Huomatkaa, että tämä on Pertti Louneston sivuista peilattu kopio. Pertin perikunta on ystävällisesti antanut luvan pitää näitä sivuja esillä. Näiden sivustojen URL on
Pertti Lounesto

Pertti Lounesto

My research focuses on algebras emerging from problems in geometry and physics, called Clifford algebras. In physics, the concept of Clifford algebra, as such or in a disguise, is a necessity in the description of electron spin, because spinors cannot be constructed by tensorial methods, in terms of exterior powers of the vector space. In geometry, information about orientation of subspaces can be encoded in simple multivectors, which can be added and multiplied. Physicists are familiar with this tool in the special case of one-dimensional subspaces, which they manipulate by vectors (not by projection operators, which lose information about orientations).
I am also interested in misconceptions of research mathematicians, while they enter unexplored domains.
This Spring 2002, I delivered lectures on Clifford algebras.
Clifford's geometric algebras have applications in robotics, computer vision, image processing, signal processing and space dynamics.

I have authored/co-edited the following books:

Clifford Algebras and Spinors Pertti Lounesto:
Clifford Algebras and Spinors,
Cambridge University Press, 1997, 306 pages.
Second edition, 2001, 338 pages.
Changes between editions 1/2. Reviews in English, German.
Clifford Algebras with Numeric and Symbolic Computations Rafal Ablamowicz, Pertti Lounesto, Josep Parra (eds.):
Clifford Algebras with Numeric and Symbolic Computations,
Birkhäuser, 1996, 322 pages.
Software to accompany the book. Review by Dongming Wang.
Clifford Numbers and Spinors; with Riesz's private lectures to E. Folke Bolinder and a historical review by Pertti Lounesto Marcel Riesz (lecture notes, delivered in 1957-58), E. Folke Bolinder, Pertti Lounesto (eds.):
Clifford Numbers and Spinors; with Riesz's private lectures to E. Folke Bolinder and a historical review by Pertti Lounesto,
Kluwer, 1993, 241 pages. Review by S. Rogosin.
Clifford Algebras and Spinor Structure: A special volume dedicated to the memory of Albert Crumeyrolle, 1919-1992 Rafal Ablamowicz, Pertti Lounesto (eds.):
Clifford Algebras and Spinor Structures: A special volume dedicated to the memory of Albert Crumeyrolle (1919-1992),
Kluwer, 1995, 421 pages.

CLICAL User Manual: Complex Number, Vector Space and Clifford Algebra Calculator for MS-DOS Personal Computers Pertti Lounesto, Risto Mikkola, Vesa Vierros:
CLICAL User Manual: Complex Number, Vector Space and Clifford Algebra Calculator for MS-DOS Personal Computers,
Institute of Mathematics, Helsinki University of Technology, 1987, 80 pages.
CLICAL is free to download.

Comparison to other books on Clifford algebras and spinors:

I.R. Porteous: Clifford algebras and the classical groups. Cambridge UP, 1995.
J. Snygg: Clifford algebra, a computational tool for physicists. Oxford UP, 1997.
D. Hestenes, G. Sobczyk: Clifford algebra to geometric calculus. Reidel, 1984, 1987.
D. Hestenes: New foundations for classical mechanics. Reidel, 1986, 1987, (2nd ed.) 1999.
B. Jancewicz: Multivectors and Clifford algebra in electrodynamics. World Scientific, 1988.
W.E. Baylis: Electrodynamics: a modern geometric approach. Birkhäuser, 1999.
W.E. Baylis (ed.): Clifford (geometric) algebras with applications to physics, mathematics, and engineering. Springer, 1996.
I.M. Benn, R.W. Tucker: An introduction to spinors and geometry with applications in physics. Adam Hilger, 1987.
F.R. Harvey: Spinors and calibrations. Academic Press, 1990.
J. Gilbert, M. Murray: Clifford algebras and Dirac operators in harmonic analysis, CUP, 1991.
G. Sommer: Geometric computing with Clifford algebras; theoretical foundations and applications in computer vision and robotics. Springer, 2001.

See errata of the books. See commentary of a paper, where the authors prove a conjecture of one of the above authors, although the conjecture was a corollory of Ado's theorem.

