Bessis–Moussa–Villani conjecture

Let $A$ and $B$ be Hermitian $n\times n$ matrices, with $B$ positive semidefinite. Let $\operatorname{Tr}$ denote the trace. Bessis–Moussa–Villani conjecture. The function

$$\lambda\mapsto\operatorname{Tr}\exp(A-\lambda B)$$

is the Laplace transform of a positive measure on $[0,\infty)$. The conjecture was subsequently solved, so the assertion is now a theorem.