Jens von Bergmann (University of Calgary),
Bibliographic record
Abstract
Symplectic manifolds are the natural domains of the modern mathematical formulation of classical mechanics, and they play a prominent role in many areas of mathematics. Since the introduction of the theory J–holomorphic curves by Gromov in [5], a tremendous amount of progress has been made in the study of these manifolds and the maps which preserve their symplectic structures. This progress has been particularly dramatic in the case of symplectic manifolds of dimension 4. For example, for S 2 ×S 2, the symplectic forms are classified (see [5], [14]), their symplectomorphism groups are well understood (see [5], [3]), and their Lagrangian spheres are all known to be symplectically equivalent (see [6]). Ruled symplectic 4–manifolds have also been completely classified (see [11]). The proofs of these results all have the same starting point; the existence of foliations by J–holomorphic spheres for all tamed almost complex structures. There have been several attempts to establish the existence of such foliations in more general settings. It was recently shown in [7] that such existence statements do not hold for foliations by J–holomorphic spheres of manifolds of dimension greater than 4. For index reasons, one can also not expect to find foliations by J–holomorphic curves of higher genus. To overcome this latter limitation, in the setting of symplectizations of a contact 3–manifolds, H. Hofer proposed in [8] to replace the standard J–holomorphic map equation with a parameterized version where the parameter takes values in the
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.006 | 0.003 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.234 | 0.176 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".