Mathematics Summer School Part 2 - Detailed Outline

This page provides a detailed outline of the Mathematics Summer School – Part Two, 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 the rigorous foundations of calculus to differential equations, mathematical proof, and the study of infinity.

The course is taught through a combination of structured problem-solving, discussion, and guided reasoning. Students are expected not only to apply mathematical techniques, but to understand the principles that justify them, working from definitions, exploring general methods, and constructing clear logical arguments.

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Day One – Analysis and the Foundations of Calculus
Day Two – Differential Equations and Mathematical Modelling
Day Three – Proof and the Nature of Infinity

Across the course, students explore how mathematics is built on precise definitions and logical reasoning. Beginning with analysis, they examine how concepts such as limits, differentiation, and integration can be defined rigorously and connected within a single framework. The course then develops these ideas through differential equations, where students model systems in terms of rates of change and explore methods for solving them. Finally, in the study of proof and infinity, students consider how mathematical results are established and how careful reasoning allows us to make sense of concepts that extend beyond everyday intuition.

Throughout, the emphasis is on clarity, structure, and justification, so that students experience mathematics not simply as a set of techniques, but as a coherent and logically grounded discipline.

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

10.00 – 1.00 Sequences, Series, and the Foundations of Calculus

The course begins by examining infinite sequences and series, focusing on the question of convergence: when does an infinite process produce a finite result? Students explore methods for determining whether a series converges or diverges, applying these ideas to examples such as the harmonic series and related constructions.

Through these examples, students begin to see how infinite sums can behave in subtle and sometimes counterintuitive ways, and how careful reasoning is required to distinguish between different types of behaviour. The session also introduces limits as a way of describing the behaviour of sequences and functions, and uses this framework to develop a precise definition of the derivative.

Students apply these ideas to differentiate a range of functions, including less familiar examples, and are introduced to techniques such as L’Hôpital’s Rule for evaluating indeterminate forms. The session also introduces integration and explores the relationship between differentiation and integration through the Fundamental Theorem of Calculus, showing how these concepts are linked within a single framework.

1.00 – 2.00 Lunch

2.00 – 3.30 Taylor Series and Approximation

The afternoon session develops these ideas further by considering how functions can be approximated using polynomials. Students are introduced to Taylor’s Theorem and explore how increasingly accurate approximations can be constructed from derivatives evaluated at a single point.

Through worked examples, students see how these expansions can be used both for approximation and for deriving exact results, and how local information about a function can be used to understand its wider behaviour. The session highlights the power of calculus as a unified framework connecting limits, differentiation, and infinite series.

10.00 – 1.00 Introducing Differential Equations

The day begins by introducing differential equations as a way of describing relationships between a function and its derivatives. Students explore how such equations arise naturally when modelling systems in which one quantity depends on the rate of change of another, and consider a range of examples to build intuition.

Students then examine how differential equations can be classified according to their order and whether they are linear or non-linear, and how this classification informs the choice of method used to solve them.

The session introduces the method of separation of variables as a first approach to solving differential equations, allowing students to transform equations into a form that can be integrated. Along the way, students begin to see that solutions are not single functions but families of functions, often involving arbitrary constants, reflecting the general nature of the underlying system.

1.00 – 2.00 Lunch

2.00 – 3.30 Solving Differential Equations

The afternoon session develops more advanced methods for solving differential equations, including exact equations and the use of integrating factors. Students work through the structure of these methods, understanding how they arise and when they can be applied.

Through a series of problems, students explore how different techniques can be used to approach increasingly complex equations, and how recognising the underlying form of an equation guides the solution strategy.

The session concludes with examples drawn from real-world contexts, illustrating how differential equations can be used to model phenomena such as growth, decay, and dynamic systems, and reinforcing the connection between abstract mathematics and practical application.

10.00 – 1.00 Mathematical Proof and Reasoning

The final day begins by examining the role of proof in mathematics: how we establish that a statement is true, and what distinguishes a convincing argument from an informal explanation. Students consider how proofs are built from previously established results and axioms, and how even simple assumptions can support complex and far-reaching conclusions.

Through a series of examples, students explore different styles of proof, beginning with geometric arguments such as proofs of Pythagoras’ Theorem and moving towards more general forms of reasoning. The emphasis is on understanding the structure of a proof: how each step follows logically from the previous one, and how different approaches can be used to establish the same result.

Students then develop key techniques including proof by contradiction and proof by induction, applying these methods to establish non-trivial results such as the existence of infinitely many prime numbers and properties of sequences and sums.

1.00 – 2.00 Lunch

2.00 – 3.30 Infinity and Mathematical Paradox

The afternoon session turns to the concept of infinity, exploring how mathematical reasoning challenges and extends everyday intuition. Students examine thought experiments such as Hilbert’s Hotel, which demonstrate that infinite sets behave in ways that differ fundamentally from finite ones.

Through these examples, students investigate how infinity can be treated rigorously within mathematics, and how careful definitions allow us to reason about seemingly paradoxical situations. The session highlights how results about infinity arise from the same logical framework developed earlier in the day, linking abstract reasoning to some of the most counterintuitive ideas in mathematics.

The course concludes by reflecting on how proof and abstraction allow mathematicians to move beyond intuition, building a coherent framework in which even the most unfamiliar concepts can be understood and explored.

This outline provides a detailed view of the themes and topics explored during the Mathematics Summer School – Part Two. The programme is designed to introduce students to more advanced areas of pure mathematics, including analysis, differential equations, proof, and the study of infinity, while also giving participants the opportunity to develop their mathematical reasoning and work with more abstract ideas.

Students who have not yet begun A-level Mathematics (or an equivalent level) should generally begin with Part One of the course, which focuses on algebra, algorithms, and number theory, and provides a foundation for the more rigorous and abstract work developed here.

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