Let $(M, \omega)$ be a compact symplectic manifold. If we choose a compatible almost complex structure $J$ (which in general is not integrable) then we can study the space of $J$-holomorphic maps $u : \Sigma \to (M, J)$ from a compact Riemann surface into $M$. By appropriately “compactifying” the space of such maps, one can obtain powerful global symplectic invariants of $M$. At the heart of such a compactification procedure is understanding the ways in which sequences of such maps can degenerate, or develop singularities. Crucial ingredients are conformal invariance and an energy identity, which lead to to a plethora of analytic consequences, including: (i) a mean value inequality, (ii) interior regularity, (iii) a removable singularity theorem, (iv) an energy gap, and (v) compactness modulo bubbling.

Riemannian manifolds with closed $G_2$ or $\mathrm{Spin}(7)$ structures share many similar properties to such almost Kahler manifolds. In particular, they admit analogues of $J$-holomorphic curves, called associative and Cayley submanifolds, respectively, which are calibrated and hence homologically volume-minimizing. A programme initiated by Donaldson-Thomas and Donaldson-Segal aims to construct similar such “counting invariants” in these cases. In 2011, a somewhat overlooked preprint of Aaron Smith demonstrated that such submanifolds can be exhibited as images of a class of maps $u : \Sigma \to M$ satisfying a conformally invariant first order nonlinear PDE analogous to the Cauchy-Riemann equation, which admits an energy identity involving the integral of higher powers of the pointwise norm $|du|$. I will discuss joint work with Da Rong Cheng (Chicago) and Jesse Madnick (McMaster) in which we establish the analogous analytic results of (i)-(v) in this setting. arXiv:1909.03512