Moduli Spaces - Geometry and Arithmetic

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The PI investigates several questions regarding moduli spaces by using techniques of algebraic and analytic nature. One main direction of Laza's research program is the construction of geometric compactifications for moduli spaces and the development of tools for studying them. In particular, he is interested in the study of the behavior of the period map at the boundary of the moduli spaces. Another direction is the study of moduli spaces of certain special varieties (K3, Calabi-Yau, and Hyperkahler). In this case, the interplay between geometry, algebra, and arithmetic is the strongest, and consequently one expects a good understanding of the moduli of these varieties. Some of the questions that arise here have direct relevance to string theory. This is a proposal in algebraic geometry, with connections to the related fields of Complex Geometry and Arithmetic Geometry. Algebraic Geometry is concerned with the study of geometric properties of objects defined by algebraic equations. Within Algebraic Geometry, the PI is interested in the study of moduli spaces, which parameterizes the shapes of objects within a given topological class. This study has connections and applications to several other branches of mathematics and to mathematical physics. The PI is involved in a number of activities that broaden the impact of the proposed research: He is advising graduate and undergraduate students. He is (co)organizing several conferences, as well as a thematic semester on the geometry of Calabi-Yau varieties.