Lectures' abstracts

David Hestenes: Unification of Mathematics with Geometric Algebra (GA)

Geometric Algebra (GA) is introduced as the optimal mathematical language for encoding geometric concepts in algebraic form, equally effective at elementary and advanced levels. How it unifies diverse mathematical systems, including complex numbers, vectors, quaternions, spinors and linear algebra, is explained. Its utility in physics and engineering is demonstrated in a coordinate-free treatment of reflections, rotations and rotational dynamics. The method is equally applicable to classical and quantum physics, thereby reducing conceptual barriers between them. Geometric Algebras for vector spaces of any dimension and signature are defined. This provides an ideal arena for Lie groups and Lie algebras.

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Sebastià Xambó: Algebra and Geometry in current university curricula.

Background questions: How is algebra (particularly linear algebra) and geometry taught in university curricula? How is physics taught in mathematics curricula? How large is the gap between what we find in academic practice, particularly in relation to those two questions, and the considerable advantages that could be achieved with sounder principles? Main points: Lluís Santaló life and work. Two perspectives on the relation between mathematics and physics: H. Weyl and P. Dirac. Some Nobel laureates in theoretical physics. Some Fields medallists with impact in physics. An IAS experience. A 4-point manifesto about the benefits of a constructive dialog between mathematics and physics, and the value of GA and GC in that regard. Some background notions needed in later lectures. Algebra and geometry in Spanish university curricula. A sample of Non-GA yarstick texts. Closing remarks and references.

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Joan Lasenby: The gradient operator, reciprocal frames, curvilinear coordinates.

Having been exposed to the basics of GA and at how algebra and geometry are taught in current curricula, we now look in more detail at a few important GA concepts — which will be used extensively in subsequent lectures. Firstly we introduce the concept of a reciprocal frame, show how it is formed and give some idea of why it is so important. Next, we introduce the vector derivative/gradient operator in GA, showing how this combines the algebraic properties of a vector with the operator properties of partial derivatives. It is often convenient to work in a curvilinear coordinate system (eg spherical polars) where the frame vectors vary — we will investigate how we deal with curvilinear coordinates in GA and in particular how we can easily recover the standard conventional forms of div, grad and curl, for a few common coordinate systems.

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Anthony Lasenby: Introduction to spinors and wave equations.

The idea here is to show how with fairly elementary means we can begin to understand the nature of spinors and their role in wave equations. Starting in 2d we can consider complex numbers as the spinors appropriate to two dimensions, and their relationship with the Geometric Algebra (GA) of 2d is described. GA versions of analycity and the Cauchy-Riemann equations are then considered, and generalisation made to to higher dimensions. In 3d we investigate the role of quaternions and discuss generalisations of Cauchy's theorem and the notion of monogenics. Anticipating the GA of 4d space (the Spacetime Algebra) we discuss Weyl and Dirac spinors and their GA versions, which will allow us to make links both with the Penrose-Rindler formalism, and the wave equations of elementary particles.

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Leo Dorst: Structure Preserving Representation of Euclidean Motions through Conformal Geometric Algebra (CGA).

Conformal Geometric Algebra (CGA) is being used to encode Euclidean geometry compactly, resulting in software with fewer exceptions for the usual primitives (points, lines, planes), and extending the Euclidean primitives to spheres, circles, tangents et cetera in a consistent algebraic manner. Its power lies in being a computational framework in which constructions are represented in a structure-preserving manner: moving an element constructed from primitives is identical to moving the primitives and constructing the element (trivial, but our usual linear algebra representations fail in this). I show what the essential steps are to get from standard linear algebra to CGA, with a focus on the representation of transformations (especially Euclidean motions); the primitives then follow.

The presentation will interleave geometric equations with interactive software, and should give you a full overview of the various tricks that make CGA so effective.

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David Hestenes: Unification of Physics with Spacetime Algebra.

The entire physics curriculum can be simplified by adopting Spacetime Algebra (STA) as the standard mathematical language. That would enable early infusion of spacetime physics and give it the prominent place it deserves in the curriculum.

To show how do it, we present a coordinate-free formulation of special relativity, including Lorentz transformations and mechanics of particles and rigid bodies. Space-time splits simplify reduction to non-relativistic physics.

Coordinate-free differentiation by vectors is defined and employed to reduce electrodynamic field theory to a single equation. A spacetime split separates it into the four famous equations of Maxwell in vectorial form.

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Sebastià Xambó: On axiom systems for GA.

