The basics of homology Aaron Fenyes (IHS) Young Data Scientist - - PowerPoint PPT Presentation
The basics of homology Aaron Fenyes (IHS) Young Data Scientist Seminar Harvard, November 2020 Building geometric spaces 2 0 0 1 0 1 standard 0-simplex standard 1-simplex standard 2-simplex 2 A -complex is a space built from
Aaron Fenyes (IHÉS)
Young Data Scientist Seminar Harvard, November 2020
Building geometric spaces
standard 0-simplex 1 standard 1-simplex 1 2 standard 2-simplex 1 2 3 standard 3-simplex standard 3-simplex A Δ-complex is a space built from simplices, which attach to each other by sharing faces.
(In a simplicial complex, no two simplices have the same set of vertices.)
Investigating deformations
Can these points be deformed into each other?
Investigating deformations
Can these points be deformed into each other?
Investigating deformations
Can these points be deformed into each other?
Investigating deformations
yes Can these points be deformed into each other?
Investigating deformations
yes Can these points be deformed into each other?
Investigating deformations
yes Can these points be deformed into each other? It’s easy to decide.
Investigating deformations
(Identify opposite edges)
Can these loops be deformed into each other?
Investigating deformations
(Identify opposite edges)
Can these loops be deformed into each other?
Investigating deformations
(Identify opposite edges)
Can these loops be deformed into each other?
Investigating deformations
(Identify opposite edges)
Can these loops be deformed into each other?
Investigating deformations
(Identify opposite edges)
yes Can these loops be deformed into each other?
Investigating deformations
(Identify opposite edges)
yes Can these loops be deformed into each other?
Investigating deformations
(Identify opposite edges)
yes Can these loops be deformed into each other?
Investigating deformations
(Identify opposite edges)
yes Can these loops be deformed into each other?
Investigating deformations
(Identify opposite edges)
yes Can these loops be deformed into each other?
Investigating deformations
(Identify opposite edges)
yes Can these loops be deformed into each other?
Investigating deformations
(Identify opposite edges)
yes Can these loops be deformed into each other?
Investigating deformations
(Identify opposite edges)
yes Can these loops be deformed into each other?
Investigating deformations
(Identify opposite edges)
yes Can these loops be deformed into each other?
Investigating deformations
(Identify opposite edges)
yes Can these loops be deformed into each other?
Investigating deformations
(Identify opposite edges)
yes Can these loops be deformed into each other?
Investigating deformations
(Identify opposite edges)
no yes Can these loops be deformed into each other?
Investigating deformations
(Identify opposite edges)
no yes Can these loops be deformed into each other? Homology gives a systematic way to decide.
embedded 0-simplex standard 0-simplex An embedded simplex is a map that sends the standard n-simplex to one of the n-simplices of a Δ-complex. Ordering the vertices of the standard n-simplex makes it easy to see where the map sends each vertex.
Turning geometry into algebra
1 1 embedded 1-simplex standard 1-simplex An embedded simplex is a map that sends the standard n-simplex to one of the n-simplices of a Δ-complex. Ordering the vertices of the standard n-simplex makes it easy to see where the map sends each vertex.
Turning geometry into algebra
2 1 embedded 2-simplex 2 1 standard 2-simplex An embedded simplex is a map that sends the standard n-simplex to one of the n-simplices of a Δ-complex. Ordering the vertices of the standard n-simplex makes it easy to see where the map sends each vertex.
Turning geometry into algebra
embedded 3-simplex 1 2 3 2 1 3 standard 3-simplex standard 3-simplex An embedded simplex is a map that sends the standard n-simplex to one of the n-simplices of a Δ-complex. Ordering the vertices of the standard n-simplex makes it easy to see where the map sends each vertex.
Turning geometry into algebra
1 1 1 An n-chain is a “formal sum” of embedded n-simplices.
(Identify opposite edges)
Turning geometry into algebra
1 1 1 1 0 1 1 An n-chain is a “formal sum” of embedded n-simplices.
(Identify opposite edges)
Turning geometry into algebra
1 1 1 An n-chain is a “formal sum” of embedded n-simplices. A simplex can appear more than once in the sum.
(Identify opposite edges)
Turning geometry into algebra
1 1 1 1 1 An n-chain is a “formal sum” of embedded n-simplices. A simplex can appear more than once in the sum.
(Identify opposite edges)
Turning geometry into algebra
1 1 1 0 1 1 An n-chain is a “formal sum” of embedded n-simplices. A simplex can appear more than once in the sum. For algebraic convenience, we allow negative simplices.
(Identify opposite edges)
Turning geometry into algebra
1 1 1 0 1 1 1 An n-chain is a “formal sum” of embedded n-simplices. A simplex can appear more than once in the sum. For algebraic convenience, we allow negative simplices.
