CAIROS: Clifford Algebras International Research Open Studies

The new institute Clifford Algebras International Research Open Studies (C.A.I.R.O.S.) has just opened in the

Laboratoire de Mathématiques Emile Picard
Institut de Mathématiques de Toulouse CNRS UMR 5219
Université Paul Sabatier


Coordination of CAIROS

Principal Coordinator: Pierre Anglès (Laboratoire de Mathématiques E. Picard, IMT, Université Paul Sabatier, Toulouse, France),
Deputy Coordinator: Arkadiusz Jadczyk – : (IMP, Castelsarrasin, France)

Coordination Committee: Rafal Ablamowicz ( Tennessee Technological University, Cookeville, U.S.A), Jacques Helmstetter (Université Joseph Fourier, Grenoble, France), Jaime Keller (Universidad Nacional Autonoma de Mexico, Mexico), Waldyr Rodrigues (University of Campinas, Brazil), Wolfgang Sprößig (TU Bergakademie Freiberg, Germany),

Plus a large number of corresponding members.

The “research axis” of CAIROS


W. K. Clifford: (1845-1879) was not only a genius mathematician but also a physicist and a philosopher.
(A) The Mathematician: He was the founder of n-ways algebras (now called geometrical algebras or simply Clifford algebras). He defined these algebras as tensor product of quaternionic algebras and their extensions. He followed and improved the work initiated by Hamilton and Grassmann. He studied geometrical representations of analytic functions, non-Euclidean geometries, the theory of elliptic spaces, algebraic forms and projective geometry. His mathematical papers have been published by Macmillan and are actually republished by University of Michigan Library.
(B) The Physicist: In his paper On Space-Theory of Matter, Clifford proposed to relate matter to propagating space curvature – the idea that has found its final shape in Eintsein’s theory of gravitation.
(C) The Philosopher: As a philosopher he introduced two important concepts: mind stuff and tribal self. All his philosophical papers are subordinate to his mathematical thoughts.

Research Program:

(A) Using Cliffordian methods study: the five types of exceptional simple Lie algebras and the corresponding associated Lie groups; the singular Lie algebra D4 associated to O(8) and Spin(8); simple exceptional Jordan algebras; exceptional projective Moufang planes and non-associative composition algebras and applications to the foundations of quantum mechanics, quantum computing and cosmological models.

(B) Applications of triple systems to the study of bounded symmetric domains; relations to conformal geometry over classical pseudo-Euclidean spaces..

(C) Conformal spinors and generalized twistors; application to prequantization and quantization;

(D) Study of other spinorial algebras (Clifford-Heisenberg and orthosymplectic ones).

(E) Dirac operators on manifolds and their generalizations; new topics about Clifford analysis.

(F) Applications of Clifford algebraic methods in physics and engineering, in particular: multidimensional field theoretical models, robotics, control, image processing.

(G) Epistemological heritage of William Kingdom Clifford.

The logo C.A.I.R.O.S has been chosen in reference to Kairos, one of the three Greek gods of time.

The reference section contains a convenient list of Clifford geometric algebra books


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