Formal libraries for Algebraic Topology: status report 1 ForMath La - PowerPoint PPT Presentation

Formal libraries for Algebraic Topology: status report 1 ForMath La Rioja node (J´ onathan Heras) Departamento de Matem´ aticas y Computaci´ on Universidad de La Rioja Spain Mathematics, Algorithms and Proofs 2010 November 10, 2010 1Partially supported by Ministerio de Educaci´ on y Ciencia, project MTM2009-13842-C02-01, and by European Commission FP7, STREP project ForMath ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 1/46

Contributors Local Contributors: Jes´ us Aransay C´ esar Dom´ ınguez J´ onathan Heras Laureano Lamb´ an Vico Pascual Mar´ ıa Poza Julio Rubio ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 2/46

Contributors Local Contributors: Jes´ us Aransay C´ esar Dom´ ınguez J´ onathan Heras Laureano Lamb´ an Vico Pascual Mar´ ıa Poza Julio Rubio Contributors from INRIA - Sophia: Yves Bertot Maxime D´ en` es Laurence Rideau Contributors from Universidad de Sevilla: Francisco Jes´ us Mart´ ın Mateos Jos´ e Luis Ruiz Reina ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 2/46

Goal Formalization of libraries for Algebraic Topology ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 3/46

Goal Formalization of libraries for Algebraic Topology Application: Study of digital images ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 3/46

Applying topological concepts to analyze images F. S´ egonne, E. Grimson, and B. Fischl. Topological Correction of Subcortical Segmentation. International Conference on Medical Image Computing and Computer Assisted Intervention, MICCAI 2003, LNCS 2879, Part 2, pp. 695-702. ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 4/46

Table of Contents Mathematical concepts 1 Computing in Algebraic Topology 2 Formalizing Algebraic Topology 3 Incidence simplicial matrices formalized in SSReflect 4 Conclusions and Further Work 5 ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 5/46

Mathematical concepts Table of Contents Mathematical concepts 1 Computing in Algebraic Topology 2 Formalizing Algebraic Topology 3 Incidence simplicial matrices formalized in SSReflect 4 Conclusions and Further Work 5 ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 6/46

Mathematical concepts From General Topology to Algebraic Topology Topological Space Digital Image triangulation Simplicial Complex interpreting algebraic structure Chain Complex simplification interpreting computing Homology ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 7/46

Mathematical concepts From General Topology to Algebraic Topology Topological Space Digital Image triangulation Simplicial Complex interpreting algebraic structure Chain Complex simplification interpreting computing Homology ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 7/46

Mathematical concepts From General Topology to Algebraic Topology Topological Space Digital Image triangulation Simplicial Complex interpreting algebraic structure Chain Complex simplification interpreting computing Homology ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 7/46

Mathematical concepts From General Topology to Algebraic Topology Topological Space Digital Image triangulation Simplicial Complex interpreting algebraic structure Chain Complex simplification interpreting computing Homology ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 7/46

Mathematical concepts From General Topology to Algebraic Topology Topological Space Digital Image triangulation Simplicial Complex interpreting algebraic structure Chain Complex simplification interpreting computing Homology ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 7/46

Mathematical concepts From General Topology to Algebraic Topology Topological Space Digital Image triangulation Simplicial Complex interpreting algebraic structure Chain Complex simplification interpreting computing Homology ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 7/46

Mathematical concepts From General Topology to Algebraic Topology Topological Space Digital Image triangulation Simplicial Complex interpreting algebraic structure Chain Complex simplification interpreting computing Homology ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 7/46

Mathematical concepts Simplicial Complexes Digital Image Simplicial Complex Chain Complex Homology simplification Definition Let V be an ordered set, called the vertex set. A simplex over V is any finite subset of V . ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 8/46

Mathematical concepts Simplicial Complexes Digital Image Simplicial Complex Chain Complex Homology simplification Definition Let V be an ordered set, called the vertex set. A simplex over V is any finite subset of V . Definition Let α and β be simplices over V , we say α is a face of β if α is a subset of β . ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 8/46

Mathematical concepts Simplicial Complexes Digital Image Simplicial Complex Chain Complex Homology simplification Definition Let V be an ordered set, called the vertex set. A simplex over V is any finite subset of V . Definition Let α and β be simplices over V , we say α is a face of β if α is a subset of β . Definition An ordered (abstract) simplicial complex over V is a set of simplices K over V satisfying the property: ∀ α ∈ K , if β ⊆ α ⇒ β ∈ K Let K be a simplicial complex. Then the set S n ( K ) of n-simplices of K is the set made of the simplices of cardinality n + 1 . ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 8/46

Mathematical concepts Simplicial Complexes Digital Image Simplicial Complex Chain Complex Homology simplification 2 5 3 4 0 6 1 V = (0 , 1 , 2 , 3 , 4 , 5 , 6) K = {∅ , (0) , (1) , (2) , (3) , (4) , (5) , (6) , (0 , 1) , (0 , 2) , (0 , 3) , (1 , 2) , (1 , 3) , (2 , 3) , (3 , 4) , (4 , 5) , (4 , 6) , (5 , 6) , (0 , 1 , 2) , (4 , 5 , 6) } ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 9/46

Mathematical concepts Simplicial Complexes Digital Image Simplicial Complex Chain Complex Homology simplification Definition The facets of a simplicial complex K are the maximal simplices of the simplicial complex. 2 5 3 4 0 6 1 The facets are: { (0 , 3) , (1 , 3) , (2 , 3) , (3 , 4) , (0 , 1 , 2) , (4 , 5 , 6) } ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 10/46

Mathematical concepts Chain Complexes Digital Image Simplicial Complex Chain Complex Homology simplification Definition A chain complex C ∗ is a pair of sequences C ∗ = ( C q , d q ) q ∈ Z where: For every q ∈ Z , the component C q is an R-module, the chain group of degree q For every q ∈ Z , the component d q is a module morphism d q : C q → C q − 1 , the differential map For every q ∈ Z , the composition d q d q +1 is null: d q d q +1 = 0 ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 11/46

Mathematical concepts Homology Digital Image Simplicial Complex Chain Complex Homology simplification Definition If C ∗ = ( C q , d q ) q ∈ Z is a chain complex: The image B q = im d q +1 ⊆ C q is the (sub)module of q-boundaries The kernel Z q = ker d q ⊆ C q is the (sub)module of q-cycles Given a chain complex C ∗ = ( C q , d q ) q ∈ Z : d q − 1 ◦ d q = 0 ⇒ B q ⊆ Z q Every boundary is a cycle The converse is not generally true ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 12/46

Mathematical concepts Homology Digital Image Simplicial Complex Chain Complex Homology simplification Definition If C ∗ = ( C q , d q ) q ∈ Z is a chain complex: The image B q = im d q +1 ⊆ C q is the (sub)module of q-boundaries The kernel Z q = ker d q ⊆ C q is the (sub)module of q-cycles Given a chain complex C ∗ = ( C q , d q ) q ∈ Z : d q − 1 ◦ d q = 0 ⇒ B q ⊆ Z q Every boundary is a cycle The converse is not generally true Definition Let C ∗ = ( C q , d q ) q ∈ Z be a chain complex. For each degree n ∈ Z , the n-homology module of C ∗ is defined as the quotient module H n ( C ∗ ) = Z n B n ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 12/46