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 Functions The Hessian Morse Function Morse Lemma Morse Index 5 Transversality 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(x2) + 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 Ck -smooth. Roel Hospel (TU/e) Morse Theory May 24, 2018 11 / 29 Smooth Functions Formula Order k Derivative Smoothness 00 f (x) = x f (x) = 0 f (x) is C2-smooth 2 000 g(x) = x − 3 g (x) = 0 g(x) is C3-smooth 3 2 0000 h(x) = x + x h (x) = 0 h(x) is C4-smooth Table: Smoothness Example Formulas Roel Hospel (TU/e) Morse Theory May 24, 2018 12 / 29 Smooth Functions Formula Order k Derivative Smoothness 00 f (x) = x f (x) = 0 f (x) is C2-smooth 2 000 g(x) = x g (x) = 0 g(x) is C3-smooth 3 2 0000 h(x) = x + x h (x) = 0 h(x) is C4-smooth i(x) = sin(x) i(x) is C1-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. δf 1-Manifold: f (x) δx (p) = 0 δf δf 2-Manifold: f (x; y) δx (p) = δy (p) = 0 δf δf δf 3-Manifold: f (x; y; z) δx (p) = δy (p) = δz (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 (x1; x2; :::; xd ) in d-dimensional vector space: 2 δf δf δf 3 2 (p) (p) ··· (p) δx1 δx1δx2 δx1δxd δf δf δf 6 (p) 2 (p) ··· (p) 7 6 δx2δx1 δx2 δx2δxd 7 H(p) = 6 . 7 6 . .. 7 4 δf δf δf 5 (p) (p) ··· 2 (p) δxd δx1 δxd δx2 δxd Roel Hospel (TU/e) Morse Theory May 24, 2018 16 / 29 The Hessian, in 2D Vector Space 2 δf δf δf 3 2 (p) (p) ··· (p) δx1 δx1δx2 δx1δxd δf δf δf 6 (p) 2 (p) ··· (p) 7 6 δx2δx1 δx2 δx2δxn 7 H(p) = 6 . 7 6 . .. 7 4 δf δf δf 5 (p) (p) ··· 2 (p) δxnδx1 δxnδx2 δxd Simplified to 2-dimensionsal vector space (f (x; y)) this function would become: " δf δf # δx2 (p) δxδy (p) H(p) = δf δf δyδx (p) δy 2 (p) Roel Hospel (TU/e) Morse Theory May 24, 2018 17 / 29 Calculating the Hessian " δf δf # δx2 (p) δxδy (p) H(p) = δf δf δyδx (p) δy 2 (p) Let's calculate the Hessian over these two formulas: f (x; y) = x2 + y 2 f (x; y) = x2 + 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) 6= 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 2 M such that a Morse function f takes the form: f (x; y) = ±x2 ± y 2 Roel Hospel (TU/e) Morse Theory May 24, 2018 21 / 29 Morse Lemma Morse Lemma It is possible to choose local coordinates x1; ::; xd at a critical point p 2 M, for a vector space of dimension d, such that a Morse function f takes the form: 2 2 2 f (x1; x2; :::; xd ) = ±x1 ± x2 ::: ± xd 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 2 M, is the number of negative dimensions in the Morse function f . f (x; y) = ±x2 ± y 2 Roel Hospel (TU/e) Morse Theory May 24, 2018 23 / 29 Morse Index in Higher Dimensions 1D 2D 3D f (x) = x2 f (x; y) = x2 + y 2 f (x; y; z) = x2 + y 2 + z2 f (x) = −x2 f (x; y) = x2 − y 2 f (x; y; z) = x2 + y 2 − z2 f (x; y) = −x2 − y 2 f (x; y; z) = x2 − y 2 − z2 f (x; y; z) = −x2 − y 2 − z2 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 = lims→−∞ p(s) the origin of path p. We call dest p = lims!1 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 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 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.