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We study the Laplacian flow and coflow on contact Calabi–Yau 7-manifolds. We show that the natural initial condition leads to an ancient solution of the Laplacian flow with a finite time Type I singularity which is not a soliton, whereas it produces an immortal (though neither eternal nor self-similar) solution of the Laplacian coflow which has an infinite time singularity of Type IIb, unless the transverse Calabi–Yau geometry is flat. The flows in each case collapse (under normalised volume) to a lower-dimensional limit, which is either ℝ, for the Laplacian flow, or standard ℂ3 for the Laplacian coflow. We also study the Hitchin flow in this setting, which we show coincides with the Laplacian coflow, up to reparametrisation of time, and defines an (incomplete) Calabi–Yau structure on the spacetime track of the flow.

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

Jason Lotay explains how mathematicians studying special geometries are collaborating with physicists to explore M-theory, an 11-dimensional description of the world that unifies the various string theories

We survey the known existence and non-existence results for $G_2$-instantons on non-compact cohomogeneity-1 $G_2$-manifolds and their consequences, including an explicit example of a family of $G_2$-instantons where bubbling, removable singularities and conservation of energy phenomena occur. We also describe several open problems for future research.

The present thesis examines the role of Hermitian Yang-Mills (HYM) connections in Spin(7) instanton moduli spaces over asymptotically conical (AC) Calabi-Yau (CY) fourfolds. The starting point is Lewis' energy estimate: Lewis: if a principal G-bundle P over a closed CY fourfold X8} admits HYM solutions, then all Spin(7) instantons on P are HYM. The question we are interested in is whether this result persists in the AC setting. A first-order approach is to pass to the (Sasaki-Einstein) asymptotic link, which is not Ricci flat and could thus be homogeneous. The role of the Spin(7) instanton equation is assumed by the G2 instanton equation and that of the HYM equation by the contact instanton equation. It is easier to explore the relationship between these 7-dimensional systems instead. Imposing equivariance reduces the PDEs to representation theory. This allows us to exhibit an explicit non-contact G2 instanton on S7. This example agrees with the limiting connection of the standard octonionic instanton of Fubini and Nicolai. Prior to the results of this thesis, this was the only known non-HYM Spin(7) instanton on a CY fourfold. It originally appeared in the physics literature. We provide an alternative construction, in line with the modern framework for equivariant gauge theory. Because its limiting connection is not contact, its moduli space cannot contain HYM connections. Consequently, it does not help resolve the question we set out to answer. We extend Lewis's theorem to the AC setting, conditioning on decay rates. We construct the moduli space of SO(5)-invariant Spin(7) instantons with structure group SO(3) on the Stenzel space. These new instantons sit exactly at the slow-rate cut-off point of our extension of Lewis's theorem. They provide a negative answer to the question we set out to answer: the moduli space is one-dimensional and contains precisely two HYM connections. One of these is the epicentre of a removable singularity/ bubbling phenomenon and the development of a corresponding Fueter section. We compute this explicitly and verify (after suitable modifications) an infinite-energy version of Tian's energy conservation identity. This phenomenon hints at a possible relationship between the AC Spin(7) instanton and HYM systems.