Moduli Spaces of Abelian Varieties and Curves

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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.