Algebraic Complexity Theory: Lower bounds and Derandomization (1st of 2 parts)
Algebraic Complexity theory is a subfield of complexity theory, studying computational problems of algebraic flavor such as multiplying matrices, computing the determinant, and more generally, computing polynomials over a field, where the complexity is measured by the number of algebraic operations needed to compute it.
Two basic open problems in the area are (1) proving superpolynomial circuit lower bounds for explicit polynomials (which is the algebraic analog to the P vs. NP problem), and (2) designing deterministic efficient algorithms for the Polynomial Identity Testing problem, that asks to determine, given a circuit computing a polynomial, whether it computes the zero polynomial. Studying these questions often leads to beautiful mathematical questions of combinatorial, algebraic and geometric nature.
In this talk, we will discuss several old and new results in the area, explore the mysterious, bidirectional and intricate connections between the two problems above, and give some open problems.