Descripción del proyecto
La geometría de hiper-Kähler, fundamental para las matemáticas modernas
Comprender que la gravedad dobla el espacio-tiempo y que el universo es curvo ha impulsado la ciencia y cambiado el mundo. Hoy en día, la curvatura de hiper-Kähler es el campo más fascinante de la geometría debido a los fenómenos excepcionales que abarca. Su simetría espacial única la ha convertido en el eje del campo de las matemáticas conocido como geometría algebraica. Gracias a la precisión que ofrecen los estudios de las soluciones de las ecuaciones algebraicas, se pueden abrir ámbitos importantes para las matemáticas modernas y la ciencia en general. El objetivo del proyecto HyperK, financiado con fondos europeos, es ampliar la geometría de hiper-Kähler y hacerla coincidir con la ya asentada teoría de las superficies K3. Se propone probar las conclusiones básicas relativas a los ciclos, clasificar las estructuras de Hodge y las invariantes cohomológicas, y situar el panorama de hiper-Kähler en el corazón de las matemáticas modernas.
Objetivo
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
Ámbito científico
Programa(s)
Tema(s)
Régimen de financiación
ERC-SyG - Synergy grantInstitución de acogida
53113 Bonn
Alemania