  Computing the cohomology of the tensor product of two vector bundles is
central in the study of their moduli spaces and in applications to
representation theory, combinatorics and physics. These computations play a
fundamental role in the construction of Brill-Noether loci, birational geometry
and $S$-duality. Using recent advances in the Minimal Model Program for moduli
spaces of sheaves on $\mathbb{P}^2$, we compute the cohomology of the tensor
product of general semistable bundles on $\mathbb{P}^2$. This solves a natural
higher rank generalization of the polynomial interpolation problem. More
precisely, let $v$ and $w$ be two Chern characters of stable bundles on
$\mathbb{P}^2$ and assume that $w$ is sufficiently divisible depending on $v$.
Let $V \in M(v)$ and $W \in M(w)$ be two general stable bundles. We fully
compute the cohomology of $V \otimes W$. In particular, we show that if $W$ is
exceptional, then $V \otimes W$ has at most one nonzero cohomology group
determined by the slope and the Euler characteristic, generalizing foundational
results of Dr\'{e}zet, G\"{o}ttsche and Hirschowitz. We characterize the
invariants of effective Brill-Noether divisors on $M(v)$. We also characterize
when $V\otimes W$ is globally generated. Our computation is canonical given the
birational geometry of the moduli space, suggesting a roadmap for tackling
analogous problems on other surfaces.
