# 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.

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.

**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**Z**_{2}; 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.

## Members

Academic staff | |||||
---|---|---|---|---|---|

Name | Office | Phone | VCard | ||

Dr Kirill Mackenzie | J6a | 23745 | K.Mackenzie | vcard | Differential geometry -- Lie groupoids and Lie algebroids, connection theory, Poisson geometry, multiple structures |

Prof Ieke Moerdijk | J13 | 23843 | i.moerdijk@uu.nl | vcard | Algebraic and differential topology, category theory |

Dr Simon Willerton | J19 | 23823 | S.Willerton | vcard | quantum topology, algebraic topology, category theory, metric spaces |

Postgraduate students | |||||
---|---|---|---|---|---|

Name | Office | Phone | VCard | ||

Visakan Balakumar | VBalakumar1 | vcard | General Relativity, Quantum Field Theory in Curved Spacetime and Black Holes | ||

Malte Heuer | J14a | MAHeuer1 | vcard | ||

Mr Joseph Martin | J14a | JMartin13@sheffield.ac.uk | vcard | ||

Samuel Morgan | J18a | S.F.Morgan | vcard | Poisson geometry, Lie groupoids and Lie algebroids |

## Past grants

Visit of Professor Kosmann-Schwarzbach | LMS |

GAP XIV 'Seminaire Itinerant Geometrie et Physique' 2016, Sheffield, Graded Geometry and Applications to Physics | LMS |

## Key publications

- Mackenzie, K.C.H.

"Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids"

Journal fur die Reine und Angewandte Mathematik. [Crelle's Journal] (2011) - Willerton, S.

"On the magnitude of spheres, surfaces and other homogeneous spaces"

(2010) - Roberts, Justin and Willerton, Simon

"On the Rozansky–Witten weight systems"

Algebraic and Geometric Topology (2010) - Mackenzie, K.C.H. and Gracia-Saz, Alfonso

"Duality for triple vector bundles"

Letters in Mathematical Physics (2009) - Moerdijk, I. and Crainic, M

"Deformations of Lie brackets: cohomological aspects"

Journal of the European Mathematical Society (2008) - Mackenzie, K.C.H.

"General Theory of Lie Groupoids and Lie Algebroids"

Cambidge University Press (2005) - Bunke, Ulrich, Turner, Paul and Willerton, S.

"Gerbes and homotopy quantum field theories"

Algebraic and Geometric Topology (2004) - Moerdijk, I. and Mrcun, Janez

"Introduction to foliations and Lie groupoids"

Cambidge University Press (2003)