Weighted complete intersections and mirror symmetry for Fano varieties

In the talk we will discuss several results related to construction of new examples of prime Fano varieties and mirror symmetry for Fano varieties.
We will describe the structure of classification of smooth Fano weighted complete intersections, and discuss the existence of Landau–Ginzburg models of Givental's type for them. We will show that for any smooth Fano threefold the polarisation of its general anticanonical section by restriction of Picard lattice is mirror-dual to the polarisation of a general fibre of the Landau–Ginzburg model for the Fano threefold by monodromy invariants. If time allows, we will also discuss other topics related to weighted complete intersections: complete intersections in weighted Grassmannians and higher-dimensional versions of Belyi's theorem.