Project description
Hyperkähler geometry at the centre of modern mathematics
The understanding that gravity bends space and time and that the universe is curved has boosted science and changed the world. Today, the hyperkähler curvature is the most fascinating field in geometry due to the rare phenomena it encompasses. Its unique spaces symmetry has made it the focal point of the area of mathematics called algebraic geometry. With the precision offered by studies of solutions of algebraic equations it can open important areas for modern mathematics and science in general. The EU-funded HyperK project aims to expand hyperkähler geometry and match it with the already deep-rooted theory of K3 surfaces. The aim is to prove basic conclusions concerning cycles, classify Hodge structures and cohomological invariants and place the hyperkähler landscape at the centre of modern mathematics.
Objective
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
Fields of science
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Programme(s)
Topic(s)
Funding Scheme
ERC-SyG - Synergy grantHost institution
53113 Bonn
Germany