Description du projet
La géométrie hyperkähler au centre des mathématiques modernes
Le fait de comprendre que la gravité plie l’espace et le temps et que l’univers est courbe a stimulé la science et changé le monde. Aujourd’hui, la courbure hyperkähler est le domaine le plus fascinant de la géométrie en raison des phénomènes rares qu’elle englobe. Sa symétrie spatiale unique en a fait le point focal du domaine des mathématiques appelé géométrie algébrique. Grâce à la précision offerte par l’étude des solutions des équations algébriques, elle peut ouvrir des champs importants pour les mathématiques modernes et la science en général. Le projet HyperK, financé par l’UE, a pour objectif d’étendre la géométrie hyperkähler et de la faire correspondre à la théorie déjà bien ancrée des surfaces K3. L’objectif est de prouver les conclusions fondamentales concernant les cycles, de classer les structures de Hodge et les invariants cohomologiques et de placer le paysage hyperkähler au centre des mathématiques modernes.
Objectif
The space around us is curved. Ever since Einstein’s discovery that gravity bends space and time, mathematicians and physicists have been intrigued by the geometry of curvature. Among all geometries, the hyperkähler world exhibits some of the most fascinating phenomena. The special form of their curvature makes these spaces beautifully (super-)symmetric and the interplay of algebraic and transcendental aspects secures them a special place in modern mathematics. Algebraic geometry, the study of solutions of algebraic equations, is the area of mathematics that can unlock the secrets in this realm of geometry and that can describe its central features with great precision. HyperK combines background and expertise in different branches of mathematics to gain a deep understanding of hyperkähler geometry. A number of central conjectures that have shaped algebraic geometry as a branch of modern mathematics since Grothendieck’s fundamental work shall be tested for this particularly rich geometry.
The expertise covered by the four PIs ranges from category theory over the theory of algebraic cycles to cohomology of varieties. Any profound advance in hyperkähler geometry requires a combination of all three approaches. The concerted effort of the PIs, their collaborators, and their students will lead to major progress in this area. The goal of HyperK is to advance hyperkähler geometry to a level that matches the well established theory of K3 surfaces, the two-dimensional case of hyperkähler geometry.
We aim at proving fundamental results concerning cycles, at classifying Hodge structures and cohomological invariants, and at unifying geometry and derived categories. Specific topics in- clude the splitting conjecture, the Hodge conjecture in small degrees, moduli spaces in derived categories, geometric K3 categories, and special subvarieties.
The ultimate goal of HyperK is to draw a clear and distinctive picture of the hyperkähler landscape as a central part of mathematic
Champ scientifique
Programme(s)
Thème(s)
Régime de financement
ERC-SyG - Synergy grantInstitution d’accueil
53113 Bonn
Allemagne