String theory has helped to formulate two major new insights in the study of singular algebraic varieties. The first -- which also arose from symplectic geometry -- is that families of Kaehler metrics are an important tool in uncovering the structure of singular algebraic varieties. The second, more recent insight -- related to independent work in the representation theory of associative algebras -- is that one's understanding of a singular (affine) algebraic variety is enhanced if one can find a non- commutative ring whose center is the coordinate ring of the variety. We will describe both of these insights, and explain how they are related to string theory.