Lectures on Clifford (Geometric) Algebras and Applications

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CGA: conformal points, conformal planes, linearization of translations with translation operators translators , motion operators, Lie algebra of conformal transformations, conformal object subspaces: points, point pairs, lines, planes, circles, spheres, intersections. Filed under lectures.


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January 26, at pm. Leave a Reply Cancel reply Enter your comment here However, already before these concepts were established, Hamilton had discovered the quaternions, an algebraic system with three imaginary units which makes it possible to deal effectively with geometric transformations in three dimensions. Clifford originally introduced the notion nowadays known as Clifford algebra but which he himself called geometric algebra as a generalization of the complex numbers to arbitrarily many imaginary units.

The conceptual framework for this was laid by Grassmann already in , but it is only in recent times that one has fully begun to appreciate the algebraisation of geometry in general that the constructions of Clifford and Grassmann result in.

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Among other things, one obtains an algebraic description of geometric operations in vector spaces such as orthogonal complements, intersections, and sums of subspaces, which gives a way of proving geometric theorems that lies closer to the classical synthetic method of proof than for example Descartes's coordinate geometry. This formalism gives in addition a natural language for the formulation of classical physics and mechanics.

The best-known application of Clifford algebras is probably the "classical" theory of orthogonal maps and spinors which is used intensively in modern theoretical physics and differential geometry. The course will be given during the spring as a graduate course for PhD students in mathematics.

It will be at a level also accessible to advanced undergraduates in mathematics, physics, and other mathematical sciences, however due to administrative reasons it will not be possible to count it as part of a Master's degree but could be counted if continuing with a PhD degree.

Optional recommended literature: Delanghe, Sommen, Soucek - Clifford algebra and spinor-valued functions Doran, Lasenby - Geometric algebra for physicists Hestenes, Sobczyk - Clifford algebra to geometric calculus Lawson, Michelsohn - Spin geometry First chapter Lounesto - Clifford algebras and spinors Riesz - Clifford numbers and spinors Learning outcomes After completing this course the student should: Have a good understanding of the basic theory of Clifford algebras and the associated geometric algebras, as well as their most important applications to linear spaces and functions, orthogonal groups, spinors and multilinear analysis.

Be able to apply the formalism and tools of Clifford algebra to various problems in geometry discrete and continuous , as well as to a chosen specialization topic.