Geometry of Moduli Spaces

在几何学中，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它们的边长来分类的-这必须是正的，并且满足三角形不等式。这样的参数空间通常本身具有丰富的几何结构，并且被称为模空间。在复杂几何中，人们研究具有复杂坐标的对象-也就是说，从近距离看起来像复数。在复代数几何中，人们进一步限制于研究由复数多项式方程定义的形状-基本的分类问题是研究给定类型的所有此类形状，找到此类形状的参数，以及这些参数必须满足的条件。 模空间在代数几何中是普遍存在的，并且在最近的时代，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）渗透在代数几何的许多结构中;阿贝尔簇自然地出现在从数论到可积系统到物理学的各种背景下。拟议的研究旨在获得新的信息的几何模空间和它们之间的关系。研究者将寻求各种模问题的新的深层关系和性质，目的是为复几何和代数几何、Teichmuller理论、弦微扰理论和可积系统的研究人员提供进一步的工具。研究者将致力于进一步理解复数上模空间的几何，特别是专注于阿贝尔簇和曲线的模空间。调查人员将建立在技术和结果，他开发的Hulek，Tommasi，和Zakharov定义和研究一个扩展重言式环适当的紧化模空间的交换品种，试图确定它是否可能Gorenstein，以及是否交叉数可能满足一个有趣的递归关系。调查员将适用于真正的分析技术，他开发的Krichever，并受到启发的可积系统，研究经典问题的边界的数量尖点的平面曲线，并研究完整的子品种的模空间的曲线。研究人员将使用这项工作的几何特征的轨迹，其中的Prym地图未能单射。与Salvati Manni，研究人员将研究曲线的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.