Staying Ahead Of The Opposition - M.D. Atkinson
Number sequences are found in nearly every area of mathematics. The same sequence appearing in two different areas is often the first clue that these areas might be connected. This lecture looks at one such sequence, the Catalan sequence, from this point of view. Prerequisites: None.

Designing Experiments With Allowance For Interfering Neighbours - R.A. Bailey
Agricultural experiments are expensive. A little mathematics can improve the design of these experiments and so gain more information for the same cost. The Latin square is one standard design. However, if any treatment affects the performance on neighbouring plots, then more sophisticated designs are needed. Prerequisites: None.

Wallpaper Patterns in Different Geometries - A.F. Beardon
Wallpaper patterns arise by translating, rotating, or reflecting, a given motif to produce a tessellation of the Euclidean plane. Analogous patterns exist in hyperbolic (non-Euclidean) geometry and in spherical geometry, and the way in which the underlying geometry affects the patterns is discussed. Prerequisites: a familiarity with simple motions of the plane.

Chaology - M. Berry
In Newtonian mechanics the present state of a system determines its future, but that future need not be predictable: there are simple systems for which the trajectories can be as random as coin tosses. In the lecture some concepts of the emerging science of chaology are illustrated by means of a simple machine. Prerequisites: none.

How Mathematics gets into Knots - R. Brown
The lecture starts from the oldest known knot, dating form 7,200 BC, and shows knotting and interlacing as a motif in art and sculpture, as well as a basic practical skill. The mathematics of knots deals with their classification; with the arithmetic properties of the operation of tying one knot after another, and with the remarkable algebra which models wrapping string around a knot. Prerequisites: none.

Geometry and Computers - P. Giblin
Modern geometry interacts with computers in several ways. Computers can draw pictures for geometers to look at, helping them to formulate new results and suggesting proofs. But also geometry provides vital tools for those developing computer vision. Various interactions are described and illustrated by still and moving pictures. Prerequisites: none.

New Wine In Old Bottles: Fibonacci And Lucas Numbers Revisited - P.J. Hilton
Intriguing number tricks can be explained by analysing the properties of Fibonacci numbers and the related Lucas numbers. The explanations themselves benefit from further explanations which, in their turn, lead to further discoveries. Prerequisites: mathematical curiosity.

Games that Solve Problems - W.A. Hodges
Mathematicians don't just solve problems. They also find methods for solving new kinds of problems. How can they do this? This lecture describes various attempts to answer this question during the last 150 years. One important recent approach is based on a kind of 'spot the difference' game; simple examples are given. Prerequisites: an interest in abstract ideas and patterns.

How Should a Mathematician Think About Shape? - D.G. Kendall
The short answer is - by creating a space in which each shape 'lives' as a point, in such a way that natural questions about shapes translate into geometrical questions concerning the new space. This subject is only about 10 years old, but is already leading to new insights and has interesting applications in archaeology and astronomy. Prerequisites: none.

How To Study Random Shapes - J.T. Kent
Shape is a key aspect of geometric objects. Two objects are said to have the same shape if they are identical except for changes in location, rotation and orientation. This talk describes how to compare shapes, average shapes, and deform one shape into another. Applications will be given from biology, medicine and computer vision. Prerequisites: None, but a little knowledge of matrices would be useful.

The Rise and Fall of Matrices - W. Ledermann
A description of the revolutionary paper of 1858 by Cayley and the change of emphasis of the teaching of linear algebra from determinants to matrices and linear maps, with historical background. Prerequisites: some knowledge of matrices.

The Art Of Asymptotic Approximation - F.G. Leppington
In many mathematical problems the equations contain a small number, called a parameter. An asymptotic estimate is an approximation when this number becomes smaller and smaller. The ideas are described with reference to (i) a simple cubic equation and (ii) a differential equation. Prerequisites: None for the first part; some knowledge of differentiation would be helpful for the second part.

Games Animals Play - J. Maynard Smith
Game theory is applied by an eminent biologist to give an insight into animal contests (for instance, for mates), leading to an explanation of why there are an (almost) equal number of male and female births, the behaviour of the Hamadryas baboon, and the funnel web spider. Prerequisites: none, but a knowledge of pay-off matrix would be helpful and some idea of evolution.

Optimization Of Running And Jumping - R. McNeill Alexander
Why do we walk to go slowly and run to go faster? At what speed should a rhinoceros gallop like a cat? Why do high jumpers run up so slowly? Remarkably simple mathematical models help us to answer such questions. Prerequisites: Elementary dynamics.

Heads I Win, Tails You Lose - L. Mustoe
An introductory look at the theory of games. Emphasis is placed on the use of simple mathematical techniques to solve problems. A wide range of topics in the area is surveyed. Prerequisites: virtually None.

A Breakthrough In Algebra - P.M. Neumann
Groups are algebraic structures used for measuring symmetry. Finite groups are composed of simple groups in much the same way as integers are products of prime numbers. This lecture gives a survey of simple groups and their recent classification. Prerequisites: a little experience of the algebra of matrices.

Fermat's Last Theorem - R. Pinch
This infamous problem, which was posed in the 1630s, may now have succumbed. We describe the successes and failures along the way. Duration 25 minutes. Prerequisites: None.

Codes and Ciphers - F.C. Piper
An introduction to the art (or science) of keeping information secret. One of the main themes is how the advent of fast computers has affected cryptography. There are virtually no prerequisites.

Wild Geometry - N. Ray
Tame geometry is epitomised by familiar shapes of everyday life. Wild geometry is more mysterious, including such intricate objects as the Artin-Fox arc, the Alexander horned sphere and Antoine's necklace. Prerequisites: None.

How To See Objects In Four Dimensions - S.A. Robertson
Our experience of the physical world through sight, touch and hearing is the raw material from which classical Euclidean plane and solid geometry are derived. Geometrical structures in four-dimensional space can also be understood in visual terms, as this lecture demonstrates. Prerequisites: None.

Measuring The Marigold - P.T. Saunders
Mathematics can be applied to biology in its own right, not just as a sort of technical assistant to physics and chemistry. It can help explain why Friesian cows don't all look the same, how fruit flies get the right number of segments and why the angle 137.5° turns up so often in plants. Prerequisites: None.

Stamping through Mathematics - R.J. Wilson
This video produced by the BBC for the Open University, presents an overview of the history of mathematics, from earliest times to the modern computer age. The talk is illustrated by slides of 70 postage stamps featuring mathematics and mathematicians. Prerequisites: none.

Geometry and Perspective - E.C. Zeeman
The vanishing points and observation points of perspective are explained, and the underlying theorems in 3-dimensional Euclidean geometry are proved. The results are illustrated with renaissance paintings and a reconstruction of Brunelleschi's original experiment demonstrating his discovery of perspective in 1420. Prerequisites: none.

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