By Zeuthen Symposium, Steven L. Kleiman, H. G. Zeuthen, Anders Thorup

1989 marked the a hundred and fiftieth anniversary of the delivery of the good Danish mathematician Hieronymus Georg Zeuthen. Zeuthen's identify is identified to each algebraic geometer due to his discovery of a easy invariant of surfaces. even if, he additionally did basic examine in intersection concept, enumerative geometry, and the projective geometry of curves and surfaces. Zeuthen's striking devotion to his topic, his attribute intensity, thoroughness, and readability of proposal, and his distinct and succinct writing variety are really inspiring.

During the previous ten years or so, algebraic geometers have reexamined Zeuthen's paintings, drawing from it suggestion and new instructions for improvement within the box. The 1989 Zeuthen Symposium, held in the summertime of 1989 on the Mathematical Institute of the collage of Copenhagen, supplied a old chance for mathematicians to assemble and view these parts in modern mathematical learn that have developed from Zeuthen's fruitful principles. This quantity, containing papers provided in the course of the symposium, in addition to others encouraged through it, illuminates a few presently energetic components of analysis in enumerative algebraic geometry

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Carbon's unique affinity for bonding with other ele­ ments causes it to form long chainlike molecules like those of amino acids, proteins, and DNA. In biology, too, many temporal phenomena are describ­ able in terms of rational numbers. Each plant and animal species, for instance, appears to be characterized by a unique number of chromosomes in its individual cells. And this chro­ mosome number, like the atomic number, is always a whole number. Despite this preeminence of rational numbers, science 38 Irrational Thinking does need irrational numbers.

For example, one of these constants, the speed of light, has been measured out to nine decimal places, and the digits have yet to show any pattern. 29979245 8 . ) Another constant is one that is descriptive of dynamic behavior at the atomic level. It is called the fine-structure constant, and there is no pattern to its digits even when measured out to ten decimal places. 0072 9 73 5 0 3 . ) In physics alone there are more than a dozen of these constants, which have been measured out to anywhere from a few to eleven decimal places, and not one of them has a pattern to its digits.

34 Irrational Thinking A few years after Dedekind's achievement, the German mathematician Georg Cantor made a discovery that would lay to rest any lingering skepticism toward the seemingly impossible constitution of the real number line. He confirmed that though there are an infinite number of rational numbers, as the Greeks had suspected, there are even more real num­ bers. This confirms that the real number line is indeed some­ how more continuous, more densely packed, than the ponderably continuous rational number line.

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