Classifying matrix factorizations
Brad Hovinen
University of Toronto

Abstract:

Let f be a polynomial in n variables with, say, complex coefficients. A matrix factorization of f is a pair of square matrices of the same size, say A and B, such that AB and BA are the product of f and the identity matrix. One application of matrix factorizations is to describe matrices whose determinant is equal to f. If A is such a matrix and B is the classical adjugate of A, then (A,B) is a matrix factorization of f. In this talk I will discuss the problem of classifying matrix factorizations, focusing on the example of the classical discriminant. There is a rich and subtle interplay between the algebra of the problem and the geometric properties of the discriminant. This talk is designed to be accessible to anyone familiar with basic ring theory and linear algebra.