The many introductions to GA that one can find in the literature differ, to a greater or lesser extent, both when seen in the perspective of its natural historical evolution, including changes in individual researchers, but most importantly when considering different authors in some period of time. One way to approach the systematic appraisal of these variations is by underpinning the 'axiomatic framework' within which leading works are cast, with due regard to possible implicit assumptions that play a key role. Main points: The Grassmann algebra (outer or exterior product). The metric Grassmann algebra and the inner product. The design approach. Geometry with GA. Duality. References.

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Joan Lasenby: Multivector differentiation and Linear Algebra.

So far, we have seen how important the vector derivative/gradient operator is in spacetime physics. However, GA also provides us with a wonderfully rich calculus, in which we can differentiate any object in the algebra with respect to any other object — this is facilitated via the 'multivector derivative'. For example, we are able to differentiate an equation containing rotors (exponentials of bivectors) wrt the bivector variables. This is a powerful and underused topic in GA. We then go on to look at linear algebra, starting with the concept of linear functions mapping vectors to vectors, and extend this to linear functions acting on multivectors. We will show how this can lead to both significant simplifications and greater geometric intuition in the mathematics of physical systems that are normally treated with sophisticated tensor analysis. Finally, functional differentiation will be discussed — differentiation wrt linear functions — again, a topic which is difficult to master conventionally and which will be needed in subsequent applications in this Summer School.

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Anthony Lasenby: Lagrangian formulations and symmetries.

GA is very useful in constructing a Lagrangian-based action principle for both fields and discrete particles. The techniques of multivector differentiation allow one to take derivatives with respect to spinors in a very direct way, free of the ambiguities of conventional formulations (such as whether spinors and their adjoints should be taken as independent entities), and give streamlined versions of Noether's Theorem, and the nature of conserved quantities. Extending to multivector derivatives with respect to linear functions allows one to recover the stress-energy tensors and spin tensors of field theory in a simpler yet more general way than in the conventional approach. Additionally multivector rather than scalar Lagrangians are possible, and links are illustrated with the Berezin calculus of fermionic field theory, which becomes dramatically simpler in a GA approach.

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David Hestenes: Geometric Calculus and Differential Forms.

Cartan's theory of differential forms is reformulated and generalized in the more comprehensive language of Geometric Calculus. Coordinate-free integration is defined for manifolds of any dimension. Directed integrals are related to vector derivatives by a generalized Stokes' Theorem, shown to include as special cases all the standard conservation laws and integral theorems employed in physics and engineering.

Differentials, Adjoints and Outermorphisms are defined to describe transformations of multivector fields and their integrals. That provides a foundation for the new approach to spacetime geometry called Gauge Theory Gravity and developed by Lasenby, Doran and Gull.

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Leo Dorst: Quaternions, Rotors, Versors (I): Orthogonal Transformations as Vector Multiplications or Bivector Exponentials.

In GA, the ratio of two real vectors produces a quaternion, and the multiplication of quaternions simply implements the addition of real arcs on a sphere. We will show the geometry behind this, in the multiplicative approach (a rotation is two reflections) and in the exponential approach (a rotation is the exponential of a bivector).

We then extend both approaches to CGA (and beyond), making Cartan-Dieudonné the basis of our representation of orthogonal transformations (especially in Euclidean and Minkowski spaces). This gives us a different view of conformal transformations (and their special case, the rigid body motions). We will briefly report on the recent extension of this program of encoding orthogonal transformations to projective geometry.

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Sebastià Xambó: A view of F. Klein's geometries through GA.

In this lecture we will briefly review some of the far reaching developments, spread nearly over one century and a half, of Klein's Erlangen Program (EP, 1872). In its essence, the strength of that program lies in a unifiying principle for the geometries known at that time. The cornerstone notion being that of (Lie) group, the main goal of this lecture is to reflect on the fertility of Klein's insightful ideas by looking at some of the Lie groups that can be defined and studied with GA. Main points: What is Geometry? (an overview of the EP). Lie groups with GA (the Lipschitz approach to versors, primacy of the rotor group, topology of the rotor group). Lie algebras with GA (characterization of infinitesimal rotors as bivectors, consequences, examples). Outshoots of the EP. References.

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Anthony Lasenby: Gravity as a Gauge Theory.

