The principal investigator's research concerns the study of complex manifolds, which are higher-dimensional curved spaces that are defined using the complex numbers. The simplest examples of such spaces are called Riemann surfaces, which are complex manifolds with one complex dimension (and therefore two real dimensions), and include the familiar surfaces of the sphere and of a donut. Higher-dimensional complex manifolds include for example Calabi-Yau manifolds, which are a fundamental tool in string theory. Complex manifolds are ubiquitous objects in mathematics, and have wide-ranging applications in physics and engineering. The proposed research projects will expand our knowledge of the geometry of higher-dimensional complex manifolds using analytic techniques, and in particular partial differential equations (PDEs). These projects lie at the intersection of several mathematical disciplines, such as differential, algebraic and symplectic geometry, complex analysis and PDEs, and techniques from all these fields are necessary to attack them. Progress on these questions will not only shed some light on some basic problems in mathematics, but will also have applications in physics and other sciences.
The principal investigator proposes to use techniques from geometric analysis and nonlinear partial differential equations to investigate problems about the geometry of complex manifolds. The first project is about applications of analysis to the construction of currents on complex manifolds, which are used to study the geometry of (1,1) cohomology classes on compact Kahler manifolds. In the second project the principal investigator will develop new analytic techniques to construct special metrics on non-Kahler complex manifolds, by solving Monge-Ampere equations for (n-1,n-1) forms, building upon earlier work of the principal investigator with Szekelyhidi and Weinkove which culminated in the solution of Gauduchon's conjecture. The third project is about understanding collapsed limits of Ricci-flat Calabi-Yau manifolds. This is closely related to the theory of mirror symmetry, which was inspired by physical considerations. The fourth project is centered on Donaldson's program to extend Yau's solution of the Calabi Conjecture in Kahler geometry to symplectic four manifolds, and to its applications to symplectic topology.