KTH / Department of Mathematics
Course in mathematics:
Clifford algebra, geometric algebra, and applications
It is well known that the complex numbers form a powerful tool in the description of plane geometry. The geometry of 3-dimensional space is traditionally described with the help of the scalar product and the cross product. 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 1844, but it is only now that one has fully begun to appreciate the algebraisation of geometry in general that the constructions of Clifford and Grassmann result in. 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 (with different levels of examination)
both as a higher level course (fördjupningskurs) for undergraduate students,
and as a graduate course for PhD students
in mathematics, physics, and other mathematical sciences.
The course will be given during the spring 2009.
Time and place: Thursdays 15-17 in seminar room 3721 (Starting January 29, 2009)
Lecturers:
Lars Svensson
and Douglas Lundholm
Course contents
Introduction / overview
Foundations:
Tensor construction
Combinatorial / set theoretic construction
Algebraic operations
Standard examples (plane, space, quaternions)
Main tools:
Vector space geometry
Linear functions, outermorphisms
Classification over R and C
Representation theory
Pin and Spin groups, bivector Lie algebra, spinors
Clifford analysis in R^n (Dirac operator, vector analysis)
Other applications (depending on the interests of the participants):
Monogenic functions, Clifford-valued measures and integration, Cauchy's integral formula
Projective and conformal geometry
Various applications in physics (classical mechanics, electromagnetism, special relativity / Minkowski space, quantum mechanics)
Applications in combinatorics, discrete geometry
Division algebras, octonions
Embedded differential geometry
Course literature
We will follow these lecture notes: arXiv:0907.5356
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
Examination
All students: Home exercises.
Additionally, for PhD students: Oral/written presentation of a chosen topic.
List of suggested topics: PDF
Past lectures
| Jan 29 | | History and introduction |
| Feb 5 | | Definitions (combinatorial and tensor-algebraic), operations, universality |
| Feb 12 | | Grading, orthonormal bases, signatures, pseudoscalar, examples |
| Feb 19 | | Inner product expansion, linear independence, blades and subspace geometry |
| Feb 26 | | More basic tools, inner/outer product duality, outermorphisms |
| Mar 5 | | Dual maps, projections and rejections, fermionic operators |
| Mar 12 | | No lecture |
| Mar 19 | | Cancelled due to illness. Read on your own: Projective geometry |
| Mar 26 | | Applications in combinatorics |
| Apr 2 | | Classification |
| Apr 9 | | No lecture |
| Apr 16 | | Moved to Friday Apr 17, 10-12. Topic: Pin and Spin groups |
| Apr 23 | | Spin groups cont., linearization of the Euclidean group |
| Apr 30 | | No lecture |
| May 7 | | Representation theory and examples |
| May 14 | | Some Clifford differentiation and integration |
Homework 1: Choose 10 exercises from Chapter 2 (Foundations) in the lecture notes. Deadline: February 26.
Homework 2: Choose 10 exercises from Chapters 3 (Vector space geometry) and 4 (Discrete geometry). Deadline: April 16.
Homework 3: Choose 10 exercises from the remaining chapters (5-11). Deadline: June 4.
In case you use TEX, here is a file with some notation commands: TEX.
Preliminary plan of presentations
| June 4 | | Fridrik Freyr Gautason: | Physical applications of Clifford algebras (PDF) |
| later | | Thomas Westerbäck |
| later | | Oscar Andersson Forsman |
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