# Differential Geometry

Differential Geometry at Sheffield is concerned with new structures developed in response to recent work in mathematical physics and fundamental problems in differential geometry.

Kirill Mackenzie is primarily concerned with the multiple Lie theory which he initiated, an extension of the Lie theory of Lie groups and Lie algebras to double and multiple Lie groupoids and Lie algebroids. This work relies very much on the use of Poisson structures and in turn Poisson group(oid)s and Poisson actions give rise to double structures, the integrability of which is a major problem. Multiple Lie theory has given rise to the idea of multiple duality: the ordinary duality of vector spaces and vector bundles is involutive and may be said to have group Z2; double vector bundles have duality group the symmetric group of order 6, and 3-fold and 4-fold vector bundles have duality groups of order 96 and 3,840 respectively. An idea of double and multiple Lie theory can be obtained from Mackenzie's 2011 Crelle article (see below) and the shorter 1998 announcment, "Drinfel'd doubles and Ehresmann doubles for Lie algebroids and Lie bialgebroids" (Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 74-87) and from the 2011 paper of Th. Voronov which uses supergeometric methods. "Q-manifolds and Mackenzie Theory" Comm. Math. Phys. 315, 279-310, 2012.

Simon Willerton has worked on the role of hyper-Kähler manifolds and gerbe-connections in topological quantum field theory and is interested in how curvature relates to `magnitude', a metric space analogue of the Euler characteristic.

Ieke Moerdijk works, among many other interests, on Lie groupoids and Lie algebroids, especially étale groupoids and orbifolds and their relations with foliation theory. See in particular his 2003 book with Mrcun.

The gif above is a rotating hypercube (or tesseract) from http://en.wikipedia.org/wiki/Tesseract The outline of a 4-fold vector bundle is a hypercube.