Opened 9 months ago
Last modified 3 months ago
#31249 new enhancement
Grassmann Manifolds
Reported by: | gh-mjungmath | Owned by: | |
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Priority: | major | Milestone: | sage-9.5 |
Component: | manifolds | Keywords: | |
Cc: | egourgoulhon, tscrim, gh-tobiasdiez | Merged in: | |
Authors: | Reviewers: | ||
Report Upstream: | N/A | Work issues: | |
Branch: | Commit: | ||
Dependencies: | Stopgaps: |
Description (last modified by )
Grassmannian manifolds are extremely important in (differential) geometry. Thus, we want to add them to the catalog of manifolds in Sage.
The following notes introduce charts and their transition maps very straightforwardly:
http://www.math.toronto.edu/mgualt/courses/18-367/docs/DiffGeomNotes-2.pdf
However, the number of charts grows extremely fast, namely ncr(n,k)
. One approach could be to parallelize the initialization process to reach at least low dimensional Grassmannians in a reasonable time.
Apart from that, it could be enough to start with only one chart since this should already cover enough (in particular, everything except a set of measure zero). That should give enough freedom to do almost everything on Grassmannians already.
Part of metaticket #30189.
Change History (13)
comment:1 Changed 9 months ago by
- Description modified (diff)
comment:2 follow-up: ↓ 4 Changed 9 months ago by
comment:3 Changed 9 months ago by
Another possible approach is to have a way to construct things lazily. So you have the charts, but all of the additional initialization and constructions would not be done until you actually do something with the chart. However, this would require a major overhaul to the manifolds implementation.
comment:4 in reply to: ↑ 2 Changed 9 months ago by
comment:5 follow-up: ↓ 6 Changed 9 months ago by
Or, more generally, flag manifolds.
For the charts, maybe it is convenient to use the fact that these examples are homogeneous spaces, so that one only needs to define the chart at the identity coset. The other charts can be then generated dynamically by translation on the group.
comment:6 in reply to: ↑ 5 ; follow-up: ↓ 9 Changed 9 months ago by
Replying to gh-tobiasdiez:
Or, more generally, flag manifolds.
For the charts, maybe it is convenient to use the fact that these examples are homogeneous spaces, so that one only needs to define the chart at the identity coset. The other charts can be then generated dynamically by translation on the group.
We neither have Lie groups nor group actions at the moment. And elements of e.g. GL(R, n)
are represented by matrices in the (by nature) inexact "field" of floating-point reals. I suspect some problems here.
comment:7 follow-up: ↓ 8 Changed 9 months ago by
Using the cluster charts (namely minors of matrices) and cluster mutations may be a good way, that would give as a bonus the positive grassmanians.
comment:8 in reply to: ↑ 7 ; follow-up: ↓ 10 Changed 9 months ago by
Replying to chapoton:
Using the cluster charts (namely minors of matrices) and cluster mutations may be a good way, that would give as a bonus the positive grassmanians.
You are thinking about representing the charts in terms of Plücker coordinates, correct?
comment:9 in reply to: ↑ 6 Changed 9 months ago by
Replying to gh-mjungmath:
Replying to gh-tobiasdiez:
Or, more generally, flag manifolds.
For the charts, maybe it is convenient to use the fact that these examples are homogeneous spaces, so that one only needs to define the chart at the identity coset. The other charts can be then generated dynamically by translation on the group.
We neither have Lie groups nor group actions at the moment. And elements of e.g.
GL(R, n)
are represented by matrices in the (by nature) inexact "field" of floating-point reals. I suspect some problems here.
I think there is still something you can do by considering the free parameters of the corresponding matrices to define the coordinate chart. So you don't really need the actual Lie group implemented as a manifold I believe.
comment:10 in reply to: ↑ 8 ; follow-up: ↓ 11 Changed 9 months ago by
Replying to tscrim:
You are thinking about representing the charts in terms of Plücker coordinates, correct?
Unfortunately, I don't know Plücker coordinates. I have to read some things first...any idea where to start?
Replying to tscrim:
I think there is still something you can do by considering the free parameters of the corresponding matrices to define the coordinate chart. So you don't really need the actual Lie group implemented as a manifold I believe.
Ah, alright. That could actually work! Nice!
comment:11 in reply to: ↑ 10 Changed 9 months ago by
Replying to gh-mjungmath:
Replying to tscrim:
You are thinking about representing the charts in terms of Plücker coordinates, correct?
Unfortunately, I don't know Plücker coordinates. I have to read some things first...any idea where to start?
The short version is Plücker coordinates are the minors for a matrix representing a point in the Grassmannian and give an embedding in projective space. The transition maps are quadratic and considered as cluster relations.
comment:12 Changed 7 months ago by
- Milestone changed from sage-9.3 to sage-9.4
Sage development has entered the release candidate phase for 9.3. Setting a new milestone for this ticket based on a cursory review of ticket status, priority, and last modification date.
comment:13 Changed 3 months ago by
- Milestone changed from sage-9.4 to sage-9.5
I think this is a good opportunity to investigate possible connections to pymanopt (#30495)...