Compact moduli of marked noncommutative del Pezzo surfaces

It is known that a smooth projective variety over a field can be reconstructed from the abelian category of coherent sheaves on it. On the other hand, not all deformations of the category (noncommutative deformations) come from deformations of the variety. In this sense, usual deformations can be regarded as special NC deformations. In this talk, I would like to introduce certain construction of compactified moduli spaces of NC del Pezzo surfaces. If time permits, I would also explain its relationship to the known constructions from the traditional noncommutative algebraic geometry and how (moduli of) elliptic curves with additional structures play role in our story. This talk is based on joint works with Kazushi Ueda (Tokyo) and Tarig Abdelgadir (ICTP).