The aim of the talk is to introduce the noncommutative minimal model program (ncMMP) proposed by Halpern-Leistner and inspired by some works of Dubrovin and Kontsevich. For a given projective variety X, the classical minimal model program asks if there is a variety Y that is birational to X but has simpler geometry, such Y is called a minimal model of X. In the noncommutative context, we consider the derived category D(X) of coherent sheaves on X, and we ask if we can decompose it canonically. The biggest factor of the decomposition is then a minimal model for D(X). In order to find such a decomposition Halpern-Leistner proposes to use the quantum cohomology of X.

We will discuss some examples where Halpern-Leistner’s proposal is satisfied: Grassmannians, quadrics, and cubics in dimensions 3 and 4.