Applied algebraic topology

Topology Illustrated by Peter Saveliev

If you are familiar with algebraic topology, you can read these sections and subsections as one little applied topology book.

• 1.1 Topology around us
  • 1.1.1 Topology -- Algebra -- Geometry
  • 1.1.2 The integrity of everyday objects
  • 1.1.3 The shape of the Universe
  • 1.1.4 Data patterns
  • 1.1.5 Social choice
• 2.4 Continuous functions
  • 2.4.7 The dynamics of a market economy
• 2.5 Relative topology
  • 2.5.6 Social choice: looking for compromise
• 3.4 Simplicial complexes
  • 3.4.6 Data as a point cloud
  • 3.4.7 Social choice: the lottery of life
• 3.6 Simplicial maps and chain maps
  • 3.6.8 Social choice: no compromise
• 3.7 Homology of parametric complexe
  • 3.7.3 The homology of a gray scale image
  • 3.7.6 The "sharp" homology of a gray scale image
• 4.4 Triangulations
  • 4.4.4 Social choice: ranking
• 4.6 Products
  • 4.6.7 The Universe: 3-sphere, 3-torus, or something else?
  • 4.6.8 Configuration spaces
• 5.3 Homology theory
  • 5.3.9 Social choice: no impartiality
• 5.4 Euler and Lefschetz numbers
  • 5.4.4 An equilibrium of a market economy
  • 5.4.8 Social choice: gaming the system
• 5.5 Set-valued maps
  • 5.5.5 Motion planning in robotics
  • 5.5.6 Social choice: standstill in hostilities
• 6.1 Differential forms
  • 6.1.7 Social choice: rankings, ratings, and comparisons
• 6.3 Cohomology
  • 6.3.5 Social choice: higher order voting
• 6.4 Geometry
  • Newtonian physics, the discrete version
• 7.2 ODEs
  • 7.2.1 Motion: location from velocity
  • 7.2.2 Population growth
  • 7.2.3 Motion: location from acceleration
  • 7.2.4 Oscillating spring
  • 7.2.7 Flow simulation with a spreadsheet
  • 7.2.11 Advection with a spreadsheet
• 7.3 PDEs
  • 7.3.1 The PDE of diffusion
  • 7.3.2 Simulation in dimensions 1 and 2
  • 7.3.9 The PDE of wave propagation
  • 7.3.10 Solutions of the wave equation
  • 7.3.11 Waves in higher dimension
• 7.4 Social choice
  • 7.4.1 The paradox of social choice
  • 7.4.2 Ratings, comparisons, ranking, preferences
  • 7.4.3 The algebra of vote aggregation
  • 7.4.4 Aggregating rating votes
  • 7.4.5 Aggregating comparison votes
  • 7.4.6 Google's PageRank
  • 7.4.7 Combining ratings with comparisons
  • 7.4.8 Decycling: how to extract ratings from comparisons
  • 7.4.9 In search of fair elections
  • 7.4.10 Decycling with a spreadsheet