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Let $\varphi(t), t\in [0,T]$ be a smooth solution to the Laplacian flow for closed G_2 structures on a compact 7-manifold $M$. We show that for each fixed positive time $t\in (0,T]$, $(M,\varphi(t),g(t))$ is real analytic, where $g(t)$ is the metric induced by $\varphi(t)$. Consequently, any Laplacian soliton is real analytic and we obtain unique continuation results for the flow., Comment: 27 pages, v2: streamlined exposition, accepted version for CAG

We provide an introduction to the theory of calibrated submanifolds through the key examples related with special holonomy. We focus on calibrated geometry in Calabi-Yau, G$_2$ and Spin(7) manifolds, and describe fundamental results and techniques in the field., Comment: 24 pages. To appear in a forthcoming volume of the Fields Institute Communications, entitled "Lectures and Surveys on G2 manifolds and related topics". These notes are primarily based on a lecture course the author gave at the LMS-CMI Research School "An Invitation to Geometry and Topology via G2" at Imperial College London in July 2014

We initiate the systematic study of $G_2$-instantons with $SU(2)^2$-symmetry. As well as developing foundational theory, we give existence, non-existence and classification results for these instantons. We particularly focus on $\mathbb{R}^4\times S^3$ with its two explicitly known distinct holonomy $G_2$ metrics, which have different volume growths at infinity, exhibiting the different behaviour of instantons in these settings. We also give an explicit example of sequences of $G_2$-instantons where "bubbling" and "removable singularity" phenomena occur in the limit., 61 pages; v3 minor changes: accepted in Mathematische Annalen

Geometric flows have proved to be a powerful geometric analysis tool, perhaps most notably in the study of 3-manifold topology, the differentiable sphere theorem, Hermitian-Yang-Mills connections and canonical Kaehler metrics. In the context of G_2 geometry, there are several geometric flows which arise. Each flow provides a potential means to study the geometry and topology associated with a given class of G_2 structures. We will introduce these flows, and describe some of the key known results and open problems in the field., Comment: 20 pages. To appear in a forthcoming volume of the Fields Institute Communications, entitled "Lectures and Surveys on G2 manifolds and related topics". These notes are based primarily on a lecture given at a Minischool on "G2 Manifolds and Related Topics" at the Fields Institute, Toronto in August 2017