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.