Moduli Spaces - Geometry and Arithmetic

PI调查了几个问题，关于模空间使用技术的代数和分析性质。拉扎的研究计划的一个主要方向是建设几何紧化的模空间和发展的工具，研究他们。特别是，他感兴趣的研究行为的周期地图的边界上的模空间。另一个方向是研究某些特殊变种（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，因此人们期望很好地理解这些变体的模。这里出现的一些问题与弦理论直接相关。 这是代数几何的一个提议，与复几何和算术几何的相关领域有联系。代数几何学研究的是由代数方程定义的物体的几何性质。在代数几何学中，PI对模空间的研究感兴趣，它参数化给定拓扑类中对象的形状。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参与了一些扩大拟议研究影响的活动：他为研究生和本科生提供建议。他是（共同）组织几次会议，以及一个专题学期的几何卡拉比-丘品种。

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.