a key idea is to find a special class of minimal submanifolds, called *calibrated submanifolds*, which are instead defined by a first order partial differential equation. The definition of calibrated submanifolds is motivated by the properties of complex submanifolds in Kähler manifolds, and turns out to be useful in finding minimizers for the volume functional rather than just critical points. However, finding examples outside the classical complex setting turns out to be difficult, leading to important methods coming from a variety of sources, as well as motivating the study of the deformation theory of these objects.

Calibrated submanifolds naturally arise when the ambient manifold has *special holonomy*, including holonomy $\operatorname{G}_2$. In this situation, we would hope that the calibrated submanifolds encode even more, finer, information about the ambient manifold, potentially leading to the construction of new invariants. In this setting, there is also a relationship between calibrated submanifolds and gauge theory: specifically, connections whose curvature satisfies a natural constraint determined by the special holonomy group (so-called *instantons*). For these reasons, calibrated submanifolds form a hot topic in current research, especially in the $\operatorname{G}_2$ setting.

These notes are primarily based on a lecture course the author gave at the LMS–CMI Research 