Peaks, Passes and Pits

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Peaks, Passes and Pits

• From Topography to Topology
• (via Quantum Mechanics)

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James Clerk Maxwell, 1831-1879
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• 1861 On Physical Lines of Force
• 1864 On the Dynamical Theory of the
  Electromagnetic Field
• 1870 On Hills and Dales
• hilldale.pdf

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Scottish examples
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Peak
Pass
Pit
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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.

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Basic theorem

• ( Peaks) ( Passes) ( Pits) Euler
  Characteristic (V-EF)
• Euler Characteristic is a topological invariant
  2 for the sphere 0 for the torus.
• Does not depend on which Morse function we choose!

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

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The physics connection

• Ed Witten, Supersymmetry and Morse Theory, 1982

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Wittens 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.

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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!

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• In fact, it introduces a potential term, which
  forces our particles to congregate near the
  critical points of appropriate index.
• The potential is
• s2?h2 sXh
• where Xh is a zero order vectorial term.

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• 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.

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• 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.

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James Clerk Maxwell, 1831-1879