RECORD
On the early history of moduli and Teichm{\"u}ller spaces
History and Overview · Submitted 23 Feb 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We point out several facts which seem to be unknown to many algebraic geometers and analysts working in the theory.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
We survey some major contributions to Riemann's moduli space and Teichm{\"u}ller space. Our report has a historical character, but the stress is on the chain of mathematical ideas. We start with the introduction of Riemann surfaces, and we end with the discovery of some of the basic structures of Riemann's moduli space and Teichm{\"u}ller space. We point out several facts which seem to be unknown to many algebraic geometers and analysts working in the theory. The period we are interested in starts with Riemann, in 1851, and ends in the early 1960s, when Ahlfors and Bers confirmed that Teichm{\"u}ller's results were correct.This paper was written for the book "Lipman Bers, a life in Mathematics," edited by Linda Keen , Irwin Kra and Rubi Rodriguez (Amercian Mathematical Society, 2015). It is dedicated to the memory of Lipman Bers who was above all a complex analyst and spent a large part of his life and energy working on the analytic structure of Teichm{\"u}ller space. His work on analysis is nevertheless inseparable from geometry and topology. In this survey, we highlight the relations and the logical dependence between this work and the works of Riemann, Poincar{\'e}, Klein, Brouwer, Siegel, Teichm{\"u}ller, Weil, Grothendieck and others. We explain the motivation behind the ideas. In doing so, we point out several facts which seem to be unknown to many Teichm{\"u}ller theorists.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We survey some major contributions to Riemann's moduli space and Teichm{\"u}ller space. | contribution 0.92 | 0.41 | 0.07 | asserted 0.85 |
| s1 | Our report has a historical character, but the stress is on the chain of mathematical ideas. | contribution 0.38 | 0.35 | 0.05 | asserted 1.00 |
| s2 | We start with the introduction of Riemann surfaces, and we end with the discovery of some of the basic structures of Riemann's moduli space and Teichm{\"u}ller space. | contribution 0.56 | 0.22 | 0.10 | asserted 0.98 |
| s3 | We point out several facts which seem to be unknown to many algebraic geometers and analysts working in the theory. | gap 0.49 | 0.37 | 0.12 | hedged possibility 0.65 |
| s4 | The period we are interested in starts with Riemann, in 1851, and ends in the early 1960s, when Ahlfors and Bers confirmed that Teichm{\"u}ller's results were correct.This paper was written for the book "Lipman Bers, a life in Mathematics," edited by Linda Keen , Irwin Kra and Rubi Rodriguez (Amercian Mathematical Society, 2015). | background 0.82 | 0.09 | 0.07 | asserted 0.99 |
| s5 | It is dedicated to the memory of Lipman Bers who was above all a complex analyst and spent a large part of his life and energy working on the analytic structure of Teichm{\"u}ller space. | background 0.99 | 0.06 | 0.03 | asserted 1.00 |
| s6 | His work on analysis is nevertheless inseparable from geometry and topology. | background 0.88 | 0.06 | 0.03 | asserted 1.00 |
| s7 | In this survey, we highlight the relations and the logical dependence between this work and the works of Riemann, Poincar{\'e}, Klein, Brouwer, Siegel, Teichm{\"u}ller, Weil, Grothendieck and others. | contribution 0.84 | 0.61 | 0.07 | asserted 1.00 |
| s8 | We explain the motivation behind the ideas. | contribution 0.74 | 0.77 | 0.03 | asserted 0.95 |
| s9 | In doing so, we point out several facts which seem to be unknown to many Teichm{\"u}ller theorists. | gap 0.50 | 0.72 | 0.08 | hedged possibility 0.53 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
We survey some major contributions to Riemann's moduli space and Teichm{\"u}ller space. Our report has a historical character, but the stress is on the chain of mathematical ideas. We start with the introduction of Riemann surfaces, and we end with the discovery of some of the basic structures of Riemann's moduli space and Teichm{\"u}ller space. We point out several facts which seem to be unknown to many algebraic geometers and analysts working in the theory. The period we are interested in starts with Riemann, in 1851, and ends in the early 1960s, when Ahlfors and Bers confirmed that Teichm{\"u}ller's results were correct.This paper was written for the book "Lipman Bers, a life in Mathematics," edited by Linda Keen , Irwin Kra and Rubi Rodriguez (Amercian Mathematical Society, 2015). It is dedicated to the memory of Lipman Bers who was above all a complex analyst and spent a large part of his life and energy working on the analytic structure of Teichm{\"u}ller space. His work on analysis is nevertheless inseparable from geometry and topology. In this survey, we highlight the relations and the logical dependence between this work and the works of Riemann, Poincar{\'e}, Klein, Brouwer, Siegel, Teichm{\"u}ller, Weil, Grothendieck and others. We explain the motivation behind the ideas. In doing so, we point out several facts which seem to be unknown to many Teichm{\"u}ller theorists.
Human review
How the literature uses it
4 arXiv papers cite this in 4 sentences
Papers citing it
Cites it as background 4 papers
-
L'espace de modules $M_g$ des courbes lisses de genre $g⩾2$ construit par Mumford [ref] est une variété algébrique dont les points complexes sont naturellement en bijection avec les classes d'isomorphisme de courbes projectives lisses complexes de genre $g$ (nous renvoyons à [ref] et à [ref] pour un aperçu de l'histoire de ce sujet).
R\'eduction stable en dimension sup\'erieure [d'apr\`es Koll\'ar, Hacon-Xu...] · Introduction · 0.99
-
The central concept here is that of a `moduli space' $ℳ$ of objects (cf. our local models), i.e. a space whose points are in 1-1 correspondence with isomorphism classes of such objects (thus classifying them in a naive sense), and which is in addition rich enough to encode the structure of families of such objects over a base space $B$ (cf. our families of local models parameterised by a contractible cover of $M$) in the sense that such families will be in 1-1-correspondence with maps from $B$ to $ℳ$.[For a history of the development of the `moduli space' concept, see [ref] , and for an elementary introduction that stresses the notion of `classification', see [ref] .] Despite the fruitfulness of this notion, it was recognised very early on that such moduli spaces may not exist if the objects possess non-trivial automorphisms (cf. the non-trivial automorphisms of objects in $_A$), because
Why surplus structure is not superfluous · Conclusion · 0.98
-
Those two questions originated from Riemann when he set up the Riemannian geometry in the nineteenth century [ref] .
Metric topology on the moduli space · Introduction · 1.00
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.