(Identify opposite edges)
Turning geometry into algebra
Turning geometry into algebra
Sign rule: swapping two vertices of an embedded simplex is equivalent to reversing its sign. 1
Turning geometry into algebra
Sign rule: swapping two vertices of an embedded simplex is equivalent to reversing its sign. 1
Turning geometry into algebra
Sign rule: swapping two vertices of an embedded simplex is equivalent to reversing its sign. 1 2
Turning geometry into algebra
Sign rule: swapping two vertices of an embedded simplex is equivalent to reversing its sign. 2 1
Turning geometry into algebra
1 1 1 2 2 1 sign rule sign rule
Describing deformations
as substitution rules 1 You can deform a 0-simplex by pushing it across a 1-simplex.
Describing deformations
as substitution rules 1 You can deform a 0-simplex by pushing it across a 1-simplex.
Describing deformations
as substitution rules 1 You can deform a 0-simplex by pushing it across a 1-simplex.
Describing deformations
as substitution rules 1 You can deform a 0-simplex by pushing it across a 1-simplex.
Describing deformations
as substitution rules 1 You can deform a 0-simplex by pushing it across a 1-simplex.
Describing deformations
as substitution rules 1 You can deform a 0-simplex by pushing it across a 1-simplex.
Describing deformations
as substitution rules 2 1 You can deform a 1-simplex by pushing it across a 2-simplex.
Describing deformations
as substitution rules 2 1 You can deform a 1-simplex by pushing it across a 2-simplex.
Describing deformations
as substitution rules 2 1 You can deform a 1-simplex by pushing it across a 2-simplex.
Describing deformations
as substitution rules 2 1 You can deform a 1-simplex by pushing it across a 2-simplex.
Describing deformations
as substitution rules 2 1 You can deform a 1-simplex by pushing it across a 2-simplex.
Describing deformations
as substitution rules 2 1 You can deform a 1-simplex by pushing it across a 2-simplex.
Describing deformations
as substitution rules 2 1 You can deform a 1-simplex by pushing it across a 2-simplex.
Describing deformations
as substitution rules 2 1 You can deform a 1-simplex by pushing it across a 2-simplex.
Describing deformations
as substitution rules 2 1 You can deform a 1-simplex by pushing it across a 2-simplex.
Describing deformations
as substitution rules 2 1 2 1 You can deform a 1-simplex by pushing it across a 2-simplex.
Describing deformations
as substitution rules
1 2 1 1 2 1 2 1 2 1 1
Describing deformations
as substitution rules
1 2 1
1 2 1 2 1 2 1 1
boundary
2 1 1 2 1 2 1 2
1 1 1
Keeping track
1 1 1 The boundary operator sends each embedded (n+1)-simplex to the n-chain that describes deformation across the simplex.
Keeping track
2 1 1 2 1 2 1 2 The boundary operator sends each embedded (n+1)-simplex to the n-chain that describes deformation across the simplex.
Keeping track
1 2
The boundary operator, unexpectedly, characterizes closed loops, and “closed-up shapes” in higher dimensions. An n-chain C is called an n-cycle if ∂C = 0.
Deformation problems
as algebra problems
(Identify opposite edges)
1 2
The boundary operator, unexpectedly, characterizes closed loops, and “closed-up shapes” in higher dimensions. An n-chain C is called an n-cycle if ∂C = 0.
Deformation problems
as algebra problems
(Identify opposite edges)
1 2 1 2 We can deform B into A. We can turn B into A algebrically using deformation substitutions. A = B + ∂F for some (n+1)-chain F. For n-cycles A, B, these are equivalent:
Deformation problems
as algebra problems
(Identify opposite edges)
1 2 1 2 2 1 2 1
We can deform B into A. We can turn B into A algebrically using deformation substitutions. A = B + ∂F for some (n+1)-chain F. For n-cycles A, B, these are equivalent:
Deformation problems
as algebra problems
(Identify opposite edges)
1 2 2 1 1 2 1 2
We can deform B into A. We can turn B into A algebrically using deformation substitutions. A = B + ∂F for some (n+1)-chain F. For n-cycles A, B, these are equivalent:
Deformation problems
as algebra problems
(Identify opposite edges)
1 2 1 2 We can deform B into A. We can turn B into A algebrically using deformation substitutions. A = B + ∂F for some (n+1)-chain F. For n-cycles A, B, these are equivalent:
Deformation problems
as algebra problems
(Identify opposite edges)
1 2 1 2 2 1 2 1 We can deform B into A. We can turn B into A algebrically using deformation substitutions. A = B + ∂F for some (n+1)-chain F. For n-cycles A, B, these are equivalent:
Deformation problems
as algebra problems
(Identify opposite edges)
1 2 1 2 2 1 2 1 We can deform B into A. We can turn B into A algebrically using deformation substitutions. A = B + ∂F for some (n+1)-chain F. For n-cycles A, B, these are equivalent:
Deformation problems
as algebra problems
(Identify opposite edges)
1 2 1 2 2 1 1 2 We can deform B into A. We can turn B into A algebrically using deformation substitutions. A = B + ∂F for some (n+1)-chain F. For n-cycles A, B, these are equivalent:
Deformation problems
as algebra problems
(Identify opposite edges)
1 2 1 2 1 2 1 2 We can deform B into A. We can turn B into A algebrically using deformation substitutions. A = B + ∂F for some (n+1)-chain F. For n-cycles A, B, these are equivalent:
Deformation problems
as algebra problems
(Identify opposite edges)