Moduli Spaces of Abelian Varieties and Curves

Elham Izadi0071795 E.Izadi的研究目的是更好地理解主要极化的阿贝尔簇及其模空间、曲线的模空间及其与它们所映射的主要极化阿贝尔簇的模空间的关系。她的项目有两个主要部分。在第一部分中，她将霍奇关于阿贝尔变种的猜想与关于其theta除数的猜想联系起来。特别地，证明关于交换簇的Hodge猜想的步骤之一是证明它们的theta除数的本原上同调。为此，她需要在theta除数中找到合适的曲线。这些也适用于Prym-Tjuin变种理论。在第二部分，她将关于交换四折线的Hodge猜想简化为证明关于交换簇中的特定曲线族的一个具体结果：她也致力于此。这项研究是在代数几何领域，其主要研究对象是代数簇。它们被经典地定义为多项式的同时零点的集合。曲线和阿贝尔簇是一种特殊的代数簇，它们经典地出现在Abel、Jacobi和Riemann等人的工作中，他们发展了模形式理论。它们在包括数论在内的许多数学领域都有应用；特别是在Faltings对Mordell猜想的证明和Wiles对半稳态Shimura Taniyama Weil猜想的证明中使用。阿贝尔簇是代数几何中重要猜想的理想试验场，例如霍奇猜想。这些都是与代数簇的解析结构有关的深层猜想。它们最初是由霍奇以问题的形式提出的，然后被其他人重新表述和更正，其中包括本世纪一些最伟大的数学家，如格罗森迪克。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20年里，它已经发展到了解决几个世纪以来的问题的地步。今天，这个领域不仅使用代数的方法，而且还使用分析和拓扑学的方法，相反，它在这些领域被广泛使用。此外，它在物理、计算机科学、密码学、编码理论和机器人学等多个领域都证明了自己的有用。

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.