Random matrix is the study of eigenvalues of large matrices, where the entries are chosen randomly according to some probability distribution. The first result in the subject is the semicircular law for large Hermitian random matrices, where the eigenvalues have a semicircular distribution on the interval $[-2,2]$. The second result is the circular law for non-Hermitian random matrices, where the eigenvalues are uniformly distributed on the unit disk in the plane. There is a simple but intriguing connection between the two laws: twice the real part of the eigenvalues in the circular law has the same distribution as the semicircular law. Several recent results have shown that this result is part of a general phenomenon, in which one can construct pairs of random matrix models where the limiting eigenvalue distributions are related by a map of the plane to itself. In my talk, I will explain a new idea: that the relationship between the two models can be accomplished at the finite-$N$ level by applying a heat operator to the characteristic polynomial of the first model. That is, if $p_0$ and $p_1$ are the (random) characteristic polynomials of the two models, applying a heat operator to $p_0$ gives a new polynomial whose zeros conjecturally resemble the zeros of $p_1$. As an example, we conjecture that if you apply the heat operator for time $1/N$ to the characteristic polynomial of a random Hermitian matrix (semicircular law), the zeros of the new polynomial will be approximately uniform on the unit disk. This is joint work with Ching-Wei Ho.