,h In the XIX century, mathematical physics continued to be the main source of new partial differential equations and ofproblems involving them. The study ofLaplace's equation and ofthe wave equation had assumed a more systematic nature. In the beginning of the century, Fourier added the heat equation to the aforementioned two. Marvellous progress in obtaining precise solution repre- sentation formulas is connected with Poisson, who obtained formulas for the solution of the Dirichlet problem in a disc, for the solution of the Cauchy problems for the heat equation, and for the three-dimensional wave equation. The physical setting ofthe problem led to the gradual replacement ofthe search for a general solution by the study of boundary value problems, which arose naturallyfrom the physics ofthe problem. Among these, theCauchy problem was of utmost importance. Only in the context of first order equations, the original quest for general integralsjustified itself. Here again the first steps are connected with the names of D'Alembert and Euler; the theory was being intensively 1h developed all through the XIX century, and was brought to an astounding completeness through the efforts ofHamilton, Jacobi, Frobenius, and E. Cartan. In terms of concrete equations, the studies in general rarely concerned equa- tions of higher than second order, and at most in three variables. Classification 'h ofsecond orderequations was undertaken in the second halfofthe XIX century (by Du Bois-Raymond). An increase in the number of variables was not sanc- tioned by applications, and led to the little understood ultra-hyperbolic case.

One service mathematics has rendered the 'Et moi, ..., si j'avait su comment cn rcvenir, human race. It has put common sense back. je n'y serais point aile.' where it bdongs, on the topmost shelf neAt Jules Verne to the dusty canister labelled 'discarded non* sense'. The series is divergent; therefore we may be Eric T. Bdl able to do something with it. O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com- puter science .. :; 'One service category theory has rendered mathematics ...'. All a,rguably true. And all statements obtainable this way form part of the raison d'etre of this series.

I.M. Gelfand (1913 - 2009), one of the world's leading contemporary mathematicians, largely determined the modern view of functional analysis with its numerous relations to other branches of mathematics, including mathematical physics, algebra, topology, differential geometry and analysis. In this three-volume Collected Papers Gelfand presents a representative sample of his work. Gelfand's research led to the development of remarkable mathematical theories - most of which are now classics - in the field of Banach algebras, infinite-dimensional representations of Lie groups, the inverse Sturm-Liouville problem, cohomology of infinite-dimensional Lie algebras, integral geometry, generalized functions and general hypergeometric functions. The corresponding papers form the major part of the collection. Some articles on numerical methods and cybernetics as well as a few on biology are also included. A substantial number of the papers have been translated into English especially for this edition. The collection is rounded off by an extensive bibliography with almost 500 references. Gelfand's Collected Papers will be a great stimulus, especially for the younger generation, and will provide a strong incentive to researchers.