Mathematical Conversations

The permanent and determinant are polynomial functions of the entries of a matrix, differing only in the signs of their monomials. Despite their apparent similarity, these polynomials play very different roles in mathematics and computer science: the determinant appears throughout algebra and geometry, while the natural habitat of the permanent is closer to combinatorics. Are these polynomials truly different, or is there a simple way of expressing one in terms of the other? I will survey some of what is known about this problem and explain its role as an algebraic version of the P versus NP question.