There are two conformally covariant first order differential operators acting on spinor fields of a semi-Riemannian spin manifold, the Dirac operator and the twistor operator P. We discuss special non conformally flat geometries that admit solutions of the twistor equation Pf = 0. We focus our attention mainly to the case of Lorentzian signature.

We plan to give an up-to-date account on how far have the known results on the Euclidean space extend to quternionic valued functions in quaternionic spaces.

On a homogeneous manifold, the full symmetry group may be employed to study its geometry and analysis. Invariant constructions in differential geometry often come from symmetry considerations on various "flat models", namely homogeneous versions of the geometry in question. I shall discuss some examples.

In the traditional approaches to Clifford algebras, the Clifford product is evaluated by recursive application of the product of a one-vector on decomposable Graßmann multi-vectors which is later extended by bilinearity. The Hestenesian `dot' product, extending the one-vector scalar product, is even worse having exceptions for scalars and the need for applying grade operators at various times. Moreover, the multi-vector grade is not a generic Clifford algebra concept. The situation becomes even worse in geometric applications if a meet, join or contractions have to be calculated.

Starting from a naturally graded Graßmann Hopf gebra, we derive general formulæ for the products: left/right contraction, left/right co-contraction, Clifford/co-Clifford product, meet and join. All these product formulæ are valid for any grade and any inhomogeneous multi-vector factors in Clifford algebras of any bilinear form, including non-symmetric and degenerate forms. We derive the three well known formulæ of Chevalley deformation as a consequence of our approach. A co-Chevalley deformation is given and the Rota--Stein cliffordization is shown to be the generalization of Chevalley deformation. Our product formulæ are based on invariant theory and are not tied to representations/matrices and are highly computationally effective. The method is applicable to symplectic Clifford algebras too. 

Clifford algebras have played an important role in each of these branches of algebra both in the field theory and the integral theory. This talk will provide a survey of these connections.

With every symplectic space (M,y) we can associate a Weyl algebra (or symplectic Clifford algebra) Cl_s(M,y) that is isomorphic to the symmetric algebra S(M) as a vector space. Unfortunately, whereas the Lipschitz group GLip(M,q) of a quadratic space (M,q) has been defined, at least since 1946, as the group of all invertible, even or odd elements x eCl(M,q) such that xMx^-1=M, such a definition is inoperative in Cl_s(M,y) because this algebra is not large enough; and no enlarged algebra satisfies properties that would grant efficiency to such a definition. 

In order to represent a symplectic transformation g of (M,y) by means of a formal series x e S^c(M) (in such a way that g(a) = xax^-1), several researchers (independently of one another) used other methods, without being aware that they were coming back to Lipschitz's ideas of 1886, that were almost forgotten at that time (1976 or soon later). 

It is worth explaining how Lipschitz's ideas behave in a symplectic context. Whereas orthogonal transformations with eigenvalue -1 only inflicted a harmless difficulty to Lipschitz, symplectic transformations with eigenvalue -1 raised impassable obstructions (because S^c(M) contains no divisors of zero), that only allowed to define a local analytical group GLip_loc(M,y)- by means of formal series.

I review the status of a reseach program to develop a Universal Geometric Calculus for mathematics and physics, including: (1) historical roots and objectives, (2) scope and structure, (3) insights and achievements, (4) current issues and future prospects.

This lecture is devoted to the study of nonlinear partial differential equations by exploiting their special algebraic structures. These equations appear in the geometric theory of mappings in Rⁿ, principally the Cauchy-Riemann and the Beltrami systems, the nonlinear elasticity theory of deformations, calculus of variations, etc.  Here, the goal is to put these equations in a more general framework and discuss them from a number of perspectives. For instance, when lifted to the level of the exterior differential forms or Clifford algebra they linearize. Studying these linearized equations has led to major new advances in the geometric theory of nonlinear PDEs, including the multidimensional theory of quasiconformal mappings. Also, there seem to exist close connections between this approach and recent development of the methods of Clifford algebra. We shall outline some important ingredients and set up the associated equations.

