Discussion Overview
The discussion centers around the differences between projection and inclusion when mapping between lower-dimensional and higher-dimensional spaces. Participants explore the implications of these mappings in the context of mathematical concepts such as embeddings, immersions, and the specific case of time-dependent systems in contact and symplectic manifolds.
Discussion Character
- Technical explanation
- Conceptual clarification
- Debate/contested
- Mathematical reasoning
Main Points Raised
- Fred questions the difference between the projection from a four-dimensional space to a three-dimensional space and the inclusion from a three-dimensional space to a four-dimensional space, expressing confusion about their relationship and whether they are inverses.
- One participant clarifies that the projection and inclusion are not inverses due to the non-unique mapping from (x, y, z) to (x, y, z, t), as multiple values of t can correspond to the same (x, y, z).
- Another participant suggests that while a map from a lower-dimensional space to a higher-dimensional space can exist, it would not be "onto," and a linear map could map an n-dimensional space into an m-dimensional space (where n < m).
- Fred seeks clarification on terminology, asking whether a map from a lower-dimensional space to a higher-dimensional space should be called an embedding, immersion, or inclusion, and expresses uncertainty about the nature of their specific case involving symplectic manifolds.
Areas of Agreement / Disagreement
Participants generally agree that projection and inclusion are not inverses and that the terminology around mappings between different dimensional spaces is nuanced. However, there is no consensus on the specific terms to use for mappings from lower-dimensional to higher-dimensional spaces.
Contextual Notes
Participants discuss the implications of defining mappings in the context of specific mathematical structures, such as contact and symplectic manifolds, but do not resolve the terminology or the nature of these mappings in relation to Fred's original question.
Who May Find This Useful
Mathematicians, physicists, and students interested in differential geometry, manifold theory, and the relationships between different dimensional spaces may find this discussion relevant.