Bibliographic record
Abstract
Modular stuffThis chapter introduces modular functions and forms, a subject central to the remainder of the book.Some earlier parts of this chapter are beautifully covered in [414].Section 2.1 supplies the underlying geometry, but can be skimmed on a first reading.In spite of this background material, the theory of modular forms and functions discussed in Sections 2.2 and 2.3 will probably appear as somewhat arbitrary to the uninitiated reader.Section 2.4.1 addresses some of this apparent artificiality, by developing the broader context of automorphic forms.As explained in the introductory chapter, Moonshine involves unexpected occurrences of modularity.The modularity of Moonshine functions follows from Zhu's Theorem (Theorem 5.3.8).However, the complexity of the underlying mathematics begs the question: Can modularity be established in a more elementary way?The simplest example of Moonshine involves theta functions.Hence we explore the limits and potentials of four classical strategies for proving the modularity of theta functions: Poisson summation, Dirichlet series, the heat kernel and representations of Heisenberg groups (Sections 2.2.3, 2.3.1, 2.3.4 and 2.4.2, respectively).Moonshine has really only been worked out in genus 1, 1 but conformal field theory tells us that there is an analogue for every genus (Section 6.3.1).It will be much more complicated, but it will be more rewarding because the number theoretic side is much less developed.In other words, we will find traces of, for example, the Monster in automorphic forms for the higher mapping class groups g,n and Sp 2n (Z).We include Sections 2.1.4and 2.3.5 in anticipation of this most natural and significant future development. The underlying geometry The hyperbolic planeThe birth of hyperbolic geometry is one of the most remarkable and instructive in the history of mathematics.Euclid's Fifth Postulate 2 was noticeably more complicated than the other axioms, looking more like a theorem than a self-evident proposal.Indeed, its converse was a theorem proved by Euclid.For example, compare it with Euclid's First 1 There are two possible meanings of 'genus' in a phrase like 'higher genus Moonshine'.Ordinary Monstrous Moonshine is genus 0 in the sense that the j-function is a Hauptmodul, i.e. a function on a sphere.It is genus 1 in the sense that the argument τ of j parametrises different tori.In this paragraph we are anticipating Moonshine's extension to higher genus in this second sense. 2 Also called the Parallel Postulate, it is equivalent to the simpler statement: Given any line L and a point p not on L, there is a unique line parallel to L that passes through p.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.603 | 0.450 |
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; the direct Gemma label and the distilled Codex classifier agree on what is shown here.
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".