Bibliographic record
Abstract
Stationary geodesic nets are embedded graphs in a Riemannian manifold $(M^{n},g)$ which are stationary with respect to the length functional. In this thesis, we study the distribution of closed geodesics and stationary geodesic nets in Riemannian manifolds. We prove that for a generic set of metrics on a closed manifold $M^{n}$, $n\geq 2$, the union of all the embedded stationary geodesic nets in $(M^{n},g)$ forms a dense subset of $M^{n}$. For $n=2$, we prove that for generic metrics on $M^{2}$ we can obtain an equidistributed sequence of closed geodesics. This means that there exists a sequence of closed geodesics $\{\gamma_{i}\}_{i\in\mathbb{N}}$ such that for every open subset $U$ of $M^{2}$, \begin{equation*} \lim_{k\to\infty}\frac{\sum_{i=1}^{k}\length_{g}(\gamma_{i}\cap U)}{\sum_{i=1}^{k}\length_{g}(\gamma_{i})}=\frac{\Vol_{g}(U)}{\Vol_{g}(M)}. \end{equation*} We show that the previous equidistribution result also holds for $n\geq 3$ but replacing closed geodesics by stationary geodesic nets. The main tool that we use is Almgren-Pitts Min-Max Theory, in particular the Weyl law for the volume spectrum. We also prove a Structure Theorem for stationary geodesic nets analogous to that of Brian White for minimal submanifolds, which is used to prove the density and equidistribution results. The density result was obtained in collaboration with Yevgeny Liokumovich, and the equidistribution result in dimensions $2$ and $3$ is joint work with Xinze Li.
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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 | 0.012 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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".