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 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. 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 2016 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).

Time and place: Tuesdays 15-17 in seminar room 3418 (Starting February 2, 2016)

Lecturers: Lars Svensson and Douglas Lundholm

Course contents
Introduction / overview

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

A revised version is being continuously updated here: PDF
(Please inform us of any additional typos or other suggestions.)

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.
  • Be able to independently read, understand and present advanced mathematics.
  • Be able to discuss and synthesize mathematics.
Eligibility and prerequisites
The course requires basic knowledge of several-variable calculus (preferably a solid background such as SF2713 Foundations of Analysis) and linear algebra and geometry. A basic course in abstract algebra (such as SF2719 Groups and Rings) is also recommended. Mathematical maturity (as expected on Ph.D. level) is assumed.

Home exercises and oral/written presentation of a chosen topic.
List of suggested topics: PDF

Homework 1: Choose 10 exercises from Chapter 2 (Foundations) in the lecture notes. Deadline: March 29.

Homework 2: Choose 10 exercises from Chapters 3 (Vector space geometry) and 4 (Discrete geometry). Deadline: April 26.

Homework 3: Choose 10 exercises from the remaining chapters (5-11, Appendix A). Deadline: May 24.

In case you use TEX, here is a file with some notation commands: TEX.

Preliminary plan of presentations
11/5Emil Jacobsen & Erika Lind:Triangles in particle interactions and applications of Clifford algebra
17/5Mitja Nedic:Hypercomplex analysis (PDF)
24/5Eric Larsson:Killing spinors (PDF)
later?Hamad Chaudhry
later?Simon Larson
later?Katharina Radermacher
later?Rune Suhr