| 7:12p |
Математическое, разное
1. Редко какой из моих тестов, по нынешним временам, обходится без ссылки на книжку про квадратичные алгебры. Не обошелся и текст про контрагерентные копучки. При чем тут, казалось бы? А при том, что вялые пучки (скажем, на нетеровых аффинных схемах) -- это такие дистрибутивные решетки подмодулей. Ковялые копучки, тем более.
2. Открыл следующий паттерн рассуждения. Докажем сначала утверждение для полуотделимых (или полуотделимых квазикомпактных) схем, пользуясь аффинными покрытиями и аффинностью пересечений. Потом тем же способом докажем то же утверждение для произвольных (или квазиотделимых квазикомпактных) схем, пользуясь полуотделимыми покрытиями и полуотделимостью пересечений. |
| 9:24p |
Рота о Грассмане
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It is not surprising that Grassmann was not entirely welcome among mathematicians. Anyone coming up with a new definition is likely to make enemies. New ideas are always unwelcome and regarded as intrusive. Grassmann made a number of enemies, and the animosity against his great definition has not entirely died out.
The reactions against Grassmann make a humorous chapter in the history of mathematics. For example, Professor Pringsheim, dean of German mathematicians and author of over one hundred substantial papers on the theory of infinite series, both convergent and divergent, kept insisting that Grassmann should be doing something relevant instead of writing up his maniacal ravings. "Why doesn't he do something useful, like discovering some new criterion for the convergence of infinite series!" Pringsheim asserted, with all the authority that his position conferred.
The invariant theory community led by Clebsch and Gordan also loudly protested Grassmann's work as pointless since it did not contribute one single result to the invariant theory of binary forms. They were dead wrong, but would not be proved so for another fifty years.
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Evil tongues whispered that there was really nothing new in Grassmann's exterior algebra, that it was just a mixture of Moebius' barycentric calculus, Pluecker's coordinates, and von Staudt's algebra of throws. The standard objection was expressed by the notorious question, "What can you prove with the exterior algebra that you cannot prove without it?" Whenever you hear this question raised about some new piece of mathematics, be assured that you are likely to be in the presence of something important. In my time, I have heard it repeated for random variables, Laurent Schwartz' theory of distributions, ideles and Grothendieck's schemes, to mention only a few. A proper retort might be: "You are right. There is nothing in yesterday's mathematics that you can prove with exterior algebra that could not also be proved without it. Exterior algebra is not meant to prove old facts, it is meant to disclose a new world. Disclosing new worlds is as worthwile a mathematical enterprise as proving old conjectures."
The first mathematician to understand the importance of exterior algebra was Peano who published a beautiful short introduction to the subject. At the time Peano was teaching at the Pinerolo military school. His audiences for what must have been beautiful lectures on exterior algebra consisted of Italian cavalry officers and cadets. No one living beyound the Alps read Peano's book until Elie Cartan came along. Three hundred copies were printed of the first and only edition.
It took almost one hundred years before mathematicians realized the greatness of Grassmann's discovery. Such is the fate meted out to mathematicians who make their living on definitions.
(G.-C. Rota, "Hermann Grassmann and Exterior Algebra", Indiscrete Thoughts, p.46-48.) |