On-line tutorials, lecture courses and books on Clifford algebras, etc.:

Tony Smith: What are Clifford algebras and spinors?
Bill Pezzaglia: International Clifford Algebra Society.
Flow chart on the history of Clifford's geometric algebras.
Perttu Puska: Electromagnetism formulated in Clifford algebra.
Kirby Urner: Clifford Algebra for high schoolers?
Tennessee TU: 6th Conference on Clifford Algebras, May 18-25, 2002.
Courses on Clifford algebras applied to analysis and physics.
Octonions multiplied, explained and applied.
Clifford algebra might make its curriculum debut in high schools by replacing the cross product, in undergraduate courses by replacing rotation matrices, and in graduate courses by condensing the Maxwell equations in a vacuum into a single equation in terms of Cl3.

Hamilton's quaternions are used to represent 3D rotations; quaternions enable smooth interpolation of rotation matrices in computer graphics/ games, quaternions are used to avoid a singularity, the gimbal lock in flight simulation and spacecraft navigation.

Home-pages of researchers on Clifford's geometric algebra:

Algebra: Rafal Ablamowicz, Bertfried Fauser, Alexander J. Hahn, Jacques Helmstetter, Ian R. Porteous. Analysis: Swanhild Bernstein, Freddy Brackx, Richard Delanghe, Sirkka-Liisa Eriksson-Bique, Guy Laville, Heinz Leutwiler, Enrique Ramírez de Arellano, John Ryan, Baruch Schneider, Frank Sommen, Wolfgang Sprößig. Celestial Mechanics: Jan Vrbik. Chemistry: Janne Pesonen. Computer Science: Adrian Lawrence, Stephen Mann, Gerik Scheuermann, Dongming Wang. Computer Vision: Christian Perwass, Bodo Rosenhahn. Electromagnetism: William E. Baylis, Bernard Jancewicz, Perttu Puska. Geometry: Christian Bär, Helga Baum, Sven Buchholz, Jan Cnops, Oliver Conradt, Larry Grove, Garret Sobczyk, Andrzej Trautman, Charles H.T. Wang. Physics: Timaeus Bouma, Philip Charlton, Robert Coquereaux, Chris Doran, Ramon González Calvet, Stephen Gull, David Hestenes, Heinz Krüger, Jaime Keller, Anthony Lasenby, Antony Lewis, Garrett Lisi, Nikolai Marchuk, Dennis Marks, James M. Nester, Josep Parra, William M. Pezzaglia, Charles Poole, Patrick Reany, Waldyr Rodrigues, Nikos A. Salingaros, Thalanayar S. Santhanam, John Schutz, Greg Trayling, Jose Vargas, Jayme Vaz, José Ricardo Zeni. Robotics: Eduardo Bayro-Corrochano, Curtis Collins, Leo Dorst, Seamus Garvey, Hongbo Li, Michael McCarthy, Allan McRobie, Jon Selig, Gerald Sommer. Signal Processing: Thomas Bülow, Michael Felsberg, Joan Lasenby.

Ten most influential mathematicians in Clifford algebras:

William R. Hamilton (1805-1865) invented quaternions, on Monday Oct 16, 1843.
Hermann Grassmann (1809-1877) introduced the exterior algebra, 1844/1862.
William K. Clifford (1845-1879) classified his geometric algebras (4 cases), 1878/1882.
Rudolf Lipschitz (1832-1903) represented rotations by spin groups, 1880/1886.
K. Theodor Vahlen (1869-1945): multiplication rule by binary index sets, 1897,
 .  Möbius transformations of Rn by 2x2-matrices with entries in Cln, 1902.
Elie Cartan (1869-1951): periodicity of 8, 1908, and triality of Spin(8), 1925.
Ernst Witt (1911-1991): Clifford algebra Cl(Q) of a quadratic form Q, 1937.
Claude Chevalley (1909-1984) extended the ground field to characteristic 2, 1954.
Marcel Riesz (1886-1969): exterior product via the Clifford product, 1958,
 .  x^u = ½(xu + (-1)kux)   for   x in Rn and u in /\kRn \subset Cln+ or Cln-.
Ian R. Porteous (1930- ) classified the scalar products of spinors (32 cases), 1969.
Jacques Helmstetter (1943- ) developed symplectic Clifford algebras, 1982/2002.
W.R. Hamilton H. Grassmann W.K. Clifford R. Lipschitz E. Cartan E. Witt C. Chevalley M. Riesz J. Helmstetter

Pertti.Lounesto@helsinka.fa (a -> i for e-mail)
Department of Mathematics
University of Helsinki
FIN-00014 Helsinki, Finland
My Erdös number is 3.