AlgebraicJulia / AlgebraicJulia/CombinatorialSpaces.jl
Add SimplexTree for enforcing a DeltaSet to be a Complex
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- Julia
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Descrição
Based on our discussion today. Owen is going to implement the geometric maps in terms of the simplex tree data structure, which is just a Trie on the sorted vertices of the simplex.
```julia
struct SimplicialComplex{D}
delta_set::D
complexes::Trie{Int, Int}
end
```
Then GeometricMap is a morphism between SimplicialComplexes
```julia
struct GeometricPoint
simplex::Int
coords::Vector{Float64}
end
struct GeometricMap
dom::SimplicialComplex
codom::SimplicialComplex
values::Vector{GeometricPoint}
end
```
Then we also need the pullback maps
```julia
function pullback(f::GeometricMap, u::Form)::Form
dom(u) == codom(f) || error("Must pullback along the geometric map")
...
# return value is a form over the domain(f)
end
```
Source on Whitney Forms and Interpolation for the pullbacks: https://www.sciencedirect.com/science/article/pii/S0377042721001394?via%3Dihub
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