Abstract: The talk will be split in two parts, discussing two projects broadly related to constructing and classifying special holonomy orbifolds and manifolds.
 
Firstly, motivated by finding a new compact hyperKähler manifold, we study hyperKähler orbifolds of Kummer-type (i.e. orbifolds obtained as quotients of tori) associated to the maximal torus of E8 and classify such spaces. We then study their singular sets in an attempt to find an example which can be desingularised to a smooth hyperKähler manifold. The methods developed pave the way for a larger project to classify all special holonomy Kummer orbifolds.
 
Secondly, we study 7-manifolds of holonomy SU(2) \rtimes F and partially classify them. Orbifold limits of such spaces have singular sets of the type considered by Joyce and Karigiannis who showed that (1707.09325), under certain conditions, such singular spaces can be desingularised to smooth, compact, G2-manifolds. We obtain a complete understanding of the physics of these spaces in the context of M-theory, although this will not feature much in the talk. As a bonus we discover an unexpected isomorphism between two seemingly unrelated moduli spaces: the space of Ricci flat metrics on these 7-manifolds and the space of flat connections on compact, flat 3-manifolds.