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We decompose linear $\mathrm{G}_2$-structure in canonical ways adapted to 3-dimensional subspaces, in terms of certain natural 1-forms and definite triple of 2-forms, and apply the decompositions to the study of $\mathrm{G}_2$-structure…

We consider the deformation theory of asymptotically conical (AC) and of conically singular (CS) $G_2$-manifolds. In the AC case, we show that if the rate of convergence $\nu$ to the cone at infinity is generic in a precise sense and lies…

Motivated by gauge theory under special holonomy, we present techniques to produce holomorphic bundles over certain noncompact $3-$folds, called building blocks, satisfying a stability condition `at infinity'. Such bundles are known to…

We present a new class of dualities relating non-geometric Calabi-Yau compactifications of type II string theory to T-fold compactifications of the heterotic string, both preserving four-dimensional $\mathcal{N}=2$ supersymmetry. The…

A brief, example-oriented introduction is given to special holonomy and its uses in string theory and M-theory. We discuss A_k singularities and their resolution; the construction of a K3 surface by resolving T^4/Z_2; holomorphic cycles,…

We develop some consequences of the connection between Calabi-Yau structures and torsion-free $G_2$ structures on compact and asymptotically cylindrical six- and seven-dimensional manifolds. Firstly, we improve the known proof that matching…

Numerous structural findings of homology manifolds have been derived in various ways in relation to $g_2$-values. The homology $4$-manifolds with $g_2\leq 5$ are characterized combinatorially in this article. It is well-known that all…

This paper aims to shed light on what becomes of discrete torsion within heterotic orbifolds when they are resolved to smooth geometries. Gauged Linear Sigma Models (GLSMs) possessing (0,2) worldsheet supersymmetry are employed as…

We give a general procedure for gluing together possibly noncompact manifolds of constant scalar curvature which satisfy an extra nondegeneracy hypothesis. Our aim is to provide a simple paradigm for making `analytic' connected sums. In…

We introduce new obstructions to topological knot concordance. These are obtained from amenable groups in Strebel's class, possibly with torsion, using a recently suggested $L^2$-theoretic method due to Orr and the author. Concerning…

We study complete non-compact manifolds of positive scalar curvature, with a focus on how curvature decay is constrained by topology at infinity. Our first main result shows that topological linking at infinity forces polynomial decay of…

We develop a powerful new analytic method to construct complete non-compact G2-manifolds, i.e. Riemannian 7-manifolds (M,g) whose holonomy group is the compact exceptional Lie group G2. Our construction starts with a complete non-compact…

This paper uses algebro-topological techniques such as characteristic classes and obstruction theory, together with the $h$-principles for $\widetilde{\mathrm{G}}_2$ and $\mathrm{SL}(3;\mathbb{R})^2$ forms recently established by the author…

We study M theory compactifications on manifolds of $G_2$-holonomy with gauge and matter fields supported at singularities. We show that, under certain topological conditions, the combination of background $G$-flux and background fields at…

We find new examples of compact Spin(7)-manifolds using a construction of Joyce. The essential ingredient in Joyce's construction is a Calabi-Yau 4-orbifold with particular singularities admitting an antiholomorphic involution, which fixes…

An end sum is a non-compact analogue of a connected sum. Suppose we are given two connected, oriented $n$-manifolds $M_1$ and $M_2$. Recall that to form their connected sum one chooses an $n$-ball in each $M_i$, removes its interior, and…

Fix a finite group $G$. We study the computational complexity of counting problems of the following flavor: given a group $\Gamma$, count the number of homomorphisms $\Gamma \to G$. Our first result establishes that this problem is…

We argue that G_2 manifolds for M-theory admitting string theory Calabi-Yau duals are fibered by coassociative submanifolds. Dual theories are constructed using the moduli space of M5-brane fibers as target space. Mirror symmetry and…

Let $G$ be a connected Lie group acting locally simply transitively on a manifold $M$. By connecting curves in $M$ we mean the orbits of one-parameter subgroups of $G$. To block a pair of points $m_1,m_2\in M$ is to find a finite set…

We study the dynamics of M-theory on G2 holonomy manifolds, and consider in detail the manifolds realized as the quotient of the spin bundle over S^3 by discrete groups. We analyse, in particular, the class of quotients where the triality…