**Speaker, Title,
Abstract** |

**Helga Baum, Humboldt Universität zu Berlin,***"Conformally
invariant spinor field equations and special geometric structures"*
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. |

**Carlos A. Berenstein, University of Maryland,***"Problems
of Pompeiu and Morera Type in Quaternionic Spaces"*
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. |

**Michael Eastwood, University of Adelaide,***"Symmetry
and Differential Invariants", *
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. |

**Bertfried Fauser, Universität Konstanz,***"Grade
Free Product Formulæ from Graßmann Hopf Algebras"*
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. |

**Alexander J. Hahn, University of Notre Dame,***"The
Clifford Algebra in the Theories of Algebras, Quadratic Forms, and Classical
Groups",*
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. |

**Jacques Helmstetter, Université
de Grenoble I,** *"Lipschitz's Methods of 1886 Applied to Symplectic
Clifford Algebras"*
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* e*Cl(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. |

**David Hestenes, Arizona State University,***"The
Development of Geometric Algebra and Calculus -- A Personal Retrospective"*
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. |

**Tadeusz Iwaniec, Syracuse University,***"A
Study of Nonlinear PDE's via Exterior Algebra"*
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 **R**^{n}, 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. |

**Palle Jorgensen, University of Iowa,***"Quantum
Wavelet Algorithms: Factorization, and Use of Clifford Groups and Algebras"*
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. |

**Jan J. Koenderink, Universiteit Utrecht,***"Geometry
of Image Space"*
``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. |

**Heinz Krüger, Universität Kaiserslautern,***"The
Lepton as a Lightlike Point"*
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. |

**Anthony Lasenby, Cambridge University,***"Applications
of Geometric Algebra in Electromagnetism, Quantum Mechanics and Gravity"*
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. |

**Shahn Majid, Queen Mary, University of London,***"Electromagnetism
and Gravity on Finite Noncommutative Spaces"*
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. |

**Marius Mitrea, University of Missouri,***"Decomposition
Theorems for General Dirac Operators on Nonsmooth Manifolds"*
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. |

**Victor Nistor, Pennsylvania State University,***"Dirac
Operators and Manifolds with a Uniform Structure at Infinity"*
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. |

**Zbigniew Oziewicz, UNAM,** *"Clifford
Coalgebra"*
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]. |

**Tao Qian, University of Macau,** *"Paley-Wiener
Theorem in the Clifford algebra setting" *
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. |

**S.L. Woronowicz, Warsaw University,***"Entropy
Uncertainty Relations in Quantum Groups"*
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. |

**Joseph C. Varilly, Universidad de Costa Rica,***"The
Interface of Noncommutative Geometry and Physics"*
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". |