Introduction All conics in the plane make up a P 5 as there are six quadratic monomials in three ... more Introduction All conics in the plane make up a P 5 as there are six quadratic monomials in three variables. The singular conics form a cubic hypersurface referred to as the discriminant, whose singular locus consists of all double lines and make up 2dimensional subvariety of the discriminant, in fact isomorphic with P 2 via the Veronese embedding. In fact there is a 1-1correspondence between trinary quadratic forms and symmetric 3× 3 matrices. The singular ones correspond to singular matrices, given by the determinant, and hence by a cubic equation. It can be parametrized by P 2×P 2 as follows. (x0, x1, x2)× (y0, y1, y2) 7→ x0y0 x0y1+x1y0 2 x0y2+x2y0 2 x0y1+x1y0 2 x1y1 x1y2+x2y1 2 x0y2+x2y0 2 x1y2+x2y1 2 x2y2
All cubic Galois extensions of rational numbers are described by canonical minimal polynomials, w... more All cubic Galois extensions of rational numbers are described by canonical minimal polynomials, which can be computed using a "sieving" algorithm.
THE CHERN-INVARIANTS CT, x of minimal compact complex surfaces of general type satisfy certain we... more THE CHERN-INVARIANTS CT, x of minimal compact complex surfaces of general type satisfy certain well-known inequalities. They are both strictly positive and furthermore (1977), 1789-1799.
REVIEWED BY ULF PERSSON W W hen I was in my early teen years, my father suggested that I read Men... more REVIEWED BY ULF PERSSON W W hen I was in my early teen years, my father suggested that I read Men of Mathematics [1] by Eric Temple Bell. I did so and was enthralled. Few books have ever had a more lasting influence on me. The book provided me with a new set of heroes-intellectual heroes-provoking and focusing a slumbering ambition to choose, for better or for worse, a mathematical career. One should indeed be careful in recommending books to impressionable adolescents. What was so great about Bell's book? It provided potted biographies of mathematicians who were presented as fascinating individuals. Why were they fascinating? Because their lives were intertwined with mathematics, the intriguing aspects of which had only recently been indicated to me, and of which Bell provided further tantalizing glimpses. But, to be honest, the mathematical education provided by his book was subservient to the human stories told. Bell's book, more or less unique in its time, has been severely criticized on many grounds, including biographical accuracy and politically incorrect attitudes, but that I consider beside the point. Bell stops at the beginning of the twentieth century. Wouldn't it be great to have a sequel leading us into the third millennium? Does the book under review, with its promising title, fulfill this long-awaited hope? No, it does not. Bell was a real writer. His work was inspired and imbued with a single unifying author's voice. The present work is a hodgepodge of contributions by amateurs ranging from the admittedly interesting to the excruciatingly mediocre. It reads like what it is, a collection culled from independent contributions to a journal. This book starts with a discussion of David Hilbert and his 23 problems, then moves on to foundations of mathematics through Bertrand Russell, Kurt Gödel, and Alan Turing. The versatile mathematicians Andrey Kolmogorov and John von Neumann appear as bridges from foundations to mathematical physics, the latter represented by Paul Dirac, Vladimir Arnold, and Michael Atiyah. Pure mathematics is represented by G. H. Hardy, Emmy Noether, Alexander Grothendieck, Enrico Bombieri, and the relatively unknown William Lawvere on the algebraic and number-theoretic side, René Thom and Stephen Smale on the topological side, and Laurent Schwartz on analysis. To which should be added a piece on Bourbaki. Italian mathematicians are represented by the classical algebraic geometers Guido Castelnuovo, Federigo Enriques, and Francesco Severi, together with the more
Introduction All conics in the plane make up a P 5 as there are six quadratic monomials in three ... more Introduction All conics in the plane make up a P 5 as there are six quadratic monomials in three variables. The singular conics form a cubic hypersurface referred to as the discriminant, whose singular locus consists of all double lines and make up 2dimensional subvariety of the discriminant, in fact isomorphic with P 2 via the Veronese embedding. In fact there is a 1-1correspondence between trinary quadratic forms and symmetric 3× 3 matrices. The singular ones correspond to singular matrices, given by the determinant, and hence by a cubic equation. It can be parametrized by P 2×P 2 as follows. (x0, x1, x2)× (y0, y1, y2) 7→ x0y0 x0y1+x1y0 2 x0y2+x2y0 2 x0y1+x1y0 2 x1y1 x1y2+x2y1 2 x0y2+x2y0 2 x1y2+x2y1 2 x2y2
All cubic Galois extensions of rational numbers are described by canonical minimal polynomials, w... more All cubic Galois extensions of rational numbers are described by canonical minimal polynomials, which can be computed using a "sieving" algorithm.
THE CHERN-INVARIANTS CT, x of minimal compact complex surfaces of general type satisfy certain we... more THE CHERN-INVARIANTS CT, x of minimal compact complex surfaces of general type satisfy certain well-known inequalities. They are both strictly positive and furthermore (1977), 1789-1799.
REVIEWED BY ULF PERSSON W W hen I was in my early teen years, my father suggested that I read Men... more REVIEWED BY ULF PERSSON W W hen I was in my early teen years, my father suggested that I read Men of Mathematics [1] by Eric Temple Bell. I did so and was enthralled. Few books have ever had a more lasting influence on me. The book provided me with a new set of heroes-intellectual heroes-provoking and focusing a slumbering ambition to choose, for better or for worse, a mathematical career. One should indeed be careful in recommending books to impressionable adolescents. What was so great about Bell's book? It provided potted biographies of mathematicians who were presented as fascinating individuals. Why were they fascinating? Because their lives were intertwined with mathematics, the intriguing aspects of which had only recently been indicated to me, and of which Bell provided further tantalizing glimpses. But, to be honest, the mathematical education provided by his book was subservient to the human stories told. Bell's book, more or less unique in its time, has been severely criticized on many grounds, including biographical accuracy and politically incorrect attitudes, but that I consider beside the point. Bell stops at the beginning of the twentieth century. Wouldn't it be great to have a sequel leading us into the third millennium? Does the book under review, with its promising title, fulfill this long-awaited hope? No, it does not. Bell was a real writer. His work was inspired and imbued with a single unifying author's voice. The present work is a hodgepodge of contributions by amateurs ranging from the admittedly interesting to the excruciatingly mediocre. It reads like what it is, a collection culled from independent contributions to a journal. This book starts with a discussion of David Hilbert and his 23 problems, then moves on to foundations of mathematics through Bertrand Russell, Kurt Gödel, and Alan Turing. The versatile mathematicians Andrey Kolmogorov and John von Neumann appear as bridges from foundations to mathematical physics, the latter represented by Paul Dirac, Vladimir Arnold, and Michael Atiyah. Pure mathematics is represented by G. H. Hardy, Emmy Noether, Alexander Grothendieck, Enrico Bombieri, and the relatively unknown William Lawvere on the algebraic and number-theoretic side, René Thom and Stephen Smale on the topological side, and Laurent Schwartz on analysis. To which should be added a piece on Bourbaki. Italian mathematicians are represented by the classical algebraic geometers Guido Castelnuovo, Federigo Enriques, and Francesco Severi, together with the more
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