Instantons: the next frontier

Instantons, emerged in particle physics, have been intensely studied since the 1970's and had an enormous impact in mathematics since then. In this paper, we focus on one particular way in which mathematical physics has guided the development of algebraic geometry in the past 40+ years. To be precise, we examine how the notion of mathematical instanton bundles in algebraic geometry has evolved from a class of vector bundles over the complex projective 3-space both to a class of torsion free sheaves on projective varieties of arbitrary dimension, and to a class of objects in the derived category of Fano threefolds. The original results contained in this survey focus precisely on the latter direction; in particular, we prove that the classical rank 2 instanton bundles over the projective 3-space are indeed instanton objects for any suitable chamber in the space of Bridgeland stability conditions.