Didactique des mathématiques : les fondamentaux Année 2014-2015 Dr. Ruben Rodriguez Herrera Agrégé en Mathématiques ruben.rodriguez@unicaen.fr IREM, ESPE, Ifé, CEMU Université de Caen Basse-Normandie France Les modules proposés cette année sont les suivants : - 1 Les univers et les psychomorphismes - 2 Les transpositions didactiques et les variables didactiques - 3 Les situations-problèmes, les obstacles : épistémologiques, didactiques, ontogénétiques et psychomorphiques - 4 Les situations a-didactiques - 5 Le contrat didactique : transmissif, comportementaliste, socio-constructiviste - 6 La dialectique outil-objet - 7 La progressivité des apprentissages
Mot(s) clés libre(s) : Didactique des mathématiques, irem

The counting function associated to the moduli space of stable pairs on a 3-fold X is conjectured to give the Laurent expansion of a rational function. For toric X , this conjecture can be proven by a careful grouping of the box con gurations appearing in the stable pairs equivariant descendent vertex. I will describe this approach and then say a little about how it might also be used to study the Donaldson{Thomas vertex. This talk presents joint work with Rahul Pandharipande.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, courbes, summer school, Gromov-Witten, isntitut fourier

Voyage en Mathématique - Michel Guillemot (IREM de Toulouse) - Ahmès, un scribe égyptien www.voyage-mathematique.com L'exposition « Voyage en Mathématique » permet de voyager avec des mathématiciennes et des mathématiciens de renom à travers l'histoire des mathématiques. Elle a été réalisée par l'association Fermat Science en étroite collaboration avec son conseil scientifique en lien avec des spécialistes en mathématiques notamment des IPR et la mission culturelle de l'Académie de Toulouse. Elle est composée de 13 panneaux. Chaque panneau est lié à une manipulation interactive et à une vidéo d'un mathématicien.
Mot(s) clés libre(s) : voyage, mathématiques, Egypte, ahmes, scribe, fermat

Voyage en Mathématique - Ahmed Djebbar - Professeur émérite d'histoire des mathématiques à l'université des sciences et technologies de Lille. www.voyage-mathematique.com
Mot(s) clés libre(s) : mathématiques, civilisation arabe, Fermat

In this talk we address some issues concerning the wave propagation in the 4D+1 anti de Sitter space time : the role of the conformal boundary, the representation of the fields in term of Kaluza Klein tower, the existence of new dynamics associated with a family of novel boundary conditions, the linear stability of a De Sitter brane.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

Mirror symmetry is a phenomenon which inspired fundamental progress in a wide range of disciplines in mathematics and physics in the last twenty years; we will review here a number of results going from the enumerative geometry of curves to homological algebra. These advances justify the introduction of new techniques, which are interesting in their own right. Among them, Gromov{Witten theory and its variants allow us to provide are ned statement of mirror symmetry. Of course this leads to further open questions (despite much e ort and progress, Gromov{Witten theory remains unknown in high genus for the quintic threefold). In this course, we will illustrate the natural problem of moving beyond the local mirror symmetry statement and completing a framework of global mirror symmetry which is gradually taking shape. We will show how the missing piece in this picture comes unexpectedly from a classical subject in algebraic geometry: the theory of curves with level structures.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, courbes, institut fourier, summer school, Gromov-Witten

Mirror symmetry is a phenomenon which inspired fundamental progress in a wide range of disciplines in mathe- matics and physics in the last twenty years; we will review here a number of results going from the enumerative geometry of curves to homological algebra. These advances justify the introduction of new techniques, which are interesting in their own right. Among them, Gromov{Witten theory and its variants allow us to provide a re ned statement of mirror symmetry. Of course this leads to further open questions (despite much e ort and progress, Gromov{Witten theory remains unknown in high genus for the quintic threefold). In this course, we will illustrate the natural problem of moving beyond the local mirror symmetry statement and completing a framework of global mirror symmetry which is gradually taking shape. We will show how the missing piece in this picture comes unexpectedly from a classical subject in algebraic geometry: the theory of curves with level structures.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, courbes, institut fourier, summer school, Gromov-Witten

Mirror symmetry is a phenomenon which inspired fundamental progress in a wide range of disciplines in mathe- matics and physics in the last twenty years; we will review here a number of results going from the enumerative geometry of curves to homological algebra. These advances justify the introduction of new techniques, which are interesting in their own right. Among them, Gromov{Witten theory and its variants allow us to provide a re ned statement of mirror symmetry. Of course this leads to further open questions (despite much e ort and progress, Gromov{Witten theory remains unknown in high genus for the quintic threefold). In this course, we will illustrate the natural problem of moving beyond the local mirror symmetry statement and completing a framework of global mirror symmetry which is gradually taking shape. We will show how the missing piece in this picture comes unexpectedly from a classical subject in algebraic geometry: the theory of curves with level structures.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, courbes, institut fourier, summer school, Gromov-Witten

indisponible
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

The fundamental problem in the theory of Diophantine approximation is to understand how well points in the Euclidean space can be approximated by rational vectors with given bounds on denominators. It turns out that Diophantine properties of points can be encoded using flows on homogeneous spaces, and in this course we explain how to use techniques from the theory of dynamical systems to address some of questions in Diophantine approximation. In particular, we give a dynamical proof of Khinchin’s theorem and discuss Sprindzuk’s question regarding Diophantine approximation with dependent quantities, which was solved using non-divergence properties of unipotent flows. In conclusion we explore the problem of Diophantine approximation on more general algebraic varieties.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, dynamics, institut fourier, summer school, number theory