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Computing the cohomology of the tensor product of two vector bundles iscentral in the study of their moduli spaces and in applications torepresentation theory, combinatorics and physics. These computations play afundamental role in the construction of Brill-Noether loci, birational geometryand $S$-duality. Using recent advances in the Minimal Model Program for modulispaces of sheaves on $\mathbb{P}^2$, we compute the cohomology of the tensorproduct of general semistable bundles on $\mathbb{P}^2$. This solves a naturalhigher rank generalization of the polynomial interpolation problem. Moreprecisely, 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 fullycompute the cohomology of $V \otimes W$. In particular, we show that if $W$ isexceptional, then $V \otimes W$ has at most one nonzero cohomology groupdetermined by the slope and the Euler characteristic, generalizing foundationalresults of Dr\'{e}zet, G\"{o}ttsche and Hirschowitz. We characterize theinvariants of effective Brill-Noether divisors on $M(v)$. We also characterizewhen $V\otimes W$ is globally generated. Our computation is canonical given thebirational geometry of the moduli space, suggesting a roadmap for tacklinganalogous problems on other surfaces.

The cohomology of general tensor products of vector bundles on the projective plane