Abstract

This talk is an introduction to the theory of mutations. We will discuss two closely related viewpoints on the subject: the algebraic approach interprets mutations as birational transformations, and is concerned with their action on Laurent polynomials. The combinatorial approach sees mutations as operations on lattice polytopes, and allows one to construct deformations of the corresponding toric varieties. We will explain the role played by algebraic mutations in the program to classify Fano 4-folds. We also discuss recent results concerning combinatorial mutations of weighted projective surfaces.

The contents of this talk are joint work with Tom Coates, Alexander Kasprzyk and Sergey Galkin.