Notes prepared by
Professor Marcus du Sautoy, University of Oxford

The Building Blocks of Graphs

In the mid 1960s Carleson completed a project that had begun during the heat of the French Revolution by a French mathematician that Abel met whilst working in Paris: Jean Baptiste Fourier. Fourier was a great friend of Napoleon and had been taken by the Emperor on his invasion of Egypt to entertain him with interesting science lectures on board his ship the Orient. Whilst in the deserts of North Africa, Fourier became addicted to the medicinal effect of the searing heat. But it was the mathematics of heat that immortalized Fourier.

Many natural phenomena can be represented in graphical form: for example the evolution of heat over time or the pictorial representation of a sound wave. Fourier proposed that just as water can be broken down into its atomic components, complicated graphs could be reduced to adding up very basic or atomic waves. Fourier conjectured that the Hydrogen and Oxygen of the world of graphs is the simple oscillating sine wave. The sine wave is the graphical representation of the sound of a tuning fork and the purity of the sound is one reason we use a tuning fork to tune an orchestra or piano.

The sound of a violin in contrast is more complex; its graph looks like the teeth on a saw. But Fourier believed that by adding up the heights of simple sine waves of different frequencies you could reproduce the saw-tooth shape of the violin.

Fourier realised that even complicated graphs like those depicting the sound wave produced by an orchestra playing music could be reduced to adding up an infinite number of simple sine functions. This realisation is at the heart of how a CD or MP3 player reproduces the sound of an orchestra. The CD tells your speaker to vibrate to create all the sine waves making up the sound of the music. Played simultaneously it miraculously creates the sensation of having an orchestra in your living room.

But could every finite graph of a general type be captured by Fourier's analysis? It was not clear that there might not be some pathological graphs that were beyond the scope of Fourier's theory. As generations of mathematicians battled with the ideas, the difficulty of the problem began to make mathematicians suspect that there might indeed be strange graphs that couldn't be built out of Fourier's sine waves. But no-one could find such a graph.

In 1966 Carleson published a paper that explained why. All finite graphs of the general type considered by mathematicians could be captured by adding up the heights of sine waves of appropriate frequencies. The proof was tough and involved showing why, when you add up the infinitely many numbers that came out of Fourier's analysis, the answer didn't spiral off to infinity but honed in on the graph you were trying to capture. The ideas in the proof were so tough that it is only in recent decades that mathematicians have appreciated how influential these ideas really are. Fourier's problem however is still open for higher dimensional graphs and represents a major goal for mathematicians in this area.

HomeNews ArchiveCalendar      Editor: Anne Marie Astad  The Norwegian Academy of Science and Letters  E-mail: dnva@online.no