Comparing combinatorial models of moduli space and their compactifications
Bibliographic record
Abstract
Comparing combinatorial models of moduli space and their compactifications 597 and f is said to be quasiconformal if K f is finite.If QC.Œg 1 ; Œg 2 / denotes the set of all quasiconformal homeomorphisms between .S; Œg 1 / and .S; Œg 2 / fixing the points p i , then we can define the Teichmüller distance between Œg 1 and Œg 2 as follows:The moduli space of two-dimensional oriented cobordisms isomorphic to S is then defined to be the metric spaceconformal classes of metrics on S with good boundary conformal diffeomorphisms fixing the points p i ; d T Ã :For S that are not connected, we take the product of these spaces over all components.An alternative definition of these spaces is as the quotient of Teichmüller space (the space of quasiconformal maps modulo conformal equivalence) by the action of the mapping class group Mod.S; @S /, ie the group of components of the diffeomorphism group Diff.S; @S /.This is a free proper action on a contractible space, and hence M g .n;m/ ' B Mod.S; @S /.All connected components of Diff.S; @S / are contractible, and we can thus conclude thatThis explains why M g .n;m/ is a model for the moduli space of two-dimensional oriented cobordisms; any bundle of cobordisms over a paracompact space B with transition functions given by diffeomorphisms can be obtained up to isomorphism by pulling back a universal bundle over M g .n;m/ along a map B ! M g .n;m/.This universal bundle is the quotient of the space consisting of pairs .Œg; x/ of a conformal class of metrics and a point x 2 S , by conformal diffeomorphisms acting diagonally. Combinatorial models of moduli spaceWe discuss several combinatorial models of M g .n;m/, as well as certain compactifications.The following diagram spells out the relations between them (we fix g, n and m and drop them from the notation)
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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 | 0.000 | 0.001 |
| 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 | 0.002 |
| Scholarly communication | 0.004 | 0.004 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.007 | 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".