Basic algebraic geometry (for example R. Hartshorne, Algebraic Geometry, 1977, Springer, or I. Shafarevich, Basic Algebraic Geometry 1 and 2, 2007, Springer).
The object of this course is to introduce a new perspective on the classification of Fano manifolds coming from work in mirror symmetry. The quantum period of a Fano manifold X is a generating function for certain Gromov-Witten invariants of X which plays an important role in mirror symmetry. Indeed, we say that a Fano manifold X is mirror to a Laurent polynomial f if the quantum period of X equals the classical period integral of f.  shows by explicit computation that the 3-dimensional Fano manifolds with very ample anticanonical bundle have mirrors given by a combinatorially defined collection of Laurent polynomials called Minkowski polynomials. This strongly suggests a new approach to the classification of Fano manifolds: by proving an appropriate mirror theorem and then classifying Fano mirrors. The distinct advantage of this approach is that the mirrors are objects of combinatorics: given enough computing power, their classification is a trivial matter and involves none of the subtle projective and algebraic geometry that one needs in the classical approach.
The only known theoretical framework for mirror symmetry which is general enough and constructive enough for this new approach to Fano classification is the Gross-Siebert program. In the Gross-Siebert program, pairs (X; D) of a Fano manifold and anticanonical divisor D arise, together with their mirrors, as smoothings of degenerate pairs made of toric components glued torically (the Laurent polynomial f arises from a tropical disk count). We expect that, at the time of this school, we will have made some progress interpreting the work of  in the framework of the Gross-Siebert program.
The lectures will cover the following topics:
 M. Akhtar, T. Coates, S. Galkin and A. Kasprzyk, Minkowski polynomials and mutations, SIGMA Symmetry Integrability
Geom. Methods Appl. 8 (2012), Paper 094, 17 pp.
 T. Coates, A. Corti, S. Galkin, V. Golyshev and A. Kasprzyk, Mirror Symmetry and Fano Manifolds, arXiv:1212.1722, to appear in the Proceedings of the 6th European Congress of Mathematics.
 T. Coates, A. Corti, S. Galkin and A. Kasprzyk, Quantum Periods for 3-Dimensional Fano Manifolds, arXiv:1303.3288.
The object of this course is to reflect the present state of the classification theory of Fano varieties. This classification is closely related to the one of the finite subgroups of the group Bir(X) of birational automorphisms of a Fano variety X and these subgroups can be studied by a method coming from the Mori theory. In the course we will present many recent
applications of this method obtained by many mathematicians (Beauville, Blanc, Dolgachev, de Fernex, Iskovskikh, Lamy, Cantat, Cheltsov, Shramov and others).
The following topics will be covered: