GEMS is a ‘coordinated project’ , funded by the Spanish Ministry of Science, Innovation and Universities, consisting of two subprojects, DUKES (Dualities, K-stability and special holonomy) and HISMO (Higgs bundles, integrable systems and moment-map equations). The unifying theme is the theory of moduli spaces, both in its more classical algebraic-geometric form and in its analytical and gauge-theoretic aspects, as well as in its connections with higher structures and theoretical physics.

DUKES will make progress in fundamental problems in differential and algebraic geometry, combining generalised and higher geometry, geometric analysis, and birational geometry. Key topics include special holonomy, moduli spaces, the study of canonical Kähler metrics in both their algebraic and analytic versions and the mathematical study of dualities in string theory (mirror symmetry and holography). New directions are explored in special holonomy with and without torsion, generalised geometry and higher gauge theory, as well as constructions of moduli spaces via computational algebra, K-stability, uniqueness of Kähler-Yang-Mills equations, and the study of dualities: Calabi-Yau/hyperKähler mirror symmetry, Langlands duality, and the AdS/CFT correspondence, combining moduli-space theory, vertex algebras and exceptional geometry.

HISMO focuses on the theory of Higgs bundles, higher Teichmüller theory, generalizations of the Hitchin system and Langlands duality and mirror symmetry, non-commutative integrable systems, supergeometry, and moment-map equations in gauge theories and canonical parametrised theories. Key topics include the topology of moduli spaces of Higgs bundles for non-compact real Lie groups, multiplicative and cyclic Higgs bundles associated with Vinberg pairs, Langlands duality, moduli spaces of coherent sheaves in supergeometry, non-commutative integrable systems, quasi-Hamiltonian algebras, double Poisson or double Courant-Dorfman algebras, dimensional reduction and toric varieties, and moduli spaces of principal bundles with non -reductive structure group.

DUKES and HISMO are supported by Ministerio de Ciencia, Innovación y Universidades of Spain with grant numbers PID2025-174260NB-C21 and PID2025-174260NB-C22, respectively.