Bibliographic record
Abstract
This thesis studies the topological properties of momentum maps of a large family of completely integrable systems called collective completely integrable systems. The first result concerns the topological monodromy of a collective completely integrable system on a product of three two-spheres. This system is called a "Heisenberg spin-chain" for its connection with a quantum integrable system of the same name. The remainder of the thesis is concerned with collective systems that are "constructed by Thimm's trick'' with the "action coordinates of Guillemin and Sternberg.'' We observe that the open dense subset of a symplectic manifold where such systems define a Hamiltonian torus action is connected. This observation was absent from the literature until this point. We also prove that if a system is constructed in this manner from a chain of Lie subalgebras then the image is given by an explicit set of inequalities called branching inequalities. When the symplectic manifold in question is a multiplicity free Hamiltonian U(n)-manifold, the construction of Thimm's trick with the action coordinates of Guillemin and Sternberg yields a completely integrable Hamiltonian torus action on a connected open dense subset. If the momentum map is proper, then we are able to prove lower bounds for the Gromov width of the symplectic manifold from the classification of the connected open dense subset as a non-compact symplectic toric manifold. Convex multiplicity free manifolds of compact Lie groups have been classified by Knop [51] and, accordingly, our lower bounds are given in terms of the combinatorics of the classifying data: the momentum set and a lattice. This result is the first estimate for the Gromov width of general multiplicity free manifolds of a nonabelian group. This result relies crucially on connectedness of the open dense subset and the explicit description of the momentum map image. The proof is a generalization of methods used by Pabiniak [71] to prove lower bounds for the Gromov width of U(n+1) coadjoint orbits, which are an example of multiplicity free U(n)-manifolds.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.000 |
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".