make sure that you are up to speed with all the concepts you need to begin the course. Your college tutors may set you these sheets to do over the summer, but they are good to review even if your tutors don't require it. Additionally, if you've only done A-level Mathematics (or have not studied in the A-level system) you may want to look at the bridging the gap material. 

Have a look at the "How do Undergraduates do Mathematics?" booklet which gives you an idea of tutorials, how university-level mathematical study differs from school study, and how to approach problem sheets. 

Several of the books in the departmental prospectus are good for bridging the transition from school to university, we particularly recommend:

Lara Alcock - How to Study for a Mathematics Degree

Kevin Houston - How to Think Like a Mathematician: A Companion to Undergraduate Mathematics

Richard Earl - Towards Higher Mathematics

If you've done the online practice problems you might also want to look at NRICH's preparation for pure maths and preparation for applied maths. Underground Mathematics has a lot of problems encouraging you to explore various concepts more deeply. If you didn't take STEP you might also want to read through (and do the problems in) Advanced Problems in Core Mathematics: Preparing for University by Stephen Siklos which is available as a free pdf.

You are very much encouraged to view this video, which provides guidance on how to make the most of tutorials.

Introductory courses (Introduction to University Mathematics and Introduction to Complex Numbers) - M&P, M&CS

The purpose of these introductory lectures is to establish some of the basic language and notation of university mathematics, and to introduce elements of (naive) set theory and the nature of formal proof.

Linear Algebra I - M&P, M&CS

Linear algebra pervades and is fundamental to algebra, geometry, analysis, applied mathematics, statistics, and indeed most of mathematics. This course lays the foundations, concentrating mainly on vector spaces and matrices over the real and complex number systems.

Analysis I: Sequences and Series - M&P, M&CS

In these lectures we begin a rigorous treatment of the real and complex numbers, and study their properties, particularly completeness; we define and study limits of sequences, convergence of series, and power series.

Introductory Calculus - M&P

These lectures are designed to give students a gentle introduction to applied mathematics and at the end of the course, students will be able to solve a range of ordinary differential equations (ODEs). They will also be able to evaluate partial derivatives and use them in a variety of applications. 

Probability - M&P, M&CS

The aim of this introduction to probability is to develop the concept of chance in a mathematical framework. Random variables are introduced, with examples involving most of the common distributions. Probability models considered include random walks and linear difference equations.

Geometry 

The course is an introduction to some elementary ideas in the geometry of euclidean space through vectors. One focus of the course is the use of co-ordinates and an appreciation of the invariance of geometry under an orthogonal change of variable.

Computational Mathematics

Many mathematicians use general-purpose mathematical software which includes tools for symbolic and numerical computation and other features such as plotting, visualization and data analysis. In this introductory course, students will explore these ideas using the popular specialist mathematics software MATLAB.

Linear Algebra II - M&P, M&CS

This course is an introduction to determinants, linear transformations, eigenvectors and eigenvalues and the spectral theorem for real symmetric matrices. It covers both theory and applications of diagonalizability and brings in the geometrical importance of an orthogonal change of variable.

Groups and Group Actions - M&P, M&CS

Abstract algebra evolved in the twentieth century out of nineteenth century discoveries in algebra, number theory and geometry. The group is an important first example of an abstract, algebraic structure and groups permeate much of mathematics particularly where there is an aspect of symmetry involved.

Analysis II: Continuity and Differentiability - M&P, M&CS

In this term’s lectures, we study the continuity of functions of a real or complex variable, and differentiability of functions of a real variable. 

Dynamics

The subject of dynamics is about how things change with time. A major theme is the modelling of a physical system by differential equations, and one of the highlights involves using the law of gravitation to account for the motion of planets. 

Multivariable Calculus

In these lectures, students will be introduced to multi-dimensional vector calculus. They will be shown how to evaluate volume, surface and line integrals in three dimensions and how they are related via the Divergence Theorem and Stokes’ Theorem.

Fourier Series and PDEs 

Fourier series are a way of representing functions as sums of trigonometric functions. Students will then be shown how the heat equation, the wave equation and Laplace’s equation arise in physical models.

Computational Mathematics

In this term you undertake an assessed project using MATLAB and there are drop-in sessions and demonstration sessions.

Groups and Group Actions - M&P, M&CS

Continuation of the course from Hilary Term.

Analysis III - Integration - M&P, M&CS

In these lectures we define the Riemann integral and study its properties; prove the Mean Value Theorem for Integrals and the Fundamental Theorem of Calculus. This gives us the tools to justify term-by-term differentiation of power series and deduce the elementary properties of the trigonometric functions.

Statistics and Data Analysis

The course introduces the concept of likelihood for a probabilistic model and its use in estimating unknown model parameters. Techniques for finding structure in datasets are relevant to many parts of applied maths, specifically this course will cover principal components analysis and clustering techniques. 

Constructive Mathematics

This course is an introduction to mathematical algorithms; that is procedures which one can carry out to achieve a desired result. Such procedures arise throughout mathematics both Pure and Applied.