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SUMMARY:Applying Category Theory to conceptual questions in the foundation
 s of Geometric Algebra - Filip Bar
DTSTART:20120207T141500Z
DTEND:20120207T151500Z
UID:TALK35990@talks.cam.ac.uk
CONTACT:Julia Goedecke
DESCRIPTION:Generalizing the algebro-geometric two-dimensional complex pla
 ne and Hamilton's quaternions to the setting of an Euclidean vector space 
 E of arbitrary finite dimension leads to the notion of a Clifford algebra 
 of E\, its even subalgebras and its spin group with its natural action on 
 E by orthogonal transformations. This construction is remarkable in two wa
 ys. On the one hand the Clifford algebra provides a synthesis of the Eucli
 dean synthetic and Cartesian analytic approach to geometry and so becomes 
 a geometric algebra in the spirit of Leibniz. On the other hand this funct
 orial way of algebraizing geometry is manifestly covariant\, as opposed to
  the standard contravariant way of mapping a space to its ring of 'coordin
 ate functions'.\n\nOne of the most important contributors to the field of 
 Geometric Algebra was H.G. Grassmann with his theory of linear extension. 
 Surprisingly\, in his original work Grassmann defined his exterior algebra
  not only for vector spaces\, but foremost for affine spaces. Being concer
 ned with the foundations of Geometric Algebra this leads to the natural qu
 estion: "What is the geometric algebra of an affine space?"\n\nIn my talk 
 I will attempt to give several possible answers to this question using Cat
 egory Theory as a particular form of conceptual mathematics as well as the
  two leading examples of a Grassmann and Clifford algebra of an affine spa
 ce. Ranging from the general to the particular these are: the category of 
 augmented algebras\, like e.g. Hopf algebras\, certain differential graded
  algebras\, and\, following a suggestion of Lawvere\, the category of dyna
 mical algebras.\n
LOCATION:MR5\, Centre for Mathematical Sciences
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