FunctorialAlgebraicGeometry 2009-03-09 12:45:30 2009-03-09 12:50:53 Topic and physics functorial geometry functorial algebraic geometry The following is a contributed topic on functorial algebraic geometry: \PMlinkexternal{``Functorial Algebraic Geometry: An Introduction'' -by Alexander Grothendieck}{http://www.math.jussieu.fr/~leila/grothendieckcircle/FuncAlg.pdf} \subsubsection{Vol.1: Affine Algebraic Geometry} (Following the Notes in English typewritten and edited by P. Gaeta, without implying the approval by A. Grothendieck of these notes.) A century ago Algebraic Geometry could be contained in Klein's book (1880; Dover publs. 1963) : ``On RIEMANN's theory of Algebraic Functions and Their Integrals''. Furthermore, ``the distinction between pure and applied mathematics was then to a large extent artificial and unimportant'' (viz. P. Gaeta). For example in Klein's book cited above ``the study of Riemann surfaces was introduced by considering the practical physical problem of laminar flow in a plane or arbitrary surface. He even quotes Maxwell's treatise on page one. The natural continuation of such a `transcedental approach' in our times is the study of {\em complex algebraic manifolds}...'' Following Dieudonn\'e 's and Grothendieck's famous ``\'Elements de G\'eometrie Alg\'ebrique'', and Dieudonn\'e 's ``Algebraic Geometry'' and ``Fondements de la G\'eometrie Alg\'ebrique.'' Adv. in Math. (1969), Alexander Grothendieck presented in 1973 a Buffalo Summer Course entitled: ``Survey on the functorial approach to affine algebraic groups''. This was preceded by a lecture introducing the functorial `language' approach (\PMlinkexternal{Introduction au Langage Fonctoriel}{http://www.math.jussieu.fr/~karoubi/Grothendieck.Alger.pdf}) Grothendieck also organized and presented most of the four famous SGA seminars (SGA-1 to SGA-4), ``S\'eminaires de G\'eometrie Alg\'ebrique'' (Seminars of Algebraic Geometry.) . Other relevant references were: K\"ahler's ``Geometria arithmetica'' (1958), S. MacLane's ``Homology'' (1963), Manin's ``Lectures on Algebraic Geometry'', Mumford's ``Introduction to Algebraic Geometry'', and J.P. Serre's ``Faisceaux alg\'ebrique coh\'erents." (Coherent Algebraic Sheaves). In 1968 was also published by North-Holland the book ``Dix expos\'es sur la cohomologie des sch\'emes'' (Ten expositions on the cohomology of schemes) by J. Giraud and Alexander Grothendieck.