Meta-ticket: Families, posets, complexes of manifold subsets

[mkoeppe]

New subsets:

• Make open subsets of immersed/embedded submanifolds immersed/embedded submanifolds #31674 Make open subsets of immersed/embedded submanifolds immersed/embedded submanifolds (merged in Sage 9.4.beta1)
• Open subsets of arbitrary subsets of manifolds #31677 Open subsets of arbitrary subsets of manifolds (merged in Sage 9.4.beta1)
• Images of continuous maps and embedded submanifolds as manifold subsets #31653 Images of continuous maps and embedded submanifolds as manifold subsets (merged in Sage 9.4.beta1)
• DegenerateManifold: update subsets code #31765 DegenerateManifold: update subsets code
• Topological closure of manifold subsets, methods ManifoldSubset.closure, is_closed, declare_closed #31644 Topological closure of manifold subsets, methods ManifoldSubset.closure, is_closed, declare_closed (merged in Sage 9.4.beta4)
• Topological interior of manifold subsets #31757 Topological interior of manifold subsets
• Images of differentiable maps #31745 Images of differentiable maps
• Pullbacks (preimages) of manifold subsets under continuous maps #31688 Pullbacks (preimages) of manifold subsets under continuous maps (merged in Sage 9.4.beta5)
• ManifoldFiniteSubset #31724 ManifoldFiniteSubset
• ContinuousMap.image: if map is an isomorphism and domain_subset is open, create a submanifold #31685 ContinuousMap.image: if map is an isomorphism and domain_subset is open, create a submanifold
• Charts, transitions, subsets with restrictions defined using NNC_Polyhedron #31743 Charts, transitions, subsets with restrictions defined using NNC_Polyhedron

RealSet (see also Meta-ticket #20877):

• Connect RealSet to sage.manifolds #30832 Connect RealSet to sage.manifolds (merged in Sage 9.4.beta4)
• Replace OpenInterval, RealLine instances by instances of a subclass of DifferentiableManifold+RealSet #31878 Replace OpenInterval, RealLine instances by instances of a subclass of DifferentiableManifold+RealSet
• RealSet: add methods closure, interior, boundary, is_open, is_closed #21243 RealSet: add closure and interior methods (merged in Sage 9.4.beta4)
• RealSet: Extend constructors so that they can build manifold objects #31881 RealSet: Extend constructors so that they can build manifold objects (merged in Sage 9.4.rc0)

Families and posets of manifold subsets:

• Poset of manifold subsets #31680 Poset of manifold subsets (merged in Sage 9.4.beta3)
• ManifoldSubset: Change some methods to generators #31718 ManifoldSubset: Change some methods to generators (merged in Sage 9.4.beta3)
• ManifoldSubset: New methods declare_empty, declare_nonempty, is_empty, has_defined_points, open_cover_family #31732 ManifoldSubset: New methods declare_empty, declare_nonempty, is_empty, has_defined_points, open_cover_family (merged in Sage 9.4.beta3)
• ManifoldSubset: Add methods subset_family, superset_family, open_superset_family; deprecate method list_of_subsets #31727 ManifoldSubset: Add methods subset_family, superset_family, open_superset_family; deprecate method list_of_subsets (merged in Sage 9.4.beta3)
• ManifoldSubset: New methods declare_equal, equal_subsets, equal_subset_family #31736 ManifoldSubset: New methods declare_equal, equal_subsets, equal_subset_family (merged in Sage 9.4.beta3)
• ManifoldSubset: New methods declare_subset, declare_superset #31763 ManifoldSubset: New methods declare_subset, declare_superset (merged in Sage 9.4.beta3)
• ManifoldSubset: union, intersection, declare_{union,intersection} with arbitrary number of arguments #31764 ManifoldSubset: union, intersection, declare_{union,intersection} with arbitrary number of arguments (merged in Sage 9.4.beta3)
• ManifoldSubset.difference, complement #31798 ManifoldSubset.difference, complement (merged in Sage 9.4.beta3)
• ManifoldSubset: Replace use of _top_subsets by a generator #31766 ManifoldSubset: Replace use of _top_subsets by a generator
• ManifoldSubset.intersection: Compute more supersets #31738 ManifoldSubset.intersection: Compute more supersets
• manifolds: Add _test_... methods that check invariants of subset relations #31682 manifolds: Add _test_... methods that check invariants of subset relations
• declare_union yields wrong results #30263 declare_union yields wrong results

