Cubic Fourfolds, Gushel-Mukai Fourfolds, and Hyperkahler Manifolds

Cubic fourfolds and Gushel-Mukai fourfolds are bound to K3 surfaces through geometry and Hodge theory; when they contain unexpected surfaces, their middle cohomology contains a Hodge structure matching that of a K3 surface. This association contains rich geometry and involves certain Hyperkähler manifolds associated to the cubic fourfold. Hyperkähler manifolds are higher-dimensional generalizations of K3 surfaces, whose second cohomology group with integer coefficients has a lattice structure that allows us to study them with tools analogous to those used to understand K3 surfaces. The past 30 years have seen significant advances and new avenues develop in the study of the geometry of cubic fourfolds, Gushel-Mukai fourfolds, and high-dimensional Hyperkähler manifolds. The aim of this workshop is to bring together experts from algebraic and arithmetic geometry to exchange mutually beneficial points of view. For example, are there computations over finite fields that could inform or lead to a Honda-Tate theory for K3 surfaces? Are there exotic automorphism groups of high-dimensional Hyperkähler manifolds in positive characteristic? What geometric insights can be descended to number fields for applications to the study of rational points and the Brauer-Manin obstruction? We hope to study these and many other similar questions.