Moduli Spaces of Abelian Varieties and Curves

Elham Izadi0071795 E. Izadi的研究是为了更好地理解主要极化阿贝尔品种和它们的模空间，曲线的模空间和它们与它们映射的主要极化阿贝尔品种的模空间的关系。 她的项目有两个主要部分。在第一部分中，她涉及霍奇的abelian品种，以那些为它的θ因子。特别是，在证明一个阿贝尔簇的霍奇定理时，其中一个步骤是证明它的θ因子的原始上同调。 为此，她需要在θ因子中找到合适的曲线。 这些也有应用的理论Prym-Tjurin品种。在第二部分，她减少了霍奇猜想的阿贝尔fourfolds证明一个具体的结果，一个特定的家庭的曲线在阿贝尔品种：她还致力于此。这项研究是在该领域的代数几何，其主要对象的研究是代数品种。 这些是经典定义为多项式的同时零点集。曲线和交换簇是特殊的代数簇，经典地出现在阿贝尔，雅可比和黎曼等人的工作中，他们发展了模形式理论。 它们在数学的许多领域都有应用，包括数论;特别是，它们被用在法尔明斯的莫德尔猜想的证明和怀尔斯的半稳定志村谷山韦尔猜想的证明中。 阿贝尔簇为代数几何中的重要猜想（如霍奇猜想）提供了理想的测试基础。 这些都是涉及代数簇的解析结构的深刻见解。 他们最初制定的形式问题的霍奇，然后重新制定和纠正的其他人包括一些最伟大的数学家的这个世纪，如格罗滕迪克。 它们是代数几何领域的核心，代数几何是现代数学中最古老的部分之一，但在过去的20年里，它已经发展到解决了几个世纪以来的问题。 今天，该领域使用的方法不仅来自代数，而且来自分析和拓扑学，相反，它被广泛用于这些领域。此外，它已被证明在物理学、计算机科学、密码学、编码理论和机器人技术等不同领域都很有用。

Elham Izadi0071795The purpose of E. Izadi's research is to better understand principally polarized abelian varieties and their moduli spaces, the moduli spaces of curves and their relation with the moduli spaces of principally polarized abelian varieties to which they map. Her project has two main parts. In the first part she relates the Hodge conjectures for an abelian variety to those for its theta divisor. In particular, one of the steps involved in proving the Hodge conjectures for an abelian variety is to prove them for the primitive cohomology of its theta divisor. For this, she needs to find appropriate curves in the theta divisor. These also have applications to the theory of Prym-Tjurin varieties. In the second part she reduces the Hodge conjecture for abelian fourfolds to proving a specific result about a particular family of curves in the abelian variety: she also works on this.This research is in the field of algebraic geometry, whose main objects of study are algebraic varieties. These are classically defined as the sets of simultaneous zeros of polynomials. Curves and abelian varieties are special algebraic varieties which classically appear in the work of Abel, Jacobi and Riemann among others, who developed the theory of modular forms. They have applications in many areas of mathematics including number theory; in particular, they were used in Faltings' proof of the Mordell Conjecture and in Wiles' proof of the Semistable Shimura Taniyama Weil Conjecture. Abelian varieties make ideal testing grounds for important conjectures in algebraic geometry such as the Hodge Conjectures. These are deep conjectures which concern the analytic structure of algebraic varieties. They were originally formulated in the form of questions by Hodge, then reformulated and corrected by others including some of the greatest mathematicians of this century such as Grothendieck. They are central to the field of algebraic geometry, one of the oldest parts of modern mathematics, but one which has blossomed to the point where it has, in the past 20 years, solved problems that have stood for centuries. Today, the field uses methods not only from algebra, but also from analysis and topology, and conversely it is extensively used in those fields. Moreover, it has proved itself useful in fields as diverse as physics, computer science, cryptography, coding theory and robotics.