Gravity is usually considered as a theory concerning spacetime distortions, and how these link with the distribution of matter and energy. A different view is possible, one which is much more akin to the theories of the other three fundamental interactions, i.e. electromagnetism, and the strong and weak nuclear forces. These theories, coming together in the standard model of particle physics, are gauge theories involving forces in a flat spacetime background, and gravity can be treated in an analogous manner, with the symmetries gauged being those of local Lorentz rotation, and local translation. Such a description becomes particularly straightforward in a GA approach, where the differential geometry aspects of general relativity are replaced by multivector relations in flat spacetime. An introduction to this approach will be given, and it will be shown that one can rapidly reach quite advanced areas via these tools.

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Joan Lasenby: Groups, Projective Geometry and Invariants.

In previous talks we have looked both at the Kleinian view of geometry and at how a covariant approach to geometry leads us to the Conformal Geometric Algebra (CGA). However, the primary conventional approach in many computer vision applications is still projective geometry, in which we have a 4D projective description of 3D Euclidean space. Here we will look at how we treat projective geometry in GA and how the wedge/outer product enables us to fully define the operations of meet and join. We will then look at some of the invariants commonly used in computer vision (cross-ratio, Cayley-Grassmann brackets et cetera) and show how these can be easily obtained using the wedge product to form multiples of the 4D pseudoscalar. Finally, we illustrate some GA applications to groups by describing crystallographic point and space groups.

References

Leo Dorst: Quaternions, Rotors, Versors (II): Roots, Logarithms and Interpolation.

One can produce conformal transformations in a multiplicative manner in various ways (e.g., as the ratio of two circles), but a minimal parametrization is obtained by writing them as the exponential of a bivector. The linear space of bivectors is well suited to interpolation and extrapolation of motions.

We demonstrate how the extraction of the bivector (the 'rotor logarithm') is done in 3D, for rigid body motions (where it is closely related to the Chasles decompositon) and conformal motions. This brings up the decomposition of a bivector as a sum of commuting 2-blades. We demonstrate how this provides practical tools to construct conformal motions, and cyclide surfaces for blending.

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David Hestenes: Geometry of Dirac Electron Theory.

The Dirac equation is the most fundamental equation in quantum mechanics. We shall see how reformulation with STA reveals hidden geometric structure that few physicists are aware of. That provides the foundation for a new theory of the electron as a singularity in the fabric of spacetime. It unifies quantum mechanics and electrodynamics in a new way and sets the stage for modeling all elementary particles as spacetime singularities.

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S. Xambó: Enriching Abstract Algebra with GA.

The strenghth of GA has roots in a happy combination of a rather particular mathematical structure (special grammar) and an inherent capacity to model geometrical and physical systems (rich semantics). In return, GA may be used to illuminate and enrich diverse aspects of abstract algebra. In this lecture we will present some examples of this feedback which in the main will have to do with group theory and representation theory. Main points: Introductory comments on some standard texts. Classification of the non-degenerate geometric algebras and of their even subalgebras. Pin and Sping representations. Synoptic ables for dimensions up to 7. References.

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Anthony Lasenby: GA approach to topics in Gravitational Physics and Cosmology

The description of gravity as a gauge theory considered in the previous lecture provides a rapid route through to many frontier topics in gravitational physics and cosmology. Among those considered will be the Dirac equation in a black hole background, the extension of the Lagrangian of General Relativity to more general Lagrangians incorporating local scale invariance as well rotational and translation invariance, and novel insights into the cosmological constant that are achieved in such theories. Additionally, within general relativity, the enhanced understanding that a GA approach brings to the Kerr solution will be discussed, together with a new approach to de Sitter space coming out of Conformal Geometric Algebra.

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Leo Dorst: Total Least Squares Fitting of Hyperspheres and Hypercircles using Conformal Geometric Algebra.

When you need to fit a sphere or circle to 3D data points, the non-linearity of the problem seemingly precludes the use of linear algebra. We show how the problem can be reformulated into a recognizable form by means of CGA.

We derive the complete solution, using geometric differentiation. Our reward will be a 5D orthogonal basis for all spheres in 3D space, of which the first is the best fitting sphere, the intersection of the first two the best fitting circle, and the intersection of the first three the best fitting point pair.

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Joan Lasenby: Applications in Computer Vision, Fluids, Elasticity…

In this final talk we will look at how the techniques we have learned can be applied to a number of engineering areas. We will use multivector differentiation to optimise over rotors for calibration of multiple camera systems; we will see how the spacetime algebra can be used to give simplified expressions for the Doppler shift from moving acoustic sources in a fluid; we will indicate how the application of the GA treatment of linear algebra can simplify the tensor analysis in elasticity theory. If time permits, we will also investigate at some current engineering topics that might benefit from being examined from a GA perspective.

References