Morse Theory Roel Hospel Technische Universiteit Eindhoven roel.hospel@gmail.com May 24, 2018 Roel Hospel (TU/e) Morse Theory May 24, 2018 1 / 29
Overview Why Morse Theory? 1 Manifolds 2 Smooth Functions 3 Morse Functions 4 The Hessian Morse Function Morse Lemma Morse Index Transversality 5 Stable and Unstable Manifolds Morse-Smale Functions Roel Hospel (TU/e) Morse Theory May 24, 2018 2 / 29
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. Roel Hospel (TU/e) Morse Theory May 24, 2018 3 / 29
Manifolds A Manifold is a topological space that locally resembles Euclidean space near each point. Roel Hospel (TU/e) Morse Theory May 24, 2018 4 / 29
1-dimensional Manifolds Line Circle Figure-8 ? A Manifold is a topological space that locally resembles Euclidean space near each point. Roel Hospel (TU/e) Morse Theory May 24, 2018 5 / 29
1-dimensional Manifolds Line Circle Figure-8 A Manifold is a topological space that locally resembles Euclidean space near each point. Roel Hospel (TU/e) Morse Theory May 24, 2018 6 / 29
2-dimensional Manifolds Sphere Torus Boy’s Surface A Manifold is a topological space that locally resembles Euclidean space near each point. Roel Hospel (TU/e) Morse Theory May 24, 2018 7 / 29
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. Roel Hospel (TU/e) Morse Theory May 24, 2018 8 / 29
n -Manifolds n -Manifold A n -Manifold is a topological space that locally resembles n -dimensional Euclidean space near each point. Roel Hospel (TU/e) Morse Theory May 24, 2018 9 / 29
Recap: Differential Calculus Gradient (Tangent Line) Critical Points (Local) Minimum (Local) Maximum f ( x ) = x · sin( x 2 ) + 1 Roel Hospel (TU/e) Morse Theory May 24, 2018 10 / 29
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 C k -smooth. Roel Hospel (TU/e) Morse Theory May 24, 2018 11 / 29
Smooth Functions Formula Order k Derivative Smoothness f ( x ) = x f ′′ ( x ) = 0 f ( x ) is C 2 -smooth g ( x ) = x 2 − 3 g ′′′ ( x ) = 0 g ( x ) is C 3 -smooth h ( x ) = x 3 + x 2 h ′′′′ ( x ) = 0 h ( x ) is C 4 -smooth Table: Smoothness Example Formulas Roel Hospel (TU/e) Morse Theory May 24, 2018 12 / 29
Smooth Functions Formula Order k Derivative Smoothness f ( x ) = x f ′′ ( x ) = 0 f ( x ) is C 2 -smooth g ( x ) = x 2 g ′′′ ( x ) = 0 g ( x ) is C 3 -smooth h ( x ) = x 3 + x 2 h ′′′′ ( x ) = 0 h ( x ) is C 4 -smooth i ( x ) = sin( x ) i ( x ) is C ∞ -smooth j ( x ) = ... j ( x ) is non -smooth Table: Smoothness Example Formulas Roel Hospel (TU/e) Morse Theory May 24, 2018 13 / 29
Tangent Spaces on Manifolds The Tangent Space on an n -Manifold is the n -dimensional equivalent of a Tangent Line on a 1-Manifold. Roel Hospel (TU/e) Morse Theory May 24, 2018 14 / 29
Critical Points on Manifolds A point p on an n -Manifold is Critical Point iff all of its partial derivatives vanish. δ f 1-Manifold: f ( x ) δ x ( p ) = 0 δ f δ x ( p ) = δ f 2-Manifold: f ( x , y ) δ y ( p ) = 0 δ f δ x ( p ) = δ f δ y ( p ) = δ f 3-Manifold: f ( x , y , z ) δ z ( p ) = 0 etc. Roel Hospel (TU/e) Morse Theory May 24, 2018 15 / 29
