Post on 04-Jan-2016
• 1861 – “On Physical Lines of Force”
• 1864 – “On the Dynamical Theory of the Electromagnetic Field”
• 1870 – “On Hills and Dales”
• hilldale.pdf
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…
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.