Descrizione del progetto
La geometria Hyperkähler al centro della matematica moderna
L’aver compreso che la gravità piega lo spazio e il tempo e che l’universo è curvo ha potenziato la scienza e cambiato il mondo. Oggi, la curvatura di Hyperkäler è il campo più affascinante della geometria grazie ai rari fenomeni che racchiude. La sua singolare simmetria degli spazi l’ha resa il punto focale dell’area della matematica chiamata geometria algebrica. Con la precisione offerta dagli studi sulle soluzioni di equazioni algebriche, può aprire aree importanti per la matematica moderna e la scienza in generale. Il progetto HyperK, finanziato dall’UE, mira a espandere la geometria Hyperkähler e ad abbinarla alla teoria già radicata delle superfici K3. L’obiettivo è quello di dimostrare le conclusioni di base relative ai cicli, classificare le strutture di Hodge e gli invarianti coomologici e porre il quadro Hyperkähler al centro della matematica moderna.
Obiettivo
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
Campo scientifico
Programma(i)
Argomento(i)
Meccanismo di finanziamento
ERC-SyG - Synergy grantIstituzione ospitante
53113 Bonn
Germania