Peaks, Passes and Pits: From Topography to Topology via Quantum Mechanics
End : 04/02/2008 - 4:15pm
The great physicist James Clerk Maxwell was one of the first to realize that what we'd now call topological ideas were destined to play a key part in the future of physics. In an 1870 paper "On Hills and Dales" he devoted sustained attention to the apparently straightforward question of counting the "critical points" of the Earth's landscape: the mountain peaks, passes, and ocean deeps that stand out clearly on a contour map. In doing this he anticipated the key ideas of Morse theory, a central notion of 20th-century mathematics. Around 1980 Ed Witten reconnected Morse theory to physics by interpreting it in terms of the "tunneling" of certain (virtual) particles associated to the topology of a manifold. I'll explain this story in elementary terms and show how it relates to other key ideas in modern geometric analysis.