Objetivo Algebraic geometry is the study of varieties -- the zero sets of polynomial equations in several variables. The subject has a central role in mathematics with connections to number theory, representation theory, and topology. Moduli questions in algebraic geometry concern the behavior of varieties as the coefficients of the defining polynomials vary. At the end of the 20th century several fundamental links between the algebraic geometry of moduli spaces and path integrals in quantum field theory were made. I propose to study the moduli spaces of curves, sheaves, and K3 surfaces. While these moduli problems have independent roots, striking new relationships between them have been found in the past decade. I will exploit the new perspectives to attack central questions concerning the algebra of tautological classes on the moduli spaces of curves, the structure of Gromov-Witten and Donaldson-Thomas invariants of 3-folds including correspondences and Virasoro constraints, the modular properties of the invariants of K3 surfaces, and the Noether-Lefschetz loci of the moduli of K3 surfaces. The proposed approach to these questions uses a mix of new geometries and new techniques. The new geometries include the moduli spaces of stable quotients and stable pairs introduced in the past few years. The new techniques involve a combination of virtual localization, degeneration, and descendent methods together with new ideas from log geometry. The directions discussed here are fundamental to the understanding of moduli spaces in mathematics and their interplay with topology,string theory, and classical algebraic geometry. Ámbito científico natural sciencesmathematicspure mathematicstopologynatural sciencesmathematicspure mathematicsarithmeticsnatural sciencesphysical sciencestheoretical physicsstring theorynatural sciencesmathematicspure mathematicsgeometrynatural sciencesmathematicspure mathematicsalgebraalgebraic geometry Programa(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) Tema(s) ERC-AG-PE1 - ERC Advanced Grant - Mathematical foundations Convocatoria de propuestas ERC-2012-ADG_20120216 Consulte otros proyectos de esta convocatoria Régimen de financiación ERC-AG - ERC Advanced Grant Institución de acogida EIDGENOESSISCHE TECHNISCHE HOCHSCHULE ZUERICH Aportación de la UE € 2 167 997,00 Dirección Raemistrasse 101 8092 Zuerich Suiza Ver en el mapa Región Schweiz/Suisse/Svizzera Zürich Zürich Tipo de actividad Higher or Secondary Education Establishments Investigador principal Rahul Vijay Pandharipande (Prof.) Contacto administrativo Rahul Vijay Pandharipande (Prof.) Enlaces Contactar con la organización Opens in new window Sitio web Opens in new window Coste total Sin datos Beneficiarios (1) Ordenar alfabéticamente Ordenar por aportación de la UE Ampliar todo Contraer todo EIDGENOESSISCHE TECHNISCHE HOCHSCHULE ZUERICH Suiza Aportación de la UE € 2 167 997,00 Dirección Raemistrasse 101 8092 Zuerich Ver en el mapa Región Schweiz/Suisse/Svizzera Zürich Zürich Tipo de actividad Higher or Secondary Education Establishments Investigador principal Rahul Vijay Pandharipande (Prof.) Contacto administrativo Rahul Vijay Pandharipande (Prof.) Enlaces Contactar con la organización Opens in new window Sitio web Opens in new window Coste total Sin datos