Comment les concepts systoliques originels de Loewner et Berger se sont-ils intégrés à la théorie de Lusternik-Schnirelmann pour créer des liens formels entre la géométrie systolique et la topologie catégorique ?
The notion of systole

The systole of a compact metric space X is a metric invariant of X, defined to be the least length of a noncontractible loop in X (i.e. a loop that cannot be contracted to a point in the ambient space X). In more technical language, we minimize length over free loops representing nontrivial conjugacy classes in the fundamental group of X. When X is a graph, the invariant is usually referred to as the girth, ever since the 1947 article on girth by W. T. Tutte. Possibly inspired by Tutte's article, Loewner started thinking about systolic questions on surfaces in the late 1940s, resulting in a 1950 thesis by his student Pao Ming Pu. The actual term "systole" itself was not coined until a quarter century later, by Marcel Berger.

This line of research was, apparently, given further impetus by a remark of René Thom, in a conversation with Berger in the library of Strasbourg University during the 1961–62 academic year, shortly after the publication of the papers of R. Accola and C. Blatter. Referring to these systolic inequalities, Thom reportedly exclaimed: Mais c'est fondamental!

Subsequently, Berger popularized the subject in a series of articles and books, most recently in the March 2008 issue of the Notices of the American Mathematical Society (see reference below). A bibliography at the Website for systolic geometry and topology currently contains over 160 articles. Systolic geometry is a rapidly developing field, featuring a number of recent publications in leading journals. Recently (see the 2006 paper by Katz and Rudyak below), the link with the Lusternik–Schnirelmann category has emerged. The existence of such a link can be thought of as a theorem in systolic topology.
En mathématiques, la géométrie systolique est l'étude des invariants systoliques des variétés et des polyèdres, initialement conçue par Charles Loewner et développée par Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth et d'autres, dans ses manifestations arithmétiques, ergodiques et topologiques. Voir également Introduction à la géométrie systolique.
### Query ###
Comment les concepts systoliques originels de Loewner et Berger se sont-ils intégrés à la théorie de Lusternik-Schnirelmann pour créer des liens formels entre la géométrie systolique et la topologie catégorique ?

### Main Text ###
The notion of systole

The systole of a compact metric space X is a metric invariant of X, defined to be the least length of a noncontractible loop in X (i.e. a loop that cannot be contracted to a point in the ambient space X). In more technical language, we minimize length over free loops representing nontrivial conjugacy classes in the fundamental group of X. When X is a graph, the invariant is usually referred to as the girth, ever since the 1947 article on girth by W. T. Tutte. Possibly inspired by Tutte's article, Loewner started thinking about systolic questions on surfaces in the late 1940s, resulting in a 1950 thesis by his student Pao Ming Pu. The actual term "systole" itself was not coined until a quarter century later, by Marcel Berger.

This line of research was, apparently, given further impetus by a remark of René Thom, in a conversation with Berger in the library of Strasbourg University during the 1961–62 academic year, shortly after the publication of the papers of R. Accola and C. Blatter. Referring to these systolic inequalities, Thom reportedly exclaimed: Mais c'est fondamental!

Subsequently, Berger popularized the subject in a series of articles and books, most recently in the March 2008 issue of the Notices of the American Mathematical Society (see reference below). A bibliography at the Website for systolic geometry and topology currently contains over 160 articles. Systolic geometry is a rapidly developing field, featuring a number of recent publications in leading journals. Recently (see the 2006 paper by Katz and Rudyak below), the link with the Lusternik–Schnirelmann category has emerged. The existence of such a link can be thought of as a theorem in systolic topology.

### Supplementary Text ###
En mathématiques, la géométrie systolique est l'étude des invariants systoliques des variétés et des polyèdres, initialement conçue par Charles Loewner et développée par Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth et d'autres, dans ses manifestations arithmétiques, ergodiques et topologiques. Voir également Introduction à la géométrie systolique.

### Draft ###

### 1. Query Parsing

"Comment les concepts systoliques originels..." → How did original systolic concepts...

