This document provides an overview of computational topology, including its key concepts and applications. It outlines the main topics covered, such as topological spaces, surfaces, complexes, simplicial homology, and persistent homology. Computational topology involves developing algorithms to solve problems in topology, like determining if shapes are connected or embedded in a plane. Potential applications include image segmentation, analyzing gene expression data, and molecular shape matching for drug design.
This document provides an overview of computational topology, including its key concepts and applications. It outlines the main topics covered, such as topological spaces, surfaces, complexes, simplicial homology, and persistent homology. Computational topology involves developing algorithms to solve problems in topology, like determining if shapes are connected or embedded in a plane. Potential applications include image segmentation, analyzing gene expression data, and molecular shape matching for drug design.
This document provides an overview of computational topology, including its key concepts and applications. It outlines the main topics covered, such as topological spaces, surfaces, complexes, simplicial homology, and persistent homology. Computational topology involves developing algorithms to solve problems in topology, like determining if shapes are connected or embedded in a plane. Potential applications include image segmentation, analyzing gene expression data, and molecular shape matching for drug design.
This document provides an overview of computational topology, including its key concepts and applications. It outlines the main topics covered, such as topological spaces, surfaces, complexes, simplicial homology, and persistent homology. Computational topology involves developing algorithms to solve problems in topology, like determining if shapes are connected or embedded in a plane. Potential applications include image segmentation, analyzing gene expression data, and molecular shape matching for drug design.
2. Topology: James R. Munkres 3. Computational Topology: An Introduction, Herbert Edelsbrunner, John Harer, AMS, 2010. 4. Effective Computational Geometry for Curves and Surfaces, Ch 7: Computational Topology: An Introduction, Günter Rote and Gert Vegter
3/21 Overview
• What is ‘Topology’?
• Why do we require Computation (Algorithms)?
• Where are the potential applications?
4/21 Topology
• Branch of Mathematics - Systematic study of shapes
• Shapes - Topological Spaces • Goal - Classification of shapes using global properties (e.g. connectivity, holes of different dimensions) • Topological Equivalence: Homeomorphism, Isotopy, Homotopy, Homology
5/21 Topological Equivalence: Examples
6/21 Topological Equivalence: Examples
7/21 Topological Spaces: Examples
8/21 Topological Spaces: Examples
• 1D: Curves
• 2D: Surfaces
• nD: Manifolds
9/21 Classification Theorem
Any orientable closed surface is homeomorphic either to the sphere, or to
the sphere with a finite number of handles added.
10/21 How to Distinguish Topological Spaces?: Invariants
• Properties of a topological space that remain invariant under
homeomorphism - Connectedness, Euler characteristics, Homology groups etc. • Useful to show two topological spaces are not equivalent
11/21 Computational Topology
• Solving algorithmic problems in topology
• Problem: Are the black dots connected?
12/21 Problem: Parity
• Problem: Inside or outside the region enclosed by the polygon?
13/21 Problem: Equivalent Knots
• Problem: Algorithm to check whether two knots are equivalent or
not
14/21 Problem: Graph Embedding in Plane
• Problem: Algorithm to compute a planar embedding of a graph, if
exists
15/21 Surfaces: Algorithms
• Searching a triangulation of a surface
• Computing Euler characteristic of a surface • Checking Orientability of a surfaces • Computing genus and topological type of a surface • Checking Crossings • Simplification of surfaces
16/21 Complexes: Representing topological spaces
• Simplicial Complex • Convex set systems • Delaunay Complexes • Alpha Complexes
17/21 Computational Topology: Tools for Data Analysis
• Simplicial Homology: Counting Holes in the data
• Morse Theory: Study of “Critical Features”
18/21 Computational Topology: Tools for Data Analysis
• Persistent Homology: Study of "Births" and "Deaths" of holes
• Topological Skeletons: for Scalar and Vector Data
19/21 Applications
• Persistence based Image segmentation
• Measures of Gene Expression Data
• Elevation for Protein Docking
• Homology for Root Architectures
• Molecular shape matching for drug design
• From Chemical Topology to Molecular Machines (Nobel Prize in
Computational Chemistry, 2017)
20/21 Summary
• Topology: A branch of mathematics for the systematic study of
shapes based on global properties (holes) • Computational Topology: Algorithmic problems in topology (e.g. Counting holes of different dimensions in a topological space) • Applications: Data+Topology = Shape of the Data
• Applications: Topological Data Analysis, Topological Machine
