Hermann Minkowski Biography

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World of Mathematics on Hermann Minkowski

In spite of a relatively short career, Hermann Minkowski played an important role in the development of modern mathematics. His work formed the basis for modern functional analysis, and he did much to expand the knowledge of quadratic forms. Minkowski also developed the mathematical theory known as the geometry of numbers and laid the mathematical foundation for Albert Einstein's theory of relativity by pioneering the notion of a four-dimensional space-time continuum.

Minkowski was born in Alexotas, Russia, on June 22, 1864, of German parents. The family returned to their native Germany in 1872, to the city of Königsberg, where Minkowski spent the rest of his childhood and also attended university. His brother, Oskar Minkowski, became famous as the physiologist who discovered the link between diabetes and the pancreas.

Even as a student at the University of Königsberg, Minkowski demonstrated a rare mathematical talent. In 1881, the Paris Académie Royale des Sciences offered a prize, the Grand Prix des Sciences Mathématiques, for a proof describing the number of representations of an integer as a sum of five squares of integers--a proof that, unbeknownst to the Académie, the British mathematician H. J. Smith had already outlined in 1867. Minkowski produced the proof independently, while Smith filled in the details of his outline and submitted it. In 1883, both Smith and Minkowski received the prize. At that time, the 19-year-old Minkowski was two years away from receiving his doctorate from the University of Königsberg. The work contained in his 140-page solution was, in fact, considered a better formulation than Smith's because Minkowski used more natural and more general definitions in arriving at his proof.

While he was a university student, Minkowski began a lifelong friendship with fellow student David Hilbert, who would eventually edit Minkowski's collected works. After receiving his doctorate from the University of Königsberg in 1885, Minkowski taught at the University of Bonn until 1894. Returning to teach at the University of Königsberg for two years, he then taught at the University of Zurich until 1902. One of his closest colleagues at Zurich was a former teacher, A. Hurwitz, who is best known for his theorem on the composition of quadratic forms.

Throughout his life, Minkowski worked on the arithmetic of quadratic forms,particularly in n variables. According to mathematician Jean Dieudonné, who profiled him for the Dictionary of Scientific Biography, Minkowski made two important contributions to this field. One was a characterization of equivalence of quadratic forms with rational coefficients, under a linear transformation with rational coefficients. The other, published in 1905, completed Charles Hermite's theory of reduction for positive definite n-ary quadratic forms with real coefficients by finding a unique reduction form for each equivalence class. Pursuing the results of his 1905 paper, Minkowski developed a more geometric style of work, which led to what Dieudonné called his "most original achievement"--the geometry of numbers.

In 1889, Minkowski had introduced the geometrical concept of volume into his work on ternary quadratic forms. This technique involved centering ellipses on lattice pointsin the plane and looking at the areas of the ellipses as the lattice points increased in number. The limiting case as the number of non-overlapping ellipses in the plane approaches infinity gave Minkowski an estimate of the minimal solution of a specified quadratic equation in two variables. Using this type of geometrical technique, he was able to prove various theorems about numbers without performing any numerical calculations, a feat that was praised by Hilbert as "a pearl of Minkowski's creative art," reported Harris Hancock in the introduction to his book Development of the Minkowski Geometry of Numbers.

Minkowski generalized the technique to ellipsoids and other convex shapes (such as cylinders and polyhedrons) in three dimensions. This work led to investigations in packing efficiency (how to fill up space most densely with given shapes), a topic that has applications in chemistry, biology, and other sciences. Generalizing further to various types of convex objects in n-dimensional space, he produced numerous results in number theory through geometry. As Dieudonné wrote, "Long before the modern conception of a metric space was invented, Minkowski realized that a symmetric convex body in an n -dimensional space defines a new notion of 'distance' on that space and, hence, a corresponding 'geometry"'--a development that laid the foundation for modern functional analysis.

Hancock wrote of Minkowski, "His grasp of geometrical concepts seemed almost superhuman." However, he also noted that Minkowski's publications and notes were often incomplete and poorly explained. In part this was simply Minkowski's style, although it was complicated by his early death that brought an abrupt end to his works in progress. He died at the age of 44 from a ruptured appendix in Göttingen, Germany, on January 12, 1909. Hancock wrote his book to "reconstruct and clarify much that Minkowski would have done, had he lived."

At Hilbert's urging, the University of Göttingen created a new professorship for Minkowski in 1902. It was during his tenure at Göttingen that Minkowski turned his attention to relativity theory. Einstein, who published his initial work in relativity in 1905, had taken nine classes from Minkowski in Zurich, more than he had taken from anyone else (even the physics professor). The two men had no particular liking for one another. Lewis Pyenson wrote in Archive for History of Exact Sciences that "By the time he graduated in 1900 . . . Einstein had become indifferent to Minkowski's approach to mathematics and physics," and "Minkowski later thought that his own interpretation of the principle of relativity was superior to Einstein's because of Einstein's limited mathematical competence."

In 1905, Minkowski participated in an electron theory seminar that discussed the current theories of electrodynamics. With this background, he studied the competing theories of subatomic particles proposed by Einstein and Hendrik Lorentz. Minkowski was the first to realize that both theories led to the necessity of visualizing space as a four-dimensional, non-Euclidean, space-time continuum.Dieudonné wrote that Minkowski "gave a precise definition and initiated the mathematical study [of this four-space]; it became the frame of all later developments of the theory and led Einstein to his bolder conception of generalized relativity."

Rather than a mathematical adjustment of Einstein's theory, Minkowski developed his ideas as an alternate theory. As soon as Minkowski's first publication on relativity appeared, Pyenson wrote, "Einstein turned to the only part of Minkowski's paper that contained a physical prediction" and showed how it failed to account for a known phenomenