Fano manifolds are basic building blocks in algebraic geometry. There is exactly 1 one-dimensional Fano manifold: the line. There are 10 deformation families of two-dimensional Fano manifolds: the del Pezzo surfaces, known since the 19th century. There are 105 deformation families of three-dimensional Fano manifolds: these were classified by Mori–Mukai in the 1990s, as a spectacular application of Mori theory. Very little is known about Fano classification in dimension four or more.

I will describe work by Kalashnikov and joint work with Kasprzyk and Prince that constructs several hundred new four-dimensional Fano manifolds, as subvarieties of quiver flag manifolds and toric varieties. I will explain how this fits in to a program — joint work with Corti, Galkin, Golsyhev, Kasprzyk, and others — to find and classify Fano manifolds using mirror symmetry.