This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

# Applied algebraic topology

### From Mathematics Is A Science

*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
- 2.5 Relative topology
- 3.4 Simplicial complexes
- 3.6 Simplicial maps and chain maps
- 3.7 Homology of parametric complexe
- 4.4 Triangulations
- 4.4.4 Social choice: ranking

- 4.6 Products
- 5.3 Homology theory
- 5.4 Euler and Lefschetz numbers
- 5.5 Set-valued maps
- 6.1 Differential forms
- 6.3 Cohomology
- 6.4 Geometry
- 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