G. Floystad: The Exterior Algebra and Central Notions in Mathematics

Gunnar Fløystad, The Exterior Algebra and Central Notions in Mathematics, Notices of the American Mathematical Society, Vol. 62, Numb. 4, pp. 364-371, DOI: http://dx.doi.org/10.1090/noti1234
Download: http://www.ams.org/notices/201504/rnoti-p364.pdf

Dedicated to Stein Arild Strømme (1951–2014)
“The neglect of the exterior algebra is the
mathematical tragedy of our century.”
— Gian-Carlo Rota, Indiscrete Thoughts (1997)

“This note surveys how the exterior algebra and deformations or quotients of it capture essences of five domains in mathematics:
* Combinatorics
* Mathematical physics
* Topology
* Algebraic geometry
* Lie theory
The exterior algebra originated in the work of Hermann Grassmann (1809–1877) in his book Ausdehnungslehre from 1844, and the thoroughly revised 1862 version, which now exists in an English translation [20] from 2000. Grassmann worked as a professor at the gymnasium in Stettin, then Germany. Partly because Grassmann was an original thinker and maybe partly because his education had not focused much on mathematics, the first edition of his book had a more philosophical than mathematical form and therefore gained little influence in the mathematical community. The second (1862) version was strictly mathematical. Nevertheless, it also gained little influence, perhaps because it had swung too far to the other side and was scarce of motivation. Over four hundred pages it developed the exterior and interior product and the somewhat lesser-known regressive product on the exterior algebra, which intuitively corresponds to intersection of linear spaces. It relates this to geometry and it also shows how analysis may be extended to functions of extensive quantities. Only in the last two decades of the 1800s did publications inspired by Grassmann’s work achieve a certain mass. It may have been with some regret that Grassmann in his second version had an exclusively mathematical form, since he in the foreword says “[extension theory] is not simply one among the other branches of mathematics, such as algebra, combination theory or function theory, bur rather surpasses them, in that all fundamental elements are unified under this branch, which thus as it were forms the keystone of the entire structure of mathematics.”

The present note indicates that he was not quite off the mark here. We do not make any further connections to Grassmann’s original presentation, but rather present the exterior algebra in an entirely modern setting. For more on the historical context of Grassmann, see the excellent history of vector analysis [7], as well as proceedings from conferences on Grassmann’s many-faceted legacy [41] and [38]. The last fifteen years have also seen a flurry of books advocating the very effective use of the exterior algebra and its derivation, the Clifford algebra, in physics, engineering, and computer science. In the last section we report on this.” (pages 364-365)

Source: Gunnar Fløystad, The Exterior Algebra and Central Notions in Mathematics, Notices of the American Mathematical Society, Vol. 62, Numb. 4, pp. 364-371, DOI: http://dx.doi.org/10.1090/noti1234, Download: http://www.ams.org/notices/201504/rnoti-p364.pdf


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