A Gauss-Bonnet theorem, Chern classes
Bibliographic record
Abstract
My thesis is about geometry of reductive groups. The purpose is to extend some well-known geometric results that hold for the complex torus (C∗)n to the case of an arbitrary complex reductive group. The thesis consists of two parts. The first part contains an analog of the Gauss-Bonnet theorem for constructible sheaves equivariant under the adjoint action. This theorem relates the Euler char-acteristic of a sheaf to the Gaussian degrees of the components of its characteristic cycle. As a corollary I get that a perverse sheaf equivariant under the adjoint action has nonnegative Euler characteristic. In the second part, I modify one of the classical definitions of the Chern classes to construct noncompact analogs of the Chern classes for equivariant vector bundles over reductive groups. These Chern classes have the same properties as the usual Chern classes. Using the Chern classes of the tangent bundle, I obtain an adjunction formula for the Euler characteristic of hypersurfaces in arbitrary reductive groups. The common feature of the results and constructions of my thesis is that I use ii a group action to extend to noncompact setting such classical results as the Gauss-Bonnet theorem and the adjunction formula. iii Acknowledgement I am very grateful to my scientific advisors Askold Khovanskii and Mikhail Kapranov for many useful discussions. I am also grateful to the external examiner Alexander Braverman for reading my thesis and writing the report. I appreciate the discussions with Kiumars Kaveh on the topics of my thesis. I would like to thank the graduate secretary Ida Bulat for her constant help and the University of Toronto for financial support during my PhD studies. iv Table of Contents Acknowledgement iv
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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) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 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".