I'd be grateful if someone could elaborate on Abhyankar's remarks below, esp. precisely how the mathematical principles are realized physically (alas, my physics knowledge has atrophied) "On the other hand, Riemann's approach appears to have been existential: know that the formulas exist! Many of his existence arguments were based on "Dirichlet's principle," the genesis of which can be expressed by saying something like this: the real and imaginary parts of an analytic function of a complex variable satisfy the same sort of partial differential equation as a gravitational potential or an electric potential (or a fluid flow or heat conduction ...); so the existence of analytic functions or differentials with prescribed boundary behaviour or assigned singularities can be adduced from physical considerations. To carry the analogy further (or backwards), once could say that the mathematical formula "number of zeros of a function = number of its poles" corresponds to the physical fact that there is no gravitational pull inside a hollow shell, or that there is no electric intensity inside a hollow charged conductor. Riemann's approach was furthered by Hilbert who put the Dirichlet principle on firmer grounds." The above is excerpted from p.411 of Abhyankar's classic Monthly paper [1], for which he was awarded the Chauvenet Prize [0] by the MAA (for outstanding expository article on a mathematical topic). The award is well-deserved. If you haven't yet read his ramblings I highly recommend doing so. Below is its jstor link and Math Review. See also his introductory textbook Algebraic Geometry for Scientists and Engineers, AMS, 1990, where he says "I have tried to tell the story of algebraic geometry and to bring out the poetry in it". If only we had more such mathematical poets... -Bill Dubuque [0] http://www.maa.org/awards/chauvent.html ------------------------------------------------------------------------------ [1] 53 #5581 14-02 (14E15 32C45) Abhyankar, Shreeram S. Historical ramblings in algebraic geometry and related algebra. Amer. Math. Monthly 83 (1976), no. 6, 409-448. http://links.jstor.org/sici?sici=0002-9890(197606/07)83:6%3C409:HRIAGA%3E ------------------------------------------------------------------------------ The algebra that is used to formulate ideas and to prove theorems in algebraic geometry comes in several different varieties. The author distinguishes three of these, namely, "high-school algebra" (polynomials, power series), "college algebra" (rings, ideals, fields) and "university algebra" (functors, homological algebra). The fundamental thesis of this entertaining and most readable paper is, in the author's words, "The method of high-school algebra is powerful, beautiful and accessible. So let us not be overwhelmed by the groups-ring-fields or the functorial arrows of the other two algebras and thereby lose sight of the power of the explicit algorithmic processes given to us by Newton, Tschirnhausen, Kronecker, and Sylvester." The author takes as his theme the genus of a plane algebraic curve (for example over the complex numbers) and gives several interpretations in terms of the various algebras and of analysis, sketching proofs of the equivalence of the various definitions. There are also four short illustrations of the author's fundamental thesis, drawn from his own experience. (In particular he attributes the gradual progress he has made with the resolution of singularities in nonzero characteristic to a systematic weeding-out of college algebra, the better for the high-school algebra to flourish!) The definitions of genus are briefly these: (1) function-theoretic: "2g-2 = number of zeros of a differential form - number of poles"; (2) topological: "number of handles on the Riemann surface"; (3) algebraic: (N-1)(N-2)/2 - sum \d(P) for a curve of order N, where \d(P) is the number of double points concentrated at P, which varies over the curve -- four interpretations of \d(P) are used and reconciled; (4) dimension of the Jacobian variety; (5) (mentioned only) via cohomology. In the course of sketching proofs of the equivalence of the various definitions of genus -- and in particular of the various definitions of \d -- the author gives rapid introductions to a number of topics, such as abelian integrals, resultants and discriminants, conductors, blowing up singularities by means of quadratic transformations, Bezout's theorem, Puiseux expansions, valuations and the monodromy group. The bibliography of 81 items ranges from Newton ("truly the father of us all"), through Euler and Weierstrass ("a most distinguished high-school algebraist") to Salmon, Cayley, Chrystal and Macaulay ("a grand master of high-school algebra"), apart from the more predictable modern references. Reviewed by Peter Giblin