Abelian Varieties and Curves

Elham Izadi将致力于解决由关于阿贝尔变种和与之相关的曲线的Hodge猜想而产生的问题。她将使用对应关系来制作阿贝尔品种的具体曲线族。对于具有虚二次场作用的阿贝尔变种，她利用曲线族、变形和特殊化以及线性系统边界上的Hodge丛，发展了一种具体的方法。她将为学生们组织一个研究小组，研究这个项目产生的基本问题。过去，她利用这样的研究小组，成功地将女性带到了格鲁吉亚大学。这项研究是在代数几何领域，其主要研究对象是代数变种。它们被经典地定义为多项式的同时零点的集合。阿贝尔簇和曲线是一类特殊的代数簇，在数论、编码论等数学领域有着广泛的应用。经典地，它们出现在Abel、Jacobi和Riemann等人的工作中，他们发展了模形式理论。它们为代数几何中的重要猜想，如霍奇猜想，提供了理想的测试依据。这些都是深刻的猜想，对代数簇的研究具有深远的影响。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。

Elham Izadi will work on questions resulting from the Hodge conjectures for abelian varieties and curves related to them. She will use correspondences to produce concrete families of curves in abelian varieties. For abelian varieties with actions of imaginary quadratic fields, she develops a concrete approach using families of curves, deformations and specializations, and Hodge bundles at the boundaries of linear systems. She will run a research group for students on elementary questions resulting from this project. In the past, using such research groups, she has been successful in bringing women to the University of Georgia.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. Abelian varieties and curves are special algebraic varieties with applications in many areas of mathematics including number theory and coding theory. Classically they appear in the work of Abel, Jacobi and Riemann among others, who developed the theory of modular forms. They make ideal testing grounds for important conjectures in algebraic geometry such as the Hodge conjectures. These are deep conjectures with far reaching consequences for the study 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.