The Hessian The Hessian is a formula you can calculate for a point p on a given function f ( x 1 , x 2 , ..., x d ) in d -dimensional vector space: δ f δ f δ f δ x 12 ( p ) δ x 1 δ x 2 ( p ) · · · δ x 1 δ x d ( p ) δ f δ f δ f δ x 2 δ x 1 ( p ) δ x 22 ( p ) · · · δ x 2 δ x d ( p ) H ( p ) = . . . ... . . . . . . δ f δ f δ f δ x d δ x 1 ( p ) δ x d δ x 2 ( p ) · · · δ x d 2 ( p ) Roel Hospel (TU/e) Morse Theory May 24, 2018 16 / 29
The Hessian, in 2D Vector Space δ f δ f δ f δ x 12 ( p ) δ x 1 δ x 2 ( p ) δ x 1 δ x d ( p ) · · · δ f δ f δ f δ x 2 δ x 1 ( p ) δ x 22 ( p ) δ x 2 δ x n ( p ) · · · H ( p ) = . . . ... . . . . . . δ f δ f δ f δ x n δ x 1 ( p ) δ x n δ x 2 ( p ) δ x d 2 ( p ) · · · Simplified to 2-dimensionsal vector space ( f ( x , y )) this function would become: � � δ f δ f δ x 2 ( p ) δ x δ y ( p ) H ( p ) = δ f δ f δ y δ x ( p ) δ y 2 ( p ) Roel Hospel (TU/e) Morse Theory May 24, 2018 17 / 29
Calculating the Hessian � � δ f δ f δ x 2 ( p ) δ x δ y ( p ) H ( p ) = δ f δ f δ y δ x ( p ) δ y 2 ( p ) Let’s calculate the Hessian over these two formulas: f ( x , y ) = x 2 + y 2 f ( x , y ) = x 2 + y 3 For which the critical points are both located at (0 , 0). Roel Hospel (TU/e) Morse Theory May 24, 2018 18 / 29
Degeneracy A critical point p on manifold M is Non-degenerate iff it holds for the Hessian at point p that H ( p ) � = 0 Roel Hospel (TU/e) Morse Theory May 24, 2018 19 / 29
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 Roel Hospel (TU/e) Morse Theory May 24, 2018 20 / 29
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 ) = ± x 2 ± y 2 Roel Hospel (TU/e) Morse Theory May 24, 2018 21 / 29
Morse Lemma Morse Lemma It is possible to choose local coordinates x 1 , .., x d at a critical point p ∈ M , for a vector space of dimension d , such that a Morse function f takes the form: f ( x 1 , x 2 , ..., x d ) = ± x 12 ± x 22 ... ± x d 2 Roel Hospel (TU/e) Morse Theory May 24, 2018 22 / 29
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 ) = ± x 2 ± y 2 Roel Hospel (TU/e) Morse Theory May 24, 2018 23 / 29
Morse Index in Higher Dimensions 1D 2D 3D f ( x , y ) = x 2 + y 2 f ( x , y , z ) = x 2 + y 2 + z 2 f ( x ) = x 2 f ( x , y , z ) = x 2 + y 2 − z 2 f ( x , y ) = x 2 − y 2 f ( x ) = − x 2 f ( x , y ) = − x 2 − y 2 f ( x , y , z ) = x 2 − y 2 − z 2 f ( x , y , z ) = − x 2 − y 2 − z 2 Roel Hospel (TU/e) Morse Theory May 24, 2018 24 / 29
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 = lim s →−∞ p ( s ) the origin of path p . We call dest p = lim s →∞ 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 of f Roel Hospel (TU/e) Morse Theory May 24, 2018 25 / 29
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 . Roel Hospel (TU/e) Morse Theory May 24, 2018 26 / 29
Morse-Smale Functions A Morse-Smale Function is a Morse function whose stable and unstable manifolds intersect transversally Roel Hospel (TU/e) Morse Theory May 24, 2018 27 / 29
Summary Why Morse Theory? 1 Manifolds 2 Smooth Functions 3 Morse Functions 4 The Hessian Morse Function Morse Lemma Morse Index Transversality 5 Stable and Unstable Manifolds Morse-Smale Functions Roel Hospel (TU/e) Morse Theory May 24, 2018 28 / 29
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 Roel Hospel (TU/e) Morse Theory May 24, 2018 29 / 29
Recommend
More recommend
