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Morse Theory

Roel Hospel

Technische Universiteit Eindhoven

[email protected]

May 24, 2018

Roel Hospel (TU/e) Morse Theory May 24, 2018 1 / 29

Overview

1 Why Morse Theory?

2 Manifolds

3 Smooth Functions

4 Morse FunctionsThe HessianMorse FunctionMorse LemmaMorse Index

5 TransversalityStable and Unstable ManifoldsMorse-Smale Functions


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.


Manifolds

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


1-dimensional Manifolds

Line Circle Figure-8 ?




Line Circle Figure-8




Sphere Torus Boy’sSurface



n-Manifolds

We can extend manifolds to higher dimensions:

A 3-Manifold is a topological space that locallyresembles 3-dimensional Euclidean space neareach point.A 4-Manifold is a topological space that locallyresembles 4-dimensional Euclidean space neareach point.

etc.


n-Manifolds

n-Manifold

A n-Manifold is a topological space that locallyresembles n-dimensional Euclidean space neareach point.


Recap: Differential Calculus

Gradient (Tangent Line)

Critical Points

(Local) Minimum

(Local) Maximum

f (x) = x · sin(x2) + 1


Smooth Function

Smooth Function

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


Smooth Functions

Formula Order k Derivative Smoothness

f (x) = x f ′′(x) = 0 f (x) is C2-smoothg(x) = x2 − 3 g ′′′(x) = 0 g(x) is C3-smoothh(x) = x3 + x2 h′′′′(x) = 0 h(x) is C4-smooth

Table: Smoothness Example Formulas


Smooth Functions

Formula Order k Derivative Smoothness

f (x) = x f ′′(x) = 0 f (x) is C2-smoothg(x) = x2 g ′′′(x) = 0 g(x) is C3-smoothh(x) = x3 + x2 h′′′′(x) = 0 h(x) is C4-smoothi(x) = sin(x) i(x) is C∞-smoothj(x) = ... j(x) is non-smooth

Table: Smoothness Example Formulas


Tangent Spaces on Manifolds

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


Critical Points on Manifolds

A point p on an n-Manifold is Critical Point iff all of its partialderivatives vanish.

1-Manifold: f (x) δfδx (p) = 0

2-Manifold: f (x , y) δfδx (p) = δf

δy (p) = 0

3-Manifold: f (x , y , z) δfδx (p) = δf

δy (p) = δfδz (p) = 0

etc.


The Hessian

The Hessian is a formula you can calculate for a point p on a givenfunction f (x1, x2, ..., xd) in d-dimensional vector space:

H(p) =

δfδx12

(p) δfδx1δx2

(p) · · · δfδx1δxd

(p)δf

δx2δx1(p) δf

δx22(p) · · · δf

δx2δxd(p)

......

. . ....

δfδxdδx1

(p) δfδxdδx2

(p) · · · δfδxd 2

(p)


The Hessian, in 2D Vector Space

H(p) =

δfδx12

(p) δfδx1δx2

(p) · · · δfδx1δxd

(p)δf

δx2δx1(p) δf

δx22(p) · · · δf

δx2δxn(p)

......

. . ....

δfδxnδx1

(p) δfδxnδx2

(p) · · · δfδxd 2

(p)

Simplified to 2-dimensionsal vector space (f (x , y)) this function wouldbecome:

H(p) =

[δfδx2

(p) δfδxδy (p)

δfδyδx (p) δf

δy2 (p)

]


Calculating the Hessian

H(p) =

[δfδx2

(p) δfδxδy (p)

δfδyδx (p) δf

δy2 (p)

]Let’s calculate the Hessian overthese two formulas:

f (x , y) = x2 + y2

f (x , y) = x2 + y3

For which the critical points areboth located at (0, 0).


Degeneracy

A critical point p on manifold M is Non-degenerate iff it holds for theHessian at point p that H(p) 6= 0


Morse Function

Morse Function

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

i. are non-degenerate

ii. have distinct function values


Morse Lemma

The Morse Lemma states that if the have a Morse function in2-dimensional vector space:

It is possible to choose local coordinates x , y at a critical point p ∈Msuch that a Morse function f takes the form:

f (x , y) = ±x2 ± y2


Morse Lemma

Morse Lemma

It is possible to choose local coordinates x1, .., xd at a critical pointp ∈M, for a vector space of dimension d , such that a Morse function ftakes the form:

f (x1, x2, ..., xd) = ±x12 ± x22...± xd

2


Morse Index

The Morse Index i(p), of Morse function h at critical point p ∈M, is thenumber of negative dimensions in the Morse function f .

f (x , y) = ±x2 ± y2


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


Integral Lines

An Integral Line γ on a manifold M is a maximal path p whose tangentvectors 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 ofpath p.

Integral Lines have the following properties:

i. Any two integral lines are either disjoint or thesame:

ii. Integral lines cover all of M

iii. The limits org p and dest p are critical pointsof f


Stable and Unstable Manifolds

The Stable Manifold (or Ascending Manifold) for acritical point p of f is the point itself, together withall 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, togetherwith all regular points whose integral lines originateat p.


Morse-Smale Functions

A Morse-Smale Function is a Morse functionwhose stable and unstable manifolds intersecttransversally


Summary

1 Why Morse Theory?

2 Manifolds

3 Smooth Functions

4 Morse FunctionsThe HessianMorse FunctionMorse LemmaMorse Index

5 TransversalityStable and Unstable ManifoldsMorse-Smale Functions


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 Morsecomplexes

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


morse theory - eindhoven university of technologykbuchin/teaching/2ima00/2018/slides/mors… · the...

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