Introduction to Algebraic Topology and Persistent Homology Soheil Anbouhi April 26th, 2018

Algebraic topology is a branch of mathematics that uses algebraic tools to study topological spaces and classifying them up to homeomorphism, though usually most classify up to homotopy equivalence. what does it really mean? A.T concerned with properties of geometric objects that are invariant under “continuous deformations.” (i.e, bending, twisting, stretching and not tearing) Michael Lesnick

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Example: Presence of Holes!! We have different types of hole, in fact for any non-negative i we can define i- dimensional holes. 0-D holes are connected components the pair of egg shape has two 0-dimensional holes

1-D holes in 3D objects are the ones you can actually see them The donut has one 1-D hole.

2-D holes in 3-D objects are hollow spaces.

0th-Betti number= 3 1th-Betti number= 0 2th-Betti number= 0 Betti numbers Informally, the kth-Betti number refers to the number of k-dimensional holes on a topological surface X. Example 0th-Betti number= 3 1th-Betti number= 0 2th-Betti number= 0

Topological Feature = connected components, holes, voids Persistent Homology is an algebraic method to perceive topological features of data Topological Feature = connected components, holes, voids

Example: What topological shape does the following data exhibit?

2-draw ball with radius r around every point. Main idea: Choose r > 0. Let U(S,r) be the union of balls of radius r centered at the points of a sample S from X(blue dots in the figure below). 1- choose a distance r 2-draw ball with radius r around every point.

0th-Betti number= 1 1th-Betti number= 1 2th-Betti number= 0 Main idea: Choose r > 0. Let U(S,r) be the union of balls of radius r centered at the sample points S of X. When X is nice enough, for a good choice of r, B(U(S, r)) detects a topological feature of X. Cycles that are not bounding simplexes are generators of homology groups in any given dimension. The r-th Betti number is the number of algebraically independent generators of H_r(X)., the r-the homology group. 0th-Betti number= 1 1th-Betti number= 1 2th-Betti number= 0 B(U(S, r))

Problems: 1-No clear way to choose r. 2- Invariant is unstable with respect to perturbation of data or small changes in r. 3-Doesn’t distinguish small holes from big ones Answer: Consider not single choice of radius r, but all choices of r at once.

Example X=

Pick an a r and vary it over positive numbers Example Pick an a r and vary it over positive numbers 0-holes #=13 1-holes #=0 U(B(S,r1))=

Example U(B(S,r2))= 0-holes #=2 1-holes #=1

Example 0-holes #=1 1-holes #=2 U(B(S,r3))=

persistent homology is the formalization of this idea. Observation: Not only can we count holes in each space, but also we can track them in a consistent way. persistent homology is the formalization of this idea.

Thank you References: M.Lesnick Studying the Shape of Data Using Topology, 2014 F. Chazal , B. Michel An introduction to Topological Data Analysis, 2017 R. Ghrist, THE PERSISTENT TOPOLOGY OF DATA A. Hatcher, Algebraic Topology, 2001. Wikipedia.