Geometry of Moduli Spaces

在几何中，人们经常试图对给定类型的物体的所有可能形状进行分类：例如，所有三角形都是根据其边的长度进行分类的——这必须是正的，并且满足三角形不等式。这类参数空间本身往往具有丰富的几何结构，称为模空间。在复杂几何中，人们研究具有复杂坐标的物体，也就是说，从近处看，它们看起来像复数。在复杂代数几何中，进一步限制研究由复数多项式方程定义的形状-基本分类问题是研究给定类型的所有这些形状，找到这些形状的参数，以及这些参数必须满足的条件。模空间在代数几何中无处不在，近年来，它通过变形和退化为理解单个几何对象提供了一些最强大的工具。特别是，代数曲线（黎曼曲面）渗透到代数几何的许多构造中；从数论到可积系统再到物理学，阿贝尔变体自然地出现在各种环境中。本研究旨在获得模空间几何及其相互关系的新信息。研究者将寻求各种模问题的新的深层关系和性质，旨在为复杂几何和代数几何、Teichmuller理论、弦摄动理论和可积系统的研究人员提供进一步的工具。研究者将进一步了解复数上模空间的几何，特别是集中在阿贝尔变体和曲线的模空间上。研究者将建立在他与Hulek， Tommasi和Zakharov一起开发的技术和结果的基础上，定义和研究一个扩展重音环，以获得阿贝尔变体模空间的适当紧化，试图确定它是否可能是Gorenstein，以及相交数是否可能满足一个有趣的递推关系。研究者将运用他与Krichever一起开发的实解析技术，并受到可积系统的启发，研究平面曲线顶点数量的边界问题，以及曲线模空间的完全子变。研究者将利用这项工作从几何上描述Prym图谱无法注入的位点。研究者将与Salvati Manni一起研究曲线模空间的有效锥的斜率。

In geometry, one often tries to classify all possible shapes of objects of a given type: for example, all triangles are classified by the lengths of their sides - which must then be positive, and satisfy the triangle inequalities. Such parameter spaces often themselves have rich geometric structures, and are called moduli spaces. In complex geometry, one studies objects that have complex coordinates - that is, from close up look like complex numbers. In complex algebraic geometry, one further restricts to studying shapes defined by polynomial equations in complex numbers - and the basic classification problem is to study all such shapes of a given type, find the parameters for such shapes, and what conditions these parameters must satisfy. Moduli spaces are ubiquitous in algebraic geometry, and in recent times have provided some of the most powerful tools for understanding individual geometric objects, by deformation and degeneration. In particular, algebraic curves (Riemann surfaces) permeate many constructions in algebraic geometry; abelian varieties appear naturally in varied contexts ranging from number theory to integrable systems to physics. The proposed research aims to obtain new information about the geometry of moduli spaces and relations among them. The investigator will seek new deep relations and properties of various moduli problems with the aim of providing further tools that could be used by researchers in complex and algebraic geometry, Teichmuller theory, string perturbation theory, and integrable systems.The investigator will work to further understand the geometry of moduli spaces over complex numbers, especially focusing on the moduli spaces of abelian varieties, and of curves. The investigator will build on the techniques and results he developed with Hulek, Tommasi, and Zakharov to define and study an extended tautological ring for suitable compactifications of the moduli space of abelian varieties, trying to determine whether it may be Gorenstein, and whether the intersection numbers may satisfy an interesting recursion relation. The investigator will apply the real-analytic techniques he developed with Krichever, and inspired by integrable systems, to study the classical problem of bounding the number of cusps of plane curves, and to study complete subvarieties of the moduli space of curves. The investigator will use this work to characterize geometrically the locus where the Prym map fails to be injective. With Salvati Manni, the investigator will study the slope of the effective cone of the moduli space of curves.