For example, we have the following methods, which you can find out more about in [@Joycebook2] or the references provided.

-   Symmetries/evolution equations [@Goldstein; @HL; @Haskins.SLcones1; @Ionel.Minoo; @Joyce.evol; @Joyce.quadrics; @Joyce.sym; @Joyce.U1fib; @Joyce.U1a; @Joyce.U1b; @Joyce.U1c; @Lotayevol; @Lotaysym].

-   Use of integrable systems to study calibrated cones [@Carberry; @Carberry.McIntosh; @Haskins.SLcones2; @Joyce.int; @McIntosh].

-   Calibrated cones and ruled smoothings of these cones [@BryantOct; @Bryant.SL; @Fox.coass; @Fox.Cayley; @Joyce.ruled; @Lotayevol; @Lotay2R; @LotayLag].

-   Vector sub-bundle constructions [@Ionel.Kar.Minoo; @Kar.Leung; @Kar.Minoo].

-   Classification of calibrated submanifolds satisfying pointwise constraints on their second fundamental form [@Bryant.SL; @Fox.thesis; @Ionel.SL; @LotayLag; @Lotayassoc].

However, an important direction which has borne fruit in calibrated geometry and special holonomy recently has been to study the nonlinear PDE head on, especially by perturbative and gluing methods. We want to solve nonlinear PDE, so how do we tackle this? The idea is to use the linear case to help. Suppose we are on a compact manifold $N$ and recall the theory of linear *elliptic* operators $L$ of order $l$ on $N$, including:

-   the definition of ellipticity of $L$ via the *principal symbol* $\sigma_L$ (which encodes the highest order derivatives in the operator) being an isomorphism;

-   the use of *Hölder spaces* $C^{k,a}$ to give 