Morse Theory Roel Hospel Technische Universiteit Eindhoven - - PowerPoint PPT Presentation

Morse Theory

Roel Hospel

Technische Universiteit Eindhoven roel.hospel@gmail.com

May 24, 2018

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Overview

1

Why Morse Theory?

2

Manifolds

3

Smooth Functions

4

Morse Functions The Hessian Morse Function Morse Lemma Morse Index

5

Transversality Stable and Unstable Manifolds Morse-Smale Functions

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Why Morse Theory?

A lot of problems in the sciences are given as real-valued functions. Morse Theory provides us a tool to analyze these functions easily.

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Manifolds

A Manifold is a topological space that locally resembles Euclidean space near each point.

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1-dimensional Manifolds

Line Circle Figure-8 ? A Manifold is a topological space that locally resembles Euclidean space near each point.

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1-dimensional Manifolds

Line Circle Figure-8 A Manifold is a topological space that locally resembles Euclidean space near each point.

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2-dimensional Manifolds

Sphere Torus Boy’s Surface A Manifold is a topological space that locally resembles Euclidean space near each point.

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n-Manifolds

We can extend manifolds to higher dimensions: A 3-Manifold is a topological space that locally resembles 3-dimensional Euclidean space near each point. A 4-Manifold is a topological space that locally resembles 4-dimensional Euclidean space near each point. etc.

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n-Manifolds

n-Manifold

A n-Manifold is a topological space that locally resembles n-dimensional Euclidean space near each point.

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Recap: Differential Calculus

Gradient (Tangent Line) Critical Points (Local) Minimum (Local) Maximum f (x) = x · sin(x2) + 1

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Smooth Function

Smooth Function

For a function f to be Smooth Function, it has to have continuous derivatives up to a certain order k. We say that that function f is Ck-smooth.

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Smooth Functions

Formula Order k Derivative Smoothness f (x) = x f ′′(x) = 0 f (x) is C2-smooth g(x) = x2 − 3 g′′′(x) = 0 g(x) is C3-smooth h(x) = x3 + x2 h′′′′(x) = 0 h(x) is C4-smooth

Table: Smoothness Example Formulas

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Smooth Functions

Formula Order k Derivative Smoothness f (x) = x f ′′(x) = 0 f (x) is C2-smooth g(x) = x2 g′′′(x) = 0 g(x) is C3-smooth h(x) = x3 + x2 h′′′′(x) = 0 h(x) is C4-smooth i(x) = sin(x) i(x) is C∞-smooth j(x) = ... j(x) is non-smooth

Table: Smoothness Example Formulas

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Tangent Spaces on Manifolds

The Tangent Space on an n-Manifold is the n-dimensional equivalent of a Tangent Line on a 1-Manifold.

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Critical Points on Manifolds

A point p on an n-Manifold is Critical Point iff all of its partial derivatives vanish. 1-Manifold: f (x)

δf δx (p) = 0

2-Manifold: f (x, y)

δf δx (p) = δf δy (p) = 0

3-Manifold: f (x, y, z)

δf δx (p) = δf δy (p) = δf δz (p) = 0

etc.

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The Hessian

The Hessian is a formula you can calculate for a point p on a given function f (x1, x2, ..., xd) in d-dimensional vector space: H(p) =      

δf δx12 (p) δf δx1δx2 (p)

· · ·

δf δx1δxd (p) δf δx2δx1 (p) δf δx22 (p)

· · ·

δf δx2δxd (p)

. . . . . . ... . . .

δf δxdδx1 (p) δf δxdδx2 (p)

· · ·

δf δxd 2 (p)

     

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The Hessian, in 2D Vector Space

H(p) =      

δf δx12 (p) δf δx1δx2 (p)

· · ·

δf δx1δxd (p) δf δx2δx1 (p) δf δx22 (p)

· · ·

δf δx2δxn (p)

. . . . . . ... . . .

δf δxnδx1 (p) δf δxnδx2 (p)

· · ·

δf δxd 2 (p)

      Simplified to 2-dimensionsal vector space (f (x, y)) this function would become: H(p) =

• δf

δx2 (p) δf δxδy (p) δf δyδx (p) δf δy2 (p)

• Roel Hospel (TU/e)

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Calculating the Hessian

H(p) =

• δf

δx2 (p) δf δxδy (p) δf δyδx (p) δf δy2 (p)

• Let’s calculate the Hessian over

these two formulas: f (x, y) = x2 + y2 f (x, y) = x2 + y3 For which the critical points are both located at (0, 0).

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Degeneracy

A critical point p on manifold M is Non-degenerate iff it holds for the Hessian at point p that H(p) = 0

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Morse Function

Morse Function

A smooth function h : M → R is a Morse Function if all its critical points:

• i. are non-degenerate
• ii. have distinct function values

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Morse Lemma

The Morse Lemma states that if the have a Morse function in 2-dimensional vector space: It is possible to choose local coordinates x, y at a critical point p ∈ M such that a Morse function f takes the form: f (x, y) = ±x2 ± y2

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Morse Lemma

Morse Lemma

It is possible to choose local coordinates x1, .., xd at a critical point p ∈ M, for a vector space of dimension d, such that a Morse function f takes the form: f (x1, x2, ..., xd) = ±x12 ± x22... ± xd 2

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Morse Index

The Morse Index i(p), of Morse function h at critical point p ∈ M, is the number of negative dimensions in the Morse function f . f (x, y) = ±x2 ± y2

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Morse Index in Higher Dimensions

1D 2D 3D f (x) = x2 f (x, y) = x2 + y2 f (x, y, z) = x2 + y2 + z2 f (x) = −x2 f (x, y) = x2 − y2 f (x, y, z) = x2 + y2 − z2 f (x, y) = −x2 − y2 f (x, y, z) = x2 − y2 − z2 f (x, y, z) = −x2 − y2 − z2

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Integral Lines

An Integral Line γ on a manifold M is a maximal path p whose tangent vectors agree with the gradient of the manifold. We call org p = lims→−∞ p(s) the origin of path p. We call dest p = lims→∞ p(s) the destination of path p. Integral Lines have the following properties:

• i. Any two integral lines are either disjoint or the

same:

• ii. Integral lines cover all of M
• iii. The limits org p and dest p are critical points
• f f

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Stable and Unstable Manifolds

The Stable Manifold (or Ascending Manifold) for a critical point p of f is the point itself, together with all regular points whose integral lines end at p. The Unstable Manifold (or Descending Manifold) for a critical point p of f is the point itself, together with all regular points whose integral lines originate at p.

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Morse-Smale Functions

A Morse-Smale Function is a Morse function whose stable and unstable manifolds intersect transversally

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Summary

1

Why Morse Theory?

2

Manifolds

3

Smooth Functions

4

Morse Functions The Hessian Morse Function Morse Lemma Morse Index

5

Transversality Stable and Unstable Manifolds Morse-Smale Functions

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References

• H. Edelsbrunner, J. L. Harer (2010)

Computational topology. An introduction Chapter VI.1 - VI.2, p. 149 - 158.

• A. J. Zomorodian (1996)

Computing and comprehending topology: persistence and hierarchical Morse complexes Chapter 5, p. 56 - 63. Khan Academy (2016) The Hessian Matrix https://youtu.be/LbBcuZukCAw Eric W. Weisstein Manifold Definition http://mathworld.wolfram.com/Manifold.html

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