A reciprocal linear space is the image of a linear space under coordinate-wise inversion. This nice algebraic variety appears in many contexts and its structure is governed by the combinatorics of the underlying hyperplane arrangement. A reciprocal linear space is also an example of a hyperbolic variety, meaning that there is a family of linear spaces all of whose intersections with it are real. This special real structure is witnessed by a determinantal representation of its Chow form in the Grassmannian. In this talk, I will introduce reciprocal linear spaces and discuss the relation of their algebraic properties to their combinatorial and real structure

On the space of real symmetric matrices, the determinant is hyperbolic with respect to the cone of positive definite matrices. As a consequence, the determinant of a matrix of linear forms that is positive definite at some point is a hyperbolic polynomial.  A celebrated theorem of Helton and Vinnikov states that any hyperbolic polynomial in three variables has such a determinantal representation. In more variables, this is no longer the case.  During this talk, we will examine hyperbolic polynomials that do and do not have definite determinantal representations. Interlacing polynomials will play an interesting role in this story and form a convex cone in their own right.