Sobolev and Schwartz: Two Fates and Two Fames

This is a brief overview of the lives and contributions of S. L. Sobolev and L. Schwartz, the cofounders of distribution theory. In the history of mathematics there are quite a few persons whom we prefer to recollect in pairs. Listed among them are Euclid and Diophant, Newton and Leibniz, Bolyai and Lobachevskĭı, Hilbert and Poincaré, as well as Bourbaki and Arnold. In this series we enroll Sobolev and Schwartz who are inseparable from one of the most brilliant discoveries of the twentieth century, the theory of generalized functions or distribution theory, providing a revolutionary new approach to partial differential equations. The most vibrant and lasting achievements of mathematics reside in formulas and lists. There are pivotal distinctions between lists and formulas. The former deposit that which was open for us. The lists of platonic solids, elementary catastrophes, and finite simple groups are next of kin to the Almagest and herbaria. They are the objects of admiration, tremendous and awe-struck. The article of the craft of mathematics is a formula. Each formula enters into life as an instance of materialization of mathematical creativity. No formula serves only the purpose it was intended to. In part, any formula is reminiscent of household appliances, toys, or software. It is a very rare event that somebody reads the user’s guide of a new TV set or the manual for running a new computer program. Usually everyone utilizes his or her new gadgets experimentally by pressing whatever keys and switches. In much the same way we handle formulas. We painstakingly “twist and turn” them, audaciously insert new parameters, willfully interpret symbols, and so on. Mathematics is the craft of formulas and the art of calculus. If someone considers this claim as feeble and incomplete, to remind is in order that, logically speaking, set theory is just an instance of the first order predicate calculus. Distribution theory has become the calculus of today. Of such a scale and scope is the scientific discovery by Sobolev and Schwartz. Date: February 4, 2008. Partly printed in [1] with unauthorized omissions. The author thanks V. A. Aleksandrov and V. P. Golubyatnikov who helped him in better understanding of French sources. The author is especially grateful to Yu. L. Ershov who was persistent in inviting the author to make a talk at the special session of the Academic Council of the Sobolev Institute on October 14, 2003. The present article bases on this talk. The author acknowledges the subtle and deep comments of V. I. Arnold and V. S. Vladimirov on the preprint of a draft of the talk which led to many improvements.