"The focus of our proposal is on arithmetic circuit complexity. Arithmetic circuits are the most common model for computing polynomials, over arbitrary fields. This model was studied by manyresearchers in the past 40 years but still not much is known on many of the basic problems concerning this model.In this research we propose to study some of the most exciting fundamental open problems in theoretical computer science: Proving lower bounds on the size of arithmetic circuits and findingefficient deterministic algorithms for checking identity of arithmetic circuits. Proving a strong lower bound or finding efficient deterministic algorithms to the polynomial identity testing problem are the most important problems in algebraic complexity and solving either of them will be a dramatic breakthrough in theoretical computer science.The two problems that we intend to study are closely related to each other - there are several known results showing that a solution to one of the problems may lead to a solution to the other. Thus, we propose to study strongly related problems that lie in the frontier of algebraic complexity. Any advance will be a significant contributions to the field of theoretical computerscience."