Moduli Spaces - Geometry and Arithmetic

PI利用代数和分析性质的技巧研究了关于模空间的几个问题。Laza研究计划的一个主要方向是构造模空间的几何紧化，并开发研究它们的工具。特别是，他对模空间边界上周期映射的行为的研究很感兴趣。另一个方向是研究某些特殊簇(K3、Calabi-Yau和Hyperkahler)的模空间。在这种情况下，几何、代数和算术之间的相互作用是最强的，因此人们希望对这些变体的模数有很好的理解。这里提出的一些问题与弦理论有直接的关系。这是代数几何中的一项建议，与复几何和算术几何的相关领域有关。代数几何研究由代数方程定义的物体的几何性质。在代数几何中，PI对模空间的研究感兴趣，模空间将给定拓扑类中对象的形状参数化。这项研究与数学的其他几个分支和数学物理有联系和应用。PI参与了一系列活动，扩大了拟议研究的影响：他为研究生和本科生提供建议。他正在(共同)组织几个会议，以及一个关于Calabi-Yau品种几何学的专题学期。

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.