This proposal is concerned with the study of the geometry of moduli spaces and the development of tools for studying them. In particular, the PI plans to apply and adapt some newly developed techniques in abstract Hodge theory to the geometric context of moduli spaces. The PI will focus on the investigation of special classes of varieties, such as Calabi-Yau threefolds and higher dimensional Hyperkahler manifolds. In a different, but related direction, the PI aims to construct geometric compactifications for certain classes of surfaces via the KSBA approach inspired by the minimal model program. The PI is actively involved with the training of undergraduate and graduate students, and in recent years he has organized several activities with a strong educational component. He will expand these activities. In particular, he will run a summer research activity for undergraduates. Also, as part of the thematic program on Calabi-Yau varieties to be held at the Fields institute (Fall 2013), the PI will organize an introductory school for graduate students and other training activities. Additional planed activities include developing new courses for undergraduates and students in the teacher education program, organizing a workshop on Hodge theory and moduli spaces, and writing a monograph.
The general area of the proposal is algebraic geometry with connections to complex geometry and arithmetic geometry. Algebraic Geometry studies the geometric properties of objects defined by algebraic (or equivalently polynomial) equations. Within Algebraic Geometry, the PI is interested in the study of moduli spaces. A moduli space is a geometric object that parameterizes the shapes of objects within a given topological class. By studying a moduli space, one obtains important information about geometric objects of a given kind, in particular about the existence or non-existence of objects with prescribed special properties. This study has numerous applications to algebraic geometry and other related fields including mathematical physics.