Objectif "We aim to create a new and powerful theory of motivic integration which incorporates Mellin transforms. The absence of motivic Mellin transforms is a major drawback of the existing theories. Classical Mellin transforms are in essence Fourier transforms on the multiplicative group of local fields. We aim to apply this theory to study new motivic Poisson summation formulas, new transfer principles, and applications of these. All of this has so far only been studied in the presence of additive characters, and remains completely open for multiplicative characters. Understanding all this at a motivic level yields a uniform understanding when the local field varies and will require an approach using non-archimedean geometry. We will open up possibilities for applications via new transfer principles and will give access to motivic Poisson formulas of other groups than the additive group. For these applications it is important that Fubini Theorems are present at the level of the motivic integrals, which we aim to develop. We will overcome the major obstacle of the totally different nature of the dual group of the multiplicative group by a proposed sequence of germs of ideas by the author. The importance of our work on motivic Fourier transforms on the additive group is already widely recognized, and this proposal will complement it by exploring the new territory of motivic multiplicative characters. A final topic is the study of the highly non-understood exponential sums modulo powers of primes, in relation with Igusa's foundational work. We will try to discover a deeper understanding of the uniform behavior of these sums when the prime number varies. These sums are linked to geometrical concepts like the log-canonical threshold, and also to Poisson summation, after the work by Igusa. We will aim to prove a highly generalized form of Igusa's conjecture on exponential sums." Champ scientifique natural sciencesmathematicspure mathematicsgeometrynatural sciencesmathematicspure mathematicsarithmeticsprime numbers Programme(s) FP7-IDEAS-ERC - Specific programme: "Ideas" implementing the Seventh Framework Programme of the European Community for research, technological development and demonstration activities (2007 to 2013) Thème(s) ERC-CG-2013-PE1 - ERC Consolidator Grant - Mathematics Appel à propositions ERC-2013-CoG Voir d’autres projets de cet appel Régime de financement ERC-CG - ERC Consolidator Grants Institution d’accueil CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS Contribution de l’UE € 912 000,00 Adresse RUE MICHEL ANGE 3 75794 Paris France Voir sur la carte Région Ile-de-France Ile-de-France Paris Type d’activité Research Organisations Chercheur principal Raf Cluckers (Prof.) Contact administratif Bénédicte Samyn (Mrs.) Liens Contacter l’organisation Opens in new window Site web Opens in new window Coût total Aucune donnée Bénéficiaires (1) Trier par ordre alphabétique Trier par contribution de l’UE Tout développer Tout réduire CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS France Contribution de l’UE € 912 000,00 Adresse RUE MICHEL ANGE 3 75794 Paris Voir sur la carte Région Ile-de-France Ile-de-France Paris Type d’activité Research Organisations Chercheur principal Raf Cluckers (Prof.) Contact administratif Bénédicte Samyn (Mrs.) Liens Contacter l’organisation Opens in new window Site web Opens in new window Coût total Aucune donnée