Etudiant : HABBOUNE MOUAD

Filière : Master Mathématiques Pures (MMP)

Encadrant : Pr. MONIRH KARIM

Annèe : 2024

Résumé : The ultimate goal of this manuscript is to explore how algebraic geometry offers a bridge connecting different areas in mathematics. We are primarily interested in how theories are converted from an algebraic aspect to a purely geometric one, and vice versa. Historically, classical algebraic varieties, defined to be the locus of families of polynomial equations, served to give an algebro-geometric approach to many differential geometric facts, including many geo-analytic properties of standard configurations; however, some categorical equivalences made these algebraic approaches more suitable in dealing with purely algebraic and arithmetic problems; especially when schemes in algebraic geometry replaced the role of classical algebraic varieties. The essence of this transition within algebraic geometry was elaborated by Grothendieck, Dieudonn´e, Serre..., who completely renewed the concepts and methods of algebraic geometry, simplifying old results and opening new ways leading to the solution of old problems, such as Grothendieck-Riemann-Roch theorem, Grothendieck duality, ... After this period, a great amount of feedbacks was observed in the opposite sense from algebraic geometry to differential one. In particular, Mallios’s abstract version of differential manifolds, developed in his books [28],[29] can be considered as a highlight elaborated example of these feedbacks. The present work is written in a self-contained manner and is designed to show how different topics from algebra, representation theory, involutions, sheaves, algebraic and differential geometry can be seen within a complementary interplay. For this purpose, we introduce a variety of necessary backgrounds going from Hopf, Lie and central simple algebras, to differential geometry and its abstract form elaborated by Mallios. In studying parts of this interaction, especially between algebra, geometry, and sheaf theory, we consider in this work two main applications. The first consists in giving a full classification of algebras with involutions of the first kind via Severi-Brauer varieties and group schemes. Indeed, Theorem 2.7.6 in Chapter 2 gives such a classification for a fixed central simple algebra, whereas Subsection 2.8.5 in the same chapter applies some cohomological interpretations of adequate group schemes to give a full cohomological description of all classes of pairs .A; /, where A describes simple algebras and  involutions of orthogonal type (resp. symplectic type) on A. The second application consists in showing how an equivalent sheaf version of the notion of vector bundle in algebraic geometry due to Serre-Swan (see Theorem 4.3.1 and Theorem 4.4.2) is applied in the context of smooth manifolds when these differential varieties are defined by means of adequate ringed spaces (see Theorem 5.8.2.) This interpretation of vector bundles in terms of associated sheaves of modules plays a major role in Mallios’s abstract differential geometry. Indeed, it is worth pointing out that, in his abstract setting, Mallios adapted mainly this sheaf version to define vector bundles in his context of algebraized spaces. As done by Grothendieck and his collaborators (see ´El´ements de G´eom´etrie Alg´ebrique [7]) when developing scheme theory in algebraic geometry, Mallios used heavy cohomological machinery to give many classifications within his abstract context.