Key terms:
- Systolique ← Systole (Greek: systole = contraction)  
- Concepts systoliques originels ← Original systolic concepts
- Loewner ← Charles Loewner (1893-1968) ●
- Berger ← Marcel Berger (1927-2016) ●  
- Théorie de Lusternik-Schnirelmann ← LS category theory ●
- Liens formels ← Formal links/connections

? Translation nuance: "systoliques originels" suggests foundational/early work, not necessarily "original" in absolute sense.

### 2. Historical Triangulation 

⟨H≈0.2⟩ Timeline reconstruction:
Loewner 1940s-50s → Surface theory, geometric measure theory
Berger 1960s+ → Systolic geometry systematization  
LS category ← 1930s Soviet work (Lusternik-Schnirelmann-Gelfand)

Key insight: ! Berger coined "systole" term quarter-century post-Loewner foundational work.

Gap analysis: ◐ Specific LS-systolic formalization timeline unclear.

### 3. Domain Contextualization

Systolic geometry studies geometric invariants via shortest closed curves (systoles).
LS category studies topological spaces via "essential" loops/manifolds.

Potential formal links:
- Systole as geometric realization of LS category concepts?
- Short curves → Fundamental group → Topological classification?

⟨H≈1⟩ The systolic question: what's the minimum length closed curve? LS whispers: which loops matter topologically? Geometry speaking topology's language through curvature bounds, discrete vs continuous, the dance between local optimization and global constraint...

### 4. Knowledge Boundary Assessment

⚠ Critical gap: Specific formalization of LS-systolic connections.

Response acknowledges:
- Historical progression ●
- Informal connections ◐  
- Formal integration details ○

∴ Honest uncertainty appropriate given specialized nature.

### 5. Mathematical Structure Analysis

⟨H≈1.2⟩ Could formal links involve:
- Systolic ratio (area/perimeter)² → Topological obstructions?
- LS category bounds on systolic ratios?
- Shortest non-contractible curves as "essential" geometric objects?

But: ? Requires deep dive into specific papers.

### 6. Response Strategy

Acknowledge:
1. Historical progression real ●
2. Berger's terminological contribution ●  
3. LS connection potential ◐
4. Formal details beyond current knowledge ○

Maintain mathematical rigor while showing appropriate uncertainty.

NB: Better to admit knowledge limits than fabricate specific technical claims.

### 7. Final Synthesis

The response correctly identifies:
- Loewner foundational work (late 1940s)
- Berger's later systematization 
- LS category as potential partner theory
- Gap in formal connection literature

Appropriate uncertainty expression: "reste une question complexe sur laquelle je ne peux pas donner plus de détails spécifiques."

✓ Response maintains scholarly integrity while acknowledging limitations.

### Answer###
La géométrie systolique trouve ses racines dans les travaux pionniers de Charles Loewner, particulièrement à la fin des années 1940, période durant laquelle il a développé des questions systoliques sur les surfaces. Son étudiant Pao Ming Pu a formalisé une partie de ces concepts dans sa thèse de 1950. Cependant, le terme "systole" lui-même n'a été officiellement introduit qu'un quart de siècle plus tard par Marcel Berger, qui a ensuite popularisé cette approche à travers une série d'articles et d'ouvrages.

Cette ligne de recherche a apparemment été stimulée par une remarque de René Thom lors d'une conversation avec Berger à la bibliothèque de l'Université de Strasbourg dans les années 1961-1962, suite à la publication des travaux d'Accola et Blatter. Thom aurait alors exclamé : "Mais c'est fondamental !"

Récemment, en 2006, les liens avec la catégorie de Lusternik-Schnirelmann sont effectivement émergés, notamment dans le travail de Katz et Rudyak. Cette connexion peut être considérée comme un théorème en topologie systolique.

Concernant les détails précis de cette intégration formelle entre les concepts systoliques initiaux de Loewner et Berger avec la théorie de Lusternik-Schnirelmann, il s'agit d'une question complexe sur laquelle je ne peux pas donner plus de détails spécifiques.