Motivation How can we discover the shapes behind theses point clouds? Barcodes - or how to Discover Shapes in Complex Data David Meintrup University of Applied Sciences Ingolstadt September 15, 2015 David Meintrup Barcodes JMP Discovery 2015 1 / 27 David Meintrup Barcodes JMP Discovery 2015 2 / 27 Contents Geometry versus Topology I Leonhard Euler9, 1707 - 1783 Swiss Mathematician and Physicist 1 A short history of algebraic topology 866 publications 2 From point clouds to barcodes Seven Bridges of Königsberg: 3 Applications Is there a walk that crosses each bridge exactly once7? David Meintrup Barcodes JMP Discovery 2015 3 / 27 David Meintrup Barcodes JMP Discovery 2015 5 / 27 Geometry versus Topology II Allowed Operations allowed: all continuous smooth deformations The London underground map not allowed: cutting, tearing, joining (A) geometrically (B) topologically Geometry deals with distances and measures Topology deals with shapes and relations David Meintrup Barcodes JMP Discovery 2015 6 / 27 David Meintrup Barcodes JMP Discovery 2015 7 / 27 Topologically identical objects I Topologically identical objects II (A) Some versions of the (A) Square = Disk The oldest joke about a topologist Unknot6 A topologist is someone who who can’t tell the difference between a coffee mug and a doughnut2. (B) Sphere = Cube Topology is the “rubber band geometry“ David Meintrup Barcodes JMP Discovery 2015 8 / 27 David Meintrup Barcodes JMP Discovery 2015 9 / 27 Creating Shapes I Creating Shapes II What is associated to the word gluing ? Take a rectangle Mark arrows on opposite sites as shown Glue edges together What kind of shapes do you get? David Meintrup Barcodes JMP Discovery 2015 10 / 27 David Meintrup Barcodes JMP Discovery 2015 11 / 27 Creating Shapes III Creating Shapes IV (A) Do nothing: (C) Glue A on A, and B on B: (E) Glue A on A, (F) Glue A on A, and B on B, but turn the orientation but for B turn the orientation around! around! Torus Moebius band Klein bottle Disk (D) Glue all 4 edges together: (B) Glue A on A: Felix Klein9, 1849 - 1925 German Mathematician Tube learned General Relativity Theory Sphere with 70 David Meintrup Barcodes JMP Discovery 2015 12 / 27 David Meintrup Barcodes JMP Discovery 2015 13 / 27 Algebraic Invariants Betti Numbers I Betti Numbers: β , i = 0; 1; 2;::: ALGEBRAIC topology i β0: counts pieces (how many separate parts?) β : counts independent circles (how many holes?) Sir Michael Atiyah9, 1929* 1 β2: counts cavities/voids (how many empty volumes?) British Mathematician Examples: “Algebra is the offer made by the devil to the mathematician.“ We distinguish shapes by assigning numbers (so called algebraic invariants4) Idea: invariants different ) shapes different β = 1, β = 0, β = 0. β = 1, β = 1, β = 0. Example of invariants: Betti numbers10 0 1 2 0 1 2 David Meintrup Barcodes JMP Discovery 2015 14 / 27 David Meintrup Barcodes JMP Discovery 2015 15 / 27 Betti Numbers II Contents Examples:(β0 pieces, β1 holes, β2 voids) Two loops Sphere 1 A short history of algebraic topology β = , β = , β = . 0 2 1 2 2 0 2 From point clouds to barcodes Double loop β0 = 1, β1 = 0, β2 = 1. Torus 3 Applications β0 = 1, β1 = 2, β2 = 0. β0 = 1, β1 = 2, β2 = 1. David Meintrup Barcodes JMP Discovery 2015 16 / 27 David Meintrup Barcodes JMP Discovery 2015 18 / 27 TDA I TDA II - Example Topological data analysis in a nutshell: (1) Point Cloud (2) Topological shape: 1 start with a point cloud 2 inflate the points to get a topological shape 3 compute the Betti numbers 4 classify the shape (3) Betti numbers: (4) Shape: Circle β0 = 1, β1 = 1, β2 = 0. G. Seurat9 (1859-1891) David Meintrup Barcodes JMP Discovery 2015 19 / 27 David Meintrup Barcodes JMP Discovery 2015 20 / 27 TDA III TDA IV - Example Obvious question: How big should one inflate the points? Point cloud with 5 points: too small right size too big Answer: move from small to big and record the Betti numbers! Conclusion about underlying space: Circle David Meintrup Barcodes JMP Discovery 2015 21 / 27 David Meintrup Barcodes JMP Discovery 2015 22 / 27 Contents 1 A short history of algebraic topology 2 From point clouds to barcodes JMP demo 3 Applications David Meintrup Barcodes JMP Discovery 2015 24 / 27 David Meintrup Barcodes JMP Discovery 2015 25 / 27 Final Thoughts References Aleksandr Solzhenitsyn, In the First Circle References “Topology! The stratosphere of human thought! In the twenty-fourth [1] G. Carlsson, Topology and data, Bulletin of the American Mathematical Society (New century it might possibly be of use to someone . “ Series) 46 (2009), no. 2. [2] M. Coelho and J. Zigelbaum, Shape-changing interfaces, Personal Ubiquitous Comput. “The existence of the Klein bottle is 15, 2 (2011). not just a mathematical artifact. [3] R. Ghrist, Barcodes: the persistent topology of data, Bulletin of the American Mathemati- Instead, its presence is intimately tied cal Society (New Series) 45 (2008), no. 1. [ ] A. Hatcher, Algebraic topology, Cambridge University Press, New York, NY, 2002. to the geometry of cyclo-octane 4 [5] S. Martin, A. Thompson, E. A. Coutsias, and J.-P. Watson, Topology of cyclo-octane en- conformation, and . can be used to ergy landscape, Journal of Chemical Physics 132 (2010). ... explain the molecular motion of [6] popmath.org.uk, Centre for the Popularisation of Mathematics, University of Wales, Ban- cyclo-octane.“ gor, visited August 12, 2015. Topology of cyclo-octane energy [7] storyofmathematics.com/18th_euler.html, visited August 12, 2015. landscape, Martin et al. (2010) [5] [8] S. Weinberger, What is . Persistent Homology?, Notices Amer. Math. Soc. (2011). [9] Wikipedia.org, Articles and Pictures: Leonhard Euler, Felix Klein, Michael Atiyah, Klein Bottle, Georges Seurat, visited August 12, 2015. Thank you! [10] Afra Zomorodian, Topological data analysis, in: Advances in Applied and Computational Topology, Proc. Symp. Applied Math., Vol. 70. David Meintrup Barcodes JMP Discovery 2015 26 / 27 David Meintrup Barcodes JMP Discovery 2015 27 / 27.