Modern Geometric Algebra: A (Very Incomplete!) Survey by Jeff Suzuki

from the Mathematics Teacher

August 2009, volume 103, issue 1, page 26

Abstract

Representing products as rectangles can be used to introduce, connect, and reinforce concepts across mathematics.

Keywords:
Grades 9-12
Article

Clifford twist by John Bales

John W. Bales

(Submitted on 3 Aug 2011)

Gives an elementary exposition of the twisted group algebra rep- resentation of simple Clifford algebras

Subjects:	Rings and Algebras (math.RA)
Cite as:	arXiv:1108.0953v1 [math.RA]

Geometric Algebra and Applications to Physics: a book by Venzo de Sabbata and Bidyut Kumar Datta

from Amazon:

Product Description

Bringing geometric algebra to the mainstream of physics pedagogy, Geometric Algebra and Applications to Physics not only presents geometric algebra as a discipline within mathematical physics, but the book also shows how geometric algebra can be applied to numerous fundamental problems in physics, especially in experimental situations.

This reference begins with several chapters that present the mathematical fundamentals of geometric algebra. It introduces the essential features of postulates and their underlying framework; bivectors, multivectors, and their operators; spinor and Lorentz rotations; and Clifford algebra. The book also extends some of these topics into three dimensions. Subsequent chapters apply these fundamentals to various common physical scenarios. The authors show how Maxwell’s equations can be expressed and manipulated via space-time algebra and how geometric algebra reveals electromagnetic waves’ states of polarization. In addition, they connect geometric algebra and quantum theory, discussing the Dirac equation, wave functions, and fiber bundles. The final chapter focuses on the application of geometric algebra to problems of the quantization of gravity.

By covering the powerful methodology of applying geometric algebra to all branches of physics, this book provides a pioneering text for undergraduate and graduate students as well as a useful reference for researchers in the field.

Harcover, 184 pages, Taylor and Francis, 1st Edition, 2 December 2006.

Conformal geometric algebra in Wikipedia

from JHeald of Wikipedia

In mathematics, more specifically in the area of computational geometry, conformal geometric algebra (CGA) is the geometric algebra(Clifford algebra) that results if an n-dimensional Euclidean (or pseudo-Euclidean) space ℝ^p,q is projectively mapped into ℝ^p+1,q+1 in a particular way, and a geometric algebra is then constructed over that n+2 dimensional space. This allows translations of the n-dimensional space, as well as rotations, to be represented using elements of a geometric algebra; and it is found that spheres and circles as well as lines and planes gain particularly natural, as well as computationally amenable, representations.

The projective mapping from ℝ^p,q to ℝ^p+1,q+1 is constructed so that the higher dimensional space contains the full projective space of the underlying original space; so that generalized spheres in the underlying space map onto (hyper-)planes orthogonal to a particular vector inn+2 dimensional space (which can thus be used to represent them); and so that the effect of a translation (or in fact any conformal mapping) of the underlying space corresponds to the effect of an n+2 dimensional rotation in the higher dimensional space.

Introduction of the geometric algebra of this space, based on the Clifford product of vectors, allows such transformations to be represented using the algebra’s characteristic sandwich operations, similar to the use of quaternions for spatial rotation in 3D, which combine very efficiently using the Clifford product rule. A consequence of the sandwich transformation structure is that the representations of spheres, planes, circles and other geometrical objects, and equations connecting them, all transform covariantly. Some intersection operations also acquire a very tidy algebraic form: for example, applying the wedge product to the vectors corresponding to two spheres produces abivector corresponding to their circle of intersection; alternatively the tetra-vector of the hypersurface which is the dual of such a vector can be decomposed as the successive wedge product of vectors representing four points on the sphere.

Clifford Algebras, Clifford Analysis and their applications, ICNPAA Congress: Mathematical Problems in Engineering, Aerospace and Sciences, July 11-14,2012, Vienna, Austria

From GA Net Updates:

The session aims to present the latest advances in the field of Clifford (geometric) algebras and their applications in mathematics, physics, engineering and other applied sciences. The proposed session intends to gather experts working on various actually important aspects of Clifford algebras and to explore new connections between different research areas. We expect a fruitful exchange of new ideas and collaboration regarding the research development of this discipline among the participants.

We invite scientists and engineers working by means of the theory of integral equations to contribute to the session. You will find all information on the congress website:

http://icnpaa.com/index.php/icnpaa/2012

Geometric Algebra book by Emil Artin

from Google Books:

This classic text, written by one of the foremost mathematicians of the 20th century, is now available in a low-priced paperback edition. Exposition is centered on the foundations of affine geometry, the geometry of quadratic forms, and the structure of the general linear group. Context is broadened by the inclusion of projective and symplectic geometry and the structure of symplectic and orthogonal groups.

Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry (The Morgan Kaufmann Series in Computer Graphics)

by Leo Dorst; Daniel Fontijne; Stephen Mann (download ebook from System Things)

Until recently, roughly all of the interactions in between objects in practical 3D worlds have been formed upon calculations achieved regulating linear algebra. Linear algebra relies heavily upon coordinates, however, that can have many geometric programming tasks really specific as good as complex-often the lot of bid is compulsory to move about even medium opening enhancements. Although linear algebra is an fit approach to mention low-level computations, it is not the befitting high-level denunciation for geometric programming.

Geometric Algebra for Computer Science presents the constrained pick to the stipulations of linear algebra. Geometric algebra, or GA, is the compact, time-effective, as good as performance-enhancing approach to paint the geometry of 3D objects in mechanism programs. In this book we will find an pass to GA that will give we the clever sense of the attribute to linear algebra as good as the stress for your work. You will sense how to operate GA to paint objects as good as perform geometric operations upon them. And we will proceed mastering proven techniques for creation GA an constituent partial of your applications in the approach that simplifies your formula but negligence it down.

Features

• Explains GA as the healthy prolongation of linear algebra as good as conveys the stress for 3D programming of geometry in graphics, vision, as good as robotics.
• Systematically explores the concepts as good as techniques that have been pass to representing facile objects as good as geometric operators regulating GA.
• Covers in item the conformal model, the available approach to exercise 3D geometry regulating the 5D illustration space.
• Presents in outcome approaches to creation GA an constituent partial of your programming.
• Includes countless drills as good as programming exercises beneficial for both students as good as practitioners.
• Companion web site includes links to GAViewer, the module that will concede we to correlate with many of the 3D total in the book, as good as Gaigen 2, the height for the exegetic programming exercises that interpretation any chapter.

Clifford algebra in Sir Roger Penrose’s Cycles of Time

by Johannes Koelman of Science 20

Equations don’t sell. Pop science editors tell us that each equation added to a book halves its sales figure. If this is true, Sir Roger Penrose’s Cycles of Time, which was recently released in the US, and which I can testify sold at least one copy, would have sold by the billions if only the editor would have scrapped half of the equations.

With his 2004 book The Road To Reality Penrose has shown to be capable of blurring the distinction between textbooks and pop science writings.Cycles of Time continues in this tradition. I applaud Renrose’s non-populistic attitude, and admire his style. Reading ‘The Road’ and ‘Cycles’ is like listening to talks by Penrose himself. A Penrose who does not shy away from in-depth explanations and who dives deep into the beauty of mathematical physics. Penrose boldly presents spinors, twistors, Clifford algebras and conformal diagrams to the general public. I know of no other pop-science writer who dares to tread into this territory.

Read more at Science 20

Pfaffian quartic surfaces and representations of Clifford algebras

Emre Coskun, Rajesh S. Kulkarni, Yusuf Mustopa

(Submitted on 7 Jul 2011)

Given a nondegenerate ternary form $f=f(x_1,x_2,x_3)$ of degree 4 over an algebraically closed field of characteristic zero, we use the geometry of K3 surfaces and van den Bergh’s correspondence between representations of the generalized Clifford algebra $C_f$ associated to $f$ and Ulrich bundles on the surface $X_f:=\{w^{4}=f(x_1,x_2,x_3)\} \subseteq \mathbb{P}^3$ to construct a positive-dimensional family of irreducible representations of $C_f.$
The main part of our construction, which is of independent interest, uses recent work of Aprodu-Farkas on Green’s Conjecture together with a result of Basili on complete intersection curves in $\mathbb{P}^{3}$ to produce simple Ulrich bundles of rank 2 on a smooth quartic surface $X \subseteq \mathbb{P}^3$ with determinant $\mathcal{O}_X(3).$ This implies that every smooth quartic surface in $\mathbb{P}^3$ is the zerolocus of a linear Pfaffian, strengthening a result of Beauville-Schreyer on general quartic surfaces.

Comments:	This paper contains a proof of the main result claimed in the erroneous preprint arXiv:1103.0529. We also extend this result to all smooth quartic surfaces
Subjects:	Algebraic Geometry (math.AG); Rings and Algebras (math.RA)
MSC classes:	14J60, 14J28, 16G50
Cite as:	arXiv:1107.1522v1 [math.AG]