The notion of space in Grothendieck: from schemes to a geometry of forms

TL;DR

This paper traces Grothendieck's evolving ideas on the concept of space, from algebraic geometry to a broader geometry of forms, highlighting their philosophical and physical implications.

Contribution

It provides a comprehensive overview of Grothendieck's development of the notion of space, connecting mathematical, physical, and philosophical perspectives.

Findings

Grothendieck's ideas evolved from schemes to a geometry of forms.

The work links algebraic geometry to physical and philosophical questions.

It offers a unified view of the concept of space across disciplines.

Abstract

In this essay we give a general picture about the evolution of Grohendieck's ideas regarding the notion of space. Starting with his fundamental work in algebraic geometry, where he introduces schemes and toposes as generalizations of classical notions of spaces, passing through tame topology and ending with the formulation of a geometry of forms, we show how the ideas of Grothendieck evolved from pure mathematical considerations to physical and philosophical questions about the nature and structure of space and its mathematical models.