### The Invention of Numbers and Zero

This goes from the very early use of counting to the marvellous ‘place-value’ system in use today. This system is sometimes credited to the Arabic world over a thousand years ago, sometimes to India and China some two thousand years ago, but was invented by the Sumerians more than four thousand years ago. Only in the 20th century was this fully understood, and the concise nature of their floating-point, base-60 system made it a wonderful vehicle for calculations.

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### Les belles aux bois dormant

This talk, at a History of Mathematics conference on the topic of Symmetry, described the discovery of the ‘sporadic groups’ — exceptional building blocks in the mathematics of symmetry — during the 1960s and 70s

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### Symmetry and the Monster

This describes the mathematical quest for all the basic building blocks for symmetry, and the people who made it all possible. Starting with the ancient Greeks we move to a young man in early nineteenth century France who used a mathematical interpretation of symmetry to solve a great mathematical problem. From there we move rapidly to the second half of the twentieth century when mathematicians compiled a full list of the basic building blocks for symmetry, including a mysterious new one dubbed The Monster. I have given this talk in many countries on three different continents — here is a sample.

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### The Quest to go beyond Euclid

This describes the history of geometry from Euclid in about 300 BC to Europe in the eighteenth century. Euclid laid down five axioms, and geometers writing in Arabic, and later in Latin, made repeated attempts to show that the axiom on parallel lines was a consequence of the other four. I give a convincing but false proof of this. At the end we see why it is wrong, and how in the eighteenth century three mathematicians independently created a plane geometry where the parallel postulate fails. Here is a sample.

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