Recently, with work of P Shor and others, exponential speedup of algorithms has been realized in the quantum realm, i.e., when registers of qubits are used in place of the classical bits, and when quantum gates, in the form of unitary matrices acting on tensor slots, take the role of the classical logic gates: Unitary factorizations then assume the role of algorithms. If shorter than the best available (analogous) classical ones, there is a gain. But the dictates of quantum theory introduce new and serious sources of "error" called decoherence, i.e., when some qubits, that are part of the program, degenerate and behave classically. Clifford analysis is used in error correction. Wavelet algorithms split functions in a fixed resolution subspace into components, a coarser one, and detail parts. This can be turned into a quantum algorithm, and the factorization problem can be implemented effectively. 

``Image Space'' is the product of the ``image plane'' (typically a rectangular area of the Euclidian plane) and the ``image intensity'' dimension. ``Image intensity'' is a physical ``density'', for instance the ``irradiance'' (number of incident photons per "pixel" area within the "exposure" time). The natural intensity scale is logarithmic, turning the image intensity dimension into (a linear segment of) the affine line. A number of well known transformations (``gamma transformations'', ``gradients'', ``intensity scalings'', \ldots) are considered ``irrelevant'', \emph{i.e.}, one feels free to apply them in adjusting a TV~set, beamer, computer CRT screen or in printing from photographic negatives. Consequently, I identify these transformations as the congruences and similarities of image space. Then image space becomes a Cayley--Klein (3D) geometry, with two Euclidian directions and one isotropic direction (``single isotropic geometry''). 

The differential invariants of generic (no tangent plane contains the isotropic direction) surfaces in this geometry can then be identified as proper ``image features'', thus one obtains a principled framework for image processing. This framework automatically vetoes a number of  inconsist practices that currently bedevil image processing (for instance the use of Euclidean curvature, thee is no ``turning around'' in image space). Apart from the advantage of being consistent, the framework actually leads to simpler methods (for nominally the same tasks, only correct for a change) than currently practised today.

Each lepton and antilepton is modeled as a point in Lorentzian spacetime R^1,3, constrained to always move on curves with lightlike tangentvectors. This point may be charged (e, m, t) or neutral (n_e, n_m, n_t) and interacts with external electromagnetic and weak fields. A Lagrangian variational principle for the orbits is presented which is Lorentz invariant and does not depend on the choice of any particular orbit parameter. The Lagrangian is composed of a free part depending on the curvature of the orbit only and of two additional terms which generate the interactions with the electromagnetic and weak external sources. Self-interactions are not yet considered.

A resolution of the tangential lightcone restriction on the group Spin(3) raises the dimension of the configuration space by two additional internal degrees of freedom from four to six. Furthermore, it implies via a three-valued Legendre transform three different sets of canonical equations of motion in complete agreement with the number of lepton families.

The equations of motion are solved in general in the case that external fields are absent. Particular solutions are discussed for the case of an electron coupled minimally to an external magnetic field of arbitrary strength.

The observation is, that geometrical and physical properties of the solutions sensitively depend on the spacetime structure of the momentum vector associated by the variational principle to an orbit. Only timelike momenta allow to assign a restmass to lightlike trajectories. In that case however, lightlike trajectories completely may account for the wave-particle duality of matter and display a spatial spread in accord with the momentum-position uncertainty principle.

This talk will seek to review the applications of Geometric Algebra (GA) in a number of areas of physics. These will include:

(i) Application of the multiparticle spacetime algebra (MSTA) to wave equations for particles of both integer and non-integer spin, highlighting links with supersymmetry and twistors. Linked with this will be a discussion of the uses of the conformal modelof space and spacetime in geometry and physics, including new applications to non-Euclidean and projective geometry.

(ii) The uses of a GA approach to electromagnetism for performing rigorous diffraction calculations and characterizing the scattering properties of complex conducting bodies.

(iii) The bound states and scattering properties of fermions interacting with black holes, as an illustration of a gauge theory approach to gravity, and of the usefulness of GA techniques in quantum mechanics calculations.

Although somewhat disparate topics it is hoped to give a flavour of the unifications in understanding and techniques which a GA approach brings.

