FINO - Programmi di Rilevante Interesse Nazionale - Bando PRIN 2022 - Finanziamento dell’Unione Europea – NextGenerationEU – missione 4, componente 2, investimento 1.1.

This project aims to combine mathematicians whose research interests are in various aspects of geometry on real and complex manifolds and spaces. We aim to achieve results and give significant contributions in these areas of mathematics, by gathering together expertises from different researchers and promoting the interaction of various methodologies to complement and reinforce each other. The emphasis is on the systematic study of geometric properties of real and complex spaces and functions there defined, with a particular regard to holomorphic dynamics and differential geometric aspects. Geometry of manifolds concerns their analytic, algebraic and metric properties and, on the other side, the study of the topological, differential and holomorphic objects - such as maps, fiber bundles and operators - which can be defined on them. New objects appear in a natural way: self-maps, differential operators, divisors, Jacobian varieties, moduli spaces, etc. The dynamical and analytic properties of those objects reflect then on the geometrical properties of the manifolds. Starting from these considerations, the following general themes identify the scope of this project. 1. Structure of manifolds: Riemannian, Kähler, symplectic, Hermitian, CR manifolds, real and complex analytic spaces, quaternionic analogues, almost complex analogues, cohomologies of complex and almost complex manifolds, geometry of domains in the different settings, special geometries and holonomy, Special Riemannian metrics, Einstein metrics, geometric quantization. 2. Maps on manifolds: geometric function theory, discrete and continuous holomorphic dynamics of maps and automorphisms, holomorphic evolution equations, boundary regularity of biholomorphic mappings, immersions and submanifolds, pluripotential theory on complex and Hermitian manifolds, Lie groups and homogeneous spaces, representation theory.