### Seminar: *Clifford algebra, geometric algebra, and applications* (SoSe 2019)
[16096]

**Lecturer (Dozent)**: Prof. Dr. Douglas Lundholm

**Schedule**:

Fridays 16-18 (in B 132). First time (Erstes Mal): Thursday 25 April 2019.

**Note the change in time/space. If you are interested in participating, please email the lecturer!**

**Synopsis (Kurzbeschreibung):**

*"No one fully understands spinors. Their algebra is formally understood but their general significance is mysterious. In some sense they describe the "square root" of geometry and, just as understanding the square root of âˆ’1 took centuries, the same might be true of spinors."*[Atiyah]

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.

This seminar course will form an introduction to the theory of Clifford algebra, geometric algebras and their wide range of applications. It is intended to give students a universal algebraic toolbox for solving problems in geometry.

**Audience (HÃ¶rerkreis):**

Master students of Mathematics and Physics, TMP (Studierende der Mathematik, Physik, TMP).

**Credits:**

3-6 ECTS.

**Prerequisites (Vorkenntnisse):**

The course requires basic knowledge of several-variable calculus, linear algebra and geometry. A basic course in abstract algebra is also recommended. Mathematical maturity (as expected on M.Sc. level) is assumed.

**Language (Sprache):**

English.

**Content (Inhalt):**

Introduction and overview

Foundations:

- Tensor construction
- Combinatorial / set theoretic construction
- Algebraic operations
- Standard examples (plane, space, quaternions)

- 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)

- 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

**Literature (Literatur):**

We will follow

**these lecture notes**. Note that they may be revised during the course.

**Supplementary literature (ErgÃ¤nzende Literatur):**

- 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*

*Imaginary numbers are not real - the geometric algebra of spacetime*

**Office hours (Sprechstunde):**

By appointment via email.

**Program:**

Date | Speaker | Topic |

25.4 | Douglas Lundholm | Introduction |

10.5 | DL | Chapter 2.1 |

24.5 | PP | Chapter 2.2-2.3 |

14.6 | EH | Chapter 5.1-5.2 |

28.6 | LK | Chapter 3.1-3.3 |

5.7 | DL | Chapter 6.1-6.2 |

12.7 | LO | Chapter 6.6 |

19.7 | DL | Chapter 6.3-6.5 |

? | ? | Chapters 3.4 and 7 (selection) |

? | ? | Chapter 4 (selection) |

? | ? | Chapter 8 (selection) |

? | ? | Chapter 9 |

? | ? | Chapters 10 or 11 |

Wie halte ich einen Seminarvortrag? von Prof. Dr. Manfred Lehn, Johannes Gutenberg-UniversitÃ¤t Mainz.

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Letzte Ã„nderung: 19 July 2019

Douglas Lundholm