Complex Analysis out of Thin Air
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                    Michael Eastwood (Adelaide)

In this talk, `Thin Air' will usually mean Euclidean 3-space, the
space in which we lived until 1905. Sometimes, it will mean Lorentzian 4-space,
our home for the past 100 years. There's not a complex number in sight but they
are lurking unseen. In 1866, Weierstrass used Complex Analysis to describe soap
films in 3-space. Nowadays, complex numbers are everywhere thanks to Penrose's
twistor theory. I shall discuss the elementary but surprising geometry that
underlies this theory and outline some recent developments.