Geometric datastructure tools:

• Containers for families of geometric objects #32170 Containers for families of geometric objects

Topological tools:

• ManifoldSubset.separating_open_supersets #31741 ManifoldSubset.separating_open_supersets
• ManifoldSubset: Propagate emptiness/nonemptiness using LP #31744 ManifoldSubset: Propagate emptiness/nonemptiness using LP

Set_base and ConvexSet_base:

• {Polyhedron, ConvexRationalPolyhedralCone}.{interior, relative_interior} #31916 {Polyhedron, ConvexRationalPolyhedralCone}.{interior, relative_interior} (merged in Sage 9.4.beta4)
• ABC for convex sets #31919 ABC for convex sets (merged in Sage 9.4.beta4)
• PolyhedronFace: Make it a subclass of ConvexSet_closed #31959 PolyhedronFace: Make it a subclass of ConvexSet_closed (merged in Sage 9.4.beta4)
• ConvexSet_base: Add methods an_element, some_elements #31990 ConvexSet_base: Add methods an_element, some_elements (merged in Sage 9.4.beta5)
• Initialize a Set from a ConvexSet_base instance #32013 Initialize a Set from a ConvexSet_base instance (merged in Sage 9.4.beta6)
• ConditionSet #32089 ConditionSet (merged in Sage 9.4.beta6)

Application to polyhedra and optimization:

• Polyhedron.affine_hull_manifold #31659 Polyhedron.affine_hull_manifold (merged in Sage 9.4.beta3)
• Polyhedron.relative_interior_manifold, as_manifold_subset #31660 Polyhedron.relative_interior_manifold
• manifolds.Sphere: Make relation to simplicial spheres more concrete #31728 manifolds.Sphere: Make relation to simplicial spheres more concrete
• MixedIntegerLinearProgram.ambient_space, feasible_set, optimal_set #31742 MixedIntegerLinearProgram.ambient_manifold, feasible_subset, objective_scalar_field, optimal_subset
• Complex of differentiable manifolds associated with active sets of nonlinear optimization problems #31376 Complex of differentiable manifolds associated with active sets of nonlinear optimization problems
• Reformulations of optimization problems as continuous maps on manifolds #31751 Reformulations of optimization problems as continuous maps on manifolds

CC: @mjungmath @egourgoulhon @tscrim @yuan-zhou

Component: manifolds

Issue created by migration from https://trac.sagemath.org/ticket/31740

[mjungmath]

comment:13

For me, it gets harder and harder to keep track of the actual changes taking place in a particular ticket since there are a lot of cross-dependencies and no positively reviewed tickets have been merged lately, meaning the changes swiftly accumulate.

It might be helpful to state the ticket number in the commits which would make it easier to track the differences down, at least for me.

Or perhaps there is a nice git feature I am not aware of?

[mkoeppe]

comment:14

git log --first-parent goes a long way

[mjungmath]

comment:16

I think it would be nice to have something like powerset and topology that dynamically keep track of whether a (open) subset is contained in it.

This might also be a good opportunity to shift all this open_covers and subsets business to those classes and decouple it more from the manifold code. We can consider doing a similar thing for atlases, too. I can imagine that these changes could make pickling of manifolds easier.

This comes handy especially for sheaves, since it would allow to see them as (lazy) families indexed by topology.

Opinions?

[mjungmath]

comment:17

In principle, this could also lead to the more general notion of topological spaces. Manifolds are then nothing but topological spaces equipped with an atlas (roughly).

But to get there eventually, we need a lot of refactoring, i.e. something for the further future.

[mjungmath]

comment:18

This is also good for subset topologies in case of e.g. embedded manifolds.

[mkoeppe]

comment:19

See also #31785 "Category of open subsets of a topological space".