Quantum group methods lead to a general formulation of electromagnetism and Riemannian geometry that works on classical, q-deformed and also discrete spaces.  We focus on recent models of finite noncommutative spacetime possible when q is a root of unity. We also explain how Clifford algebras and octonions could  equally well be viewed in our formalism as coordinates of a finite noncommutative geometry. If time permits, we also discuss the role and experimental testability of such models in quantum gravity and particle physics. 

The aim of this talk is to discuss recent progress in the general area of boundary value problems for first-order elliptic operators, under minimal smoothness assumptions. The setting is that of variable coefficient Dirac operators on Lipschitz subdomains of Riemannian manifolds. For example, we prove that the Plemelj-Calderón-Seeley-Bojarski splittings of (boundary) functions into traces of `inner' and `outer' monogenics always lead to problems of finite index. We also consider Szegö projections and the corresponding decompositions, in the context of Hardy spaces, and explore connections with Maxwell's equations. 

I will introduce a class of non-compact Riemannian manifolds, called "manifolds with a uniform structure at infinity," a class that includes, for example, manifolds with cylindrical ends. Several results on the analysis on manifolds with cylindrical ends extend to manifolds with a uniform structure at infinity. I will present some of these results, concentrating on the results about Dirac operators on this class of spaces.

Sweedler in his monograph [1969, Chapter XII] show that Grassmann algebra is Hopf algebra, and this was generalized by Woronowicz in 1989 to any braid. Similarly the Clifford algebra (of invertible tensor, bilinear form), has the companion Clifford cogebra and vice versa. Clifford algebra and Clifford cogebra, both together, form the Clifford convolution that does not necessarily possess an antipode. The traditional view of Clifford algebra (ignoring Clifford cogebra) must be revised. The Clifford convolution gives rise to some fundamental open problems. We will present some open problems (what are the abstract axioms?, etc), and as well several applications of Cliford cogebra in physics: Dirac operator for cogebra, addition of linear momenta, spin connection from Clifford cogebra, boson from Clifford cogebra, Lie cogebra from Clifford cogebra, Clifford convolution in QFT - renormalization, Dirac operator for tangent bundle, etc. The best reference is the H\"abilitat Thesis by Bertfried Fauser [2002].

We prove the Paley-Wiener Theorem in the Clifford algebra setting.  The work is based on use of Alan McIntosh's  extension of the exponential function of two variables, and accordingly his monogenic extension formula alternative to the CK method. As an application we derive the corresponding result for conjugate harmonic functions.

The most known form of the uncertainty relation is the Heisenberg principle, where the uncertainty of position (and momentum) is measured by mean square deviation from the mean value. It says that the product of the position and momentum uncertainties must be larger than the half Planck constant. I am going to present the the classical proof of this inequality. Next I will show that the Heisenberg principle follows from the so called entropic uncertainty relation found by I. Bialynicki Birula and J. Mycielski. The proof of the latter relation is based on a sophisticated result concerning the norm of the Fourier transform on real line considered as mapping from L^p to L^q (1/p+1/q=1). Replacing real line by an arbitrary locally compact group (or by a quantum group) we obtain a number of hypothetical relations. In most cases the proofs of these relations are  open problems. Although the analog of the Young-Hausdorff theorem saying that the Fourier transform is a contraction from L^p to L^q works for quantum groups, the precise value of its norm is not known. The constant appearing in the entropic uncertainty relation is an important invariant describing quantum groups. We expect that for locally compact groups it is determined by the so called uncompact dimension of the group.

As a mathematical theory per se, noncommutative geometry (NCG) is by now well established. Over the past dozen years, its progress has been crucially influenced by quantum physics: we briefly review this development.

The Standard Model of fundamental interactions, with its central role for the Dirac operator, led to several formulations culminating in the concept of a real spectral triple. String theory then came into contact with NCG, leading to an emphasis on Moyal-like algebras and formulations of quantum field theory on noncommutative spaces. Hopf algebras have yielded an unexpected link between the noncommutative geometry of foliations and perturbative quantum field theory.

The quest for a suitable foundation of quantum gravity continues to promote fruitful ideas, among them the spectral action principle and the search for a better understanding of "noncommutative spaces".