Peaks, Passes and Pits

From Topography to Topology

(via Quantum Mechanics)

James Clerk Maxwell, 1831-1879

• 1861 – “On Physical Lines of Force”

• 1864 – “On the Dynamical Theory of the Electromagnetic Field”

• 1870 – “On Hills and Dales”

• hilldale.pdf

Scottish examples

Peak

Pass

Pit

Critical points of a function on a surface

• Peaks (local maxima)

• Passes (saddle points)

• Pits (local minima)

• Can identify by looking at 2nd derivative

• “Topology only changes when we pass through a critical point”.

Basic theorem

• (# Peaks) – (# Passes) + (# Pits) = Euler Characteristic (V-E+F)

• Euler Characteristic is a topological invariant; 2 for the sphere; 0 for the torus.

• Does not depend on which Morse function we choose!

The Hodge equations

• The Euler characteristic can also be obtained by counting solutions to certain partial differential equations – the “Hodge equations”.

• They are geometrical analogs of Maxwell’s equations!

• To see how PDE can relate to topology, think about vector fields and potentials…

The physics connection

Ed Witten, Supersymmetry and Morse Theory, 1982

Witten’s method

• Consider the Hodge equation as a quantum mechanical Hamiltonian.

• Different types of ‘particle’ according to the Morse index (‘peakons, passons and pitons’).

• Euler characteristic given by counting the low energy states of these particles.

Perturbation theory

• Replace d by esh d e-sh, where h is the Morse function and s is a real parameter.

• This perturbation does not change the number of low energy states.

• But it does change the Hodge equations!

• In fact, it introduces a potential term, which forces our particles to congregate near the critical points of appropriate index.

• The potential is

s2|h|2 + sXh

where Xh is a zero order vectorial term.

• The term Xh has a ‘zero point energy’ effect which forces each type of particle to congregate near the critical points of the appropriate index; ‘peakons’ near peaks, ‘passons’ near passes and so on.

• Thus the number of low energy n-on modes approaches the number of critical points of index n, as the parameter s becomes large.

• Appropriately formulated, this proves the fundamental result of Morse theory; peaks – passes + pits = Euler characteristic.

James Clerk Maxwell, 1831-1879