"Esquisse d'un Programme" (Sketch of a Programme) is a famous proposal for long-term mathematical research made by the German-born, French mathematician Alexander Grothendieck in 1984. He pursued the sequence of logically linked ideas in his important project proposal from 1984 until 1988, but his proposed research continues to date to be of major interest in several branches of advanced mathematics. Grothendieck's vision provides inspiration today for several developments in mathematics such as the extension and generalization of

Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...

, which is currently being extended based on his original proposal.

Brief history

Submitted in 1984, the ''Esquisse d'un Programme'' was a proposal submitted by Alexander Grothendieck for a position at the

Centre National de la Recherche Scientifique The French National Centre for Scientific Research (french: link=no, Centre national de la recherche scientifique, CNRS) is the French state research organisation and is the largest fundamental science Basic research, also called pure research o ...

. The proposal was not successful, but Grothendieck obtained a special position where, while keeping his affiliation at the University of Montpellier, he was paid by the CNRS and released of his teaching obligations. Grothendieck held this position from 1984 till 1988. This proposal was not formally published until 1997, because the author "could not be found, much less his permission requested". The outlines of ''

dessins d'enfants In mathematics, a dessin d'enfant is a type of graph embedding used to study Riemann surfaces and to provide combinatorial Invariant (mathematics), invariants for the action of the absolute Galois group of the rational numbers. The name of these e ...

'', or "children's drawings", and "

Anabelian geometry Anabelian geometry is a theory in number theory which describes the way in which the algebraic fundamental group ''G'' of a certain arithmetic variety ''X'', or some related geometric object, can help to restore ''X''. The first results for num ...

", that are contained in this manuscript continue to inspire research; thus, "Anabelian geometry is a proposed theory in

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, describing the way the algebraic fundamental group ''G'' of an

algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...

''V'', or some related geometric object, determines how ''V'' can be mapped into another geometric object ''W'', under the assumption that ''G'' is not an

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...

, in the sense of being strongly

noncommutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...

. The word ''anabelian'' (an

alpha privative An alpha privative or, rarely, privative a (from Latin ', from Ancient Greek ) is the prefix ''a-'' or ''an-'' (before vowels) that is used in Indo-European languages such as Sanskrit and Greek and in words borrowed therefrom to express negation or ...

''an-'' before ''abelian'') was introduced in ''Esquisse d'un Programme''. While the work of Grothendieck was for many years unpublished, and unavailable through the traditional formal scholarly channels, the formulation and predictions of the proposed theory received much attention, and some alterations, at the hands of a number of mathematicians. Those who have researched in this area have obtained some expected and related results, and in the 21st century the beginnings of such a theory started to be available."

Abstract of Grothendieck's programme

("''Sommaire''") *1. The Proposal and enterprise ("Envoi"). *2. " Teichmüller's Lego-game and the

Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...

of Q over Q" ("Un jeu de “Lego-Teichmüller” et le groupe de Galois de Q sur Q"). *3.

Number fields In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...

associated with dessins d'enfant". ("Corps de nombres associés à un dessin d’enfant"). *4.

Regular polyhedra A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equival ...

over

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...

s ("Polyèdres réguliers sur les corps finis"). *5.

General topology In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...

or a ' Moderated Topology' ("Haro sur la topologie dite 'générale', et réflexions heuristiques vers une topologie dite 'modérée"). *6. Differentiable theories and moderated theories ("Théories différentiables" (à la Nash) et “théories modérées"). *7.

Pursuing Stacks ''Pursuing Stacks'' (french: À la Poursuite des Champs) is an influential 1983 mathematical manuscript by Alexander Grothendieck. It consists of a 12-page letter to Daniel Quillen followed by about 600 pages of research notes. The topic of the w ...

("À la Poursuite des Champs"). *8.

Two-dimensional geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...

("Digressions de géométrie bidimensionnelle"). *9. Summary of proposed studies ("Bilan d’une activité enseignante"). *10. Epilogue. *Notes Suggested further reading for the interested mathematical reader is provided in the ''References'' section.

Extensions of Galois's theory for groups: Galois groupoids, categories and functors

Galois developed a powerful, fundamental

algebraic theory Informally in mathematical logic, an algebraic theory is a theory that uses axioms stated entirely in terms of equations between terms with free variables. Inequalities and quantifiers are specifically disallowed. Sentential logic is the subset o ...

in mathematics that provides very efficient computations for certain algebraic problems by utilizing the algebraic concept of

groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

, which is now known as the theory of

s; such computations were not possible before, and also in many cases are much more effective than the 'direct' calculations without using groups.Cartier, Pierre (1998), "La Folle Journée, de Grothendieck à Connes et Kontsevich — Évolution des Notions d'Espace et de Symétrie", ''Les Relations entre les Mathématiques et la Physique Théorique — Festschrift for the 40th anniversary of the ''IHÉS'', Institut des Hautes Études Scientifiques'', pp. 11–19 To begin with, Alexander Grothendieck stated in his proposal:'' "Thus, the group of Galois is realized as the

automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...

of a concrete, pro-finite group which respects certain structures that are essential to this group."'' This fundamental, Galois group theory in mathematics has been considerably expanded, at first to

groupoids In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: *''Group'' with a partial func ...

- as proposed in Alexander Grothendieck's ''Esquisse d' un Programme'' (''EdP'')- and now already partially carried out for groupoids; the latter are now further developed beyond groupoids to categories by several groups of mathematicians. Here, we shall focus only on the well-established and fully validated extensions of Galois' theory. Thus, EdP also proposed and anticipated, along previous Alexander Grothendieck's '' IHÉS'' seminars ( SGA1 to SGA4) held in the 1960s, the development of even more powerful extensions of the original Galois's theory for groups by utilizing categories,

functors In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...

and

natural transformations In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural ...

, as well as further expansion of the manifold of ideas presented in Alexander Grothendieck's ''

Descent Theory In mathematics, the idea of descent extends the intuitive idea of 'gluing' in topology. Since the topologists' glue is the use of equivalence relations on topological spaces, the theory starts with some ideas on identification. Descent of vecto ...

''. The notion of

motive Motive(s) or The Motive(s) may refer to: * Motive (law) Film and television * ''Motives'' (film), a 2004 thriller * ''The Motive'' (film), 2017 * ''Motive'' (TV series), a 2013 Canadian TV series * ''The Motive'' (TV series), a 2020 Israeli T ...

has also been pursued actively. This was developed into the

motivic Galois group In algebraic geometry, motives (or sometimes motifs, following French usage) is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham coho ...

Grothendieck topology In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is ca ...

and Grothendieck category . Such developments were recently extended in

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

''via''

representable functor In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets and ...

s and the fundamental groupoid functor.

References

Related works by Alexander Grothendieck

* Alexander Grothendieck. 1971, Revêtements Étales et Groupe Fondamental ( SGA1), chapter VI: '' Catégories fibrées et descente'', Lecture Notes in Math. 224, Springer-Verlag: Berlin. *Alexander Grothendieck. 1957, Sur quelques points d'algèbre homologique,'' Tohoku Mathematics Journal'', 9, 119-221. *Alexander Grothendieck and

Jean Dieudonné Jean Alexandre Eugène Dieudonné (; 1 July 1906 – 29 November 1992) was a French mathematician, notable for research in abstract algebra, algebraic geometry, and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymo ...

.: 1960,''

Éléments de géométrie algébrique The ''Éléments de géométrie algébrique'' ("Elements of algebraic geometry, Algebraic Geometry") by Alexander Grothendieck (assisted by Jean Dieudonné), or ''EGA'' for short, is a rigorous treatise, in French language, French, on algebraic ge ...

''., Publ. '' Inst. des Hautes Études Scientifiques,'' ''( IHÉS)'', 4. *Alexander Grothendieck et al.,1971. Séminaire de Géométrie Algébrique du Bois-Marie, Vol. 1-7, Berlin: Springer-Verlag. *Alexander Grothendieck. 1962. ''Séminaires en Géométrie Algébrique du Bois-Marie'', Vol. 2 - Cohomologie Locale des Faisceaux Cohèrents et Théorèmes de Lefschetz Locaux et Globaux., pp. 287. (''with an additional contributed exposé by Mme. Michele Raynaud''). (Typewritten manuscript available in French; see also a brief summary in English References Cited: **

Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...

. 1964. Cohomologie Galoisienne, Springer-Verlag: Berlin. ** J. L. Verdier. 1965. Algèbre homologiques et Catégories derivées. North Holland Publ. Cie). *Alexander Grothendieck et al. Séminaires en Géometrie Algèbrique- 4, Tome 1, Exposé 1 (or the Appendix to Exposée 1, by `

N. Bourbaki Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook in ...

) for more detail and a large number of results. AG4 is freely available in French; also available is an extensive Abstract in English. *Alexander Grothendieck, 1984
"Esquisse d'un Programme"
(1984 manuscript), finally published in " Geometric Galois Actions", L. Schneps, P. Lochak, eds., ''London Math. Soc. Lecture Notes'' 242,

Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...

, 1997, pp. 5-48
English transl.
ibid., pp. 243-283. . *Alexander Grothendieck, " La longue marche in à travers la théorie de Galois." = "The Long March Towards/Across the Theory of Galois", 1981 manuscript,

University of Montpellier The University of Montpellier (french: Université de Montpellier) is a public university, public research university located in Montpellier, in south-east of France. Established in 1220, the University of Montpellier is one of the oldest univ ...

preprint series 1996, edited by J. Malgoire.

Other related publications

*. * * *.

External links

Fundamental Groupoid Functors
{Dead link, date=August 2019 , bot=InternetArchiveBot , fix-attempted=yes , Planet Physics.
The best rejected proposal ever
Never Ending Books, Lieven le Bruyn
Notes Anabéliennes
A. Grothendieck. Group theory Algebraic geometry Category theory Algebraic topology Mathematics papers 1983 documents