Algebraic geometry is concerned with the study of geometric properties of objects that can be defined algebraically. Such objects, called algebraic varieties, are very special and have a rich structure. Consequently, techniques from other fields of mathematics, such as Arithmetic, Topology, and Differential Geometry, are employed in their study. Within algebraic geometry, this project studies the geometry of moduli spaces. A moduli space is the space of all shapes of objects with prescribed numerical invariants. The study of moduli spaces is of central importance to several branches of mathematics and theoretical physics. In particular, over the past three decades there were deep cross-influences between Mathematics and Physics centered around a special class of algebraic varieties, the Calabi-Yau threefolds, and their moduli. One of the main thrusts of this project is the development of tools for the study of moduli spaces of Calabi-Yau varieties.
In this project, the moduli spaces are studied from the perspective of period maps. The period map is an essential tool for understanding the geometry of the moduli space of abelian varieties and K3 surfaces. Beyond these classical cases, the investigation of the period maps is much more difficult due to some non-trivial and highly transcendental conditions (the Griffiths' transversality conditions) satisfied by the Variations of Hodge Structure arising in the geometric setting. Nonetheless, recent progress in the field makes possible the investigation of some non-classical situations, especially surfaces of general type with small invariants, and Calabi-Yau threefolds. In a different direction, several questions about the geometry of the moduli space of K3 surfaces and hyper-Kaehler manifolds are investigated by means of period maps.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
