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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.

Madeleine Jotz Lean, who joins SoMaS in November 2013, works in Poisson geometry, multiple structures and allied fields. She is particularly involved in extending Poisson reduction and its applications to Dirac structures, and on studying Dirac structures compatible with a Lie groupoid.

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 Email VCard  
Dr Madeleine Jotz Lean K24 23727 vcard Differential geometry, Poisson geometry, Lie algebroids and groupoids
Dr Kirill Mackenzie J6a 23745 vcard Differential geometry -- Lie groupoids and Lie algebroids, connection theory, Poisson geometry, multiple structures
Prof Ieke Moerdijk J13 23843 vcard Algebraic and differential topology, category theory
Dr Simon Willerton J19 23823 vcard quantum topology, algebraic topology, category theory, metric spaces

Postgraduate students
Name Office Phone Email VCard  
Ms Magdalini Flari J25 23767 vcard Multiple Lie theory and Poisson geometry
Mia Gleadall J26 vcard
Malte Heuer J14a vcard
Samuel Morgan J18a vcard Poisson geometry, Lie groupoids and Lie algebroids

Past grants


LMS
LMS

Key publications