The aim is to develop an emerging field of complex analysis and geometry focused on holomorphic partial differential relations (HPDR). Such a relation of order r is given by a subset of the manifold of r-jets of holomorphic maps between a pair of complex manifolds, and the main question is when does a formal solution lead to an honest analytic solution. This complex analogue of Gromov’s h-principle is highly important but poorly understood. The project will focus on the following problems. (A) Oka theory concerns the existence and approximation of holomorphic maps from Stein manifolds to complex manifolds, corresponding to HPDRs of order zero. The central notion of Oka theory is Oka manifold; this is a complex manifold such that the h-principle holds for maps from any Stein manifold into it. Recently developed techniques give a promise of major new developments on Oka manifolds and their applications to a variety of problems in complex geometry. (B) Open first order HPDRs. Oka-theoretic methods will be applied in problems concerning holomorphic immersions and locally biholomorphic maps. (C) First order HPDRs defined by analytic varieties in the jet bundle. Application of Oka-theoretic methods in holomorphic directed systems, with emphasis on complex contact manifolds and holomorphic Legendrian curves. (D) Applications of Oka theory to minimal surfaces. Development of hyperbolicity theory for minimal surfaces. The Calabi-Yau problem for minimal surfaces in general Riemannian manifolds. Study of superminimal surfaces in self-dual Einstein four-manifolds via the Penrose-Bryant correspondence. These closely interrelated topics embrace major open problems in three fields, with diverse applications.

Recent advances in computability theory have uncovered new pathways to apply computability to prove theorems in other areas of mathematics. Examples include a long-open question in topological dimension theory answered by Kihara and Pauly using computability-theoretic methods and a new proof of the 2-dimensional Kakeya conjecture by Lutz and Lutz. By bringing together the right experts, we will deepen those pathways and find more such applications. In a parallel development, we will focus on implementating algorithms working on continuous data types. This, too, has promising applications in diverse areas of mathematics. For example, tools that can compute Bloch's constant or the solution to the Lebesgue universal covering problem are within our grasp -- and both these numbers are only known up to the first post-decimal digit. The tools to be developed also have applications outside of mathematics, such as for the verification of hybrid systems. Both routes to applications built on a shared theoretical foundation, which in itself will be developed further.