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CATEGORIES:Category Theory Seminar
SUMMARY:Applying Category Theory to conceptual questions i
n the foundations of Geometric Algebra - Filip Bar
DTSTART;TZID=Europe/London:20120207T141500
DTEND;TZID=Europe/London:20120207T151500
UID:TALK35990AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/35990
DESCRIPTION:Generalizing the algebro-geometric two-dimensional
complex plane and Hamilton's quaternions to the s
etting of an Euclidean vector space E of arbitrary
finite dimension leads to the notion of a Cliffor
d algebra of E\, its even subalgebras and its spin
group with its natural action on E by orthogonal
transformations. This construction is remarkable i
n two ways. On the one hand the Clifford algebra p
rovides a synthesis of the Euclidean synthetic and
Cartesian analytic approach to geometry and so be
comes a geometric algebra in the spirit of Leibniz
. On the other hand this functorial way of algebra
izing geometry is manifestly covariant\, as oppose
d to the standard contravariant way of mapping a s
pace to its ring of 'coordinate functions'.\n\nOne
of the most important contributors to the field o
f Geometric Algebra was H.G. Grassmann with his th
eory of linear extension. Surprisingly\, in his or
iginal work Grassmann defined his exterior algebra
not only for vector spaces\, but foremost for aff
ine spaces. Being concerned with the foundations o
f Geometric Algebra this leads to the natural ques
tion: "What is the geometric algebra of an affine
space?"\n\nIn my talk I will attempt to give sever
al possible answers to this question using Categor
y Theory as a particular form of conceptual mathem
atics as well as the two leading examples of a Gra
ssmann and Clifford algebra of an affine space. Ra
nging from the general to the particular these are
: the category of augmented algebras\, like e.g. H
opf algebras\, certain differential graded algebra
s\, and\, following a suggestion of Lawvere\, the
category of dynamical algebras.\n
LOCATION:MR5\, Centre for Mathematical Sciences
CONTACT:Julia Goedecke
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