Mathematics Summer School - Part 1 - Detailed Outline

This page provides a detailed outline of the Mathematics Summer School – Part One, showing the themes and topics explored in each session across the three-day course. The programme below explains what students study on each day, from algebraic methods and polynomial equations to abstract algebra, algorithms, and number theory.

The course is taught through a combination of structured problem-solving, discussion, and guided reasoning. Students are encouraged not only to solve problems, but to understand the principles that lie behind them, working from definitions, exploring patterns, and developing general methods.

View and download a pdf of the outline here.

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Day One – Algebra and the Emergence of Structure
Day Two – Algorithms and the Limits of Computation
Day Three – Modular Arithmetic and Number Theory

Across the course, students explore how mathematics develops from concrete problems into more abstract and powerful forms of reasoning. Beginning with algebra, they learn how equations can be manipulated, simplified, and solved, and how general techniques arise from particular examples. The course then moves to the study of algorithms, where students examine how mathematical processes can be formalised and what it means for a problem to be computable. Finally, in number theory, students investigate modular arithmetic, discovering how familiar operations can be reinterpreted within new systems and applied to both theoretical questions and practical problems such as cryptography.

Throughout, the emphasis is on reasoning clearly, recognising structure, and understanding why mathematical results are true, so that students experience mathematics not simply as a set of techniques, but as a coherent and evolving discipline.

Please note that for some groups, sessions may run in a different order.

10.00 – 1.00 Polynomial Equations and Algebraic Methods

The course begins with algebra as a tool for translating problems into symbolic form and solving them systematically. Students work with polynomial equations, starting from familiar quadratic methods and extending to more complex cases, including Cardano’s solution of the cubic.

Rather than treating these results as formulas to be memorised, students are guided through the reasoning behind them. They explore how changes of variables can simplify equations, how general methods arise from particular cases, and how algebra provides a framework for solving a wide range of problems.

The session also introduces an important limitation: not all polynomial equations can be solved using algebraic methods. This provides an early insight into how mathematics develops by identifying both what is possible and what lies beyond current techniques.

1.00 – 2.00 Lunch

2.00 – 3.30 Abstract Algebra and Group Theory

The afternoon session moves from solving equations to studying the underlying structures that govern them. Students are introduced to the concept of a group as a set equipped with an operation satisfying a small number of defining axioms.

Working from these definitions, students test whether particular systems form groups, construct operation tables, and explore how algebraic structure can be used to describe symmetry and transformation.

The focus is on reasoning from first principles: understanding how abstract definitions give rise to general results, and how seemingly different mathematical systems can share the same underlying structure.

10.00 – 1.00 Algorithms and Mathematical Processes

This session introduces algorithms as precise mathematical objects: finite sequences of instructions that define how a problem is to be solved. Students explore how different types of algorithms operate, including recursive methods, divide-and-conquer strategies, and approaches that incorporate randomness.

Through worked examples, students examine how algorithms can be analysed and compared, and how mathematical reasoning can be used to understand their behaviour. The emphasis is not simply on carrying out procedures, but on understanding the structure and design of algorithms, and how complex processes can be broken down into simpler steps.

Where appropriate, students explore examples such as cellular automata, including Conway’s Game of Life, to see how simple rules can generate surprisingly rich and complex behaviour.

1.00 – 2.00 Lunch

2.00 – 3.30 Turing Machines and Computability

The afternoon session develops a formal framework for understanding computation through the introduction of the Turing machine. Students examine how a Turing machine can be used to model any algorithmic process, and how this leads to a precise definition of what it means for a problem to be computable.

Building on this, the session introduces the Church–Turing thesis, which proposes that any function that can be computed by an algorithm can be computed by a Turing machine. Students explore how this idea connects different areas of mathematics and computer science, and consider the implications for the limits of computation.

The focus throughout is on understanding computation as a mathematical concept, rather than as a practical programming task, and on recognising that there are fundamental limits to what any algorithm can achieve.

10.00 – 1.00 Modular Arithmetic and Congruences

The final day begins by developing a new way of thinking about numbers, in which integers are considered equivalent if they leave the same remainder when divided by a fixed value. Students are introduced to the language of congruences and explore how arithmetic can be carried out within this system.

Through a mixture of structured problems and exploratory exercises, students investigate residue classes, least residues, and the rules governing addition, subtraction, and multiplication in modular arithmetic. They examine how an infinite set of integers can be organised into a finite structure, and how this shift in perspective allows for new kinds of reasoning about numbers.

The session combines calculation with pattern-finding, encouraging students to experiment with examples and identify general rules within this new framework.

1.00 – 2.00 Lunch

2.00 – 3.30 Fermat’s Little Theorem, Puzzles, and Applications

The afternoon session builds on this foundation by introducing one of the central results of elementary number theory: Fermat’s Little Theorem. Students explore how this theorem arises from the structure of modular arithmetic and how it can be used to simplify calculations and solve problems involving large powers.

These ideas are then applied to a range of problems and puzzles, allowing students to test their understanding and develop confidence in working within this system. Through these examples, students see how relatively simple rules can lead to surprising and non-intuitive results.

The session also introduces the role of modular arithmetic in cryptography, illustrating how these mathematical ideas underpin real-world systems for secure communication.

Throughout, the emphasis remains on understanding why results are true and how they fit into a broader mathematical framework, even when approached through problem-solving and exploratory tasks.

This outline provides a detailed view of the themes and topics explored during the Mathematics Summer School – Part One. The programme is designed to introduce students to key areas of modern mathematics, including algebra, algorithms, and number theory, while also giving participants the opportunity to develop their problem-solving skills and explore how mathematical ideas can be generalised and applied.

For students who have studied A-level Mathematics (or an equivalent level), Part Two of the course explores more advanced topics such as calculus, proof, and higher-level mathematical structures, and is suitable for those ready to engage with mathematics in a more abstract and demanding way.

Return to the main Mathematics Summer School page for full practical details about the course and how to apply.