AustMS 2006 II
Recall the example of the Stasheff polytope in dimension three, as realised by Loday. The collection of all such associahedra is an example of a 1-operad which characterises 1-fold loop spaces. Batanin gave a history of Stasheff's ideas after a more general history discussing the need for a coherent theory of categorical coherence laws. The difficulty was in finding a combinatorial structure for the n-fold case.
He now understands n-operads by generalising the ordinals [m] that index the n-ary operations of a 1-operad to higher level trees, where an ordinal [m] is just a simple tree with one vertex and m leaves, and hence one level. By finding nice examples of 2-operads and higher n-operads one discovers the most astonishing range of polytopes. The collection of Stasheff associahedra, for example, form an n=1 example of a whole series K(n) of operads (related to the Getzler-Jones operads). In particular, the hexagons of the axioms for a braided monoidal category naturally show up here at n=2, whereas we saw the Mac Lane pentagon at n=1.
At the education session in the afternoon Terence Tao spoke about the measurement of distances in astronomical history, starting with the measurement of the Earth's radius by Eratosthenes. For example, to measure the distance from the Earth to the Moon the Greeks observed that the Earth's shadow takes a certain amount of time to pass over the Moon during a lunar eclipse. Since they already knew the radius of the Earth this gave them a fairly accurate measure of the distance using a circular orbit for the Moon about the Earth.
Aristarchus of Samos tried to measure the distance from the Earth to the Sun by observing the phases of the Moon. Since the Sun is at a finite distance, the half-Moon comes just before the halfway point in time between the times of new Moon and full Moon, because it forms a right angled triangle with the Sun and Earth.
Tao then continued increasing the scale of distance observations until he eventually mentioned briefly the poorly understood type IA supernovae. It was nice to see his great respect for Kepler's insightful method for computing the orbit of the Earth about the Sun observationally by using Mars as a reference point. From here Kepler was able to derive the third law
GM = 4 pi^2 R^3 T^(-2).
To finish a lovely day there was a friendly reception with some delicious vegetable savouries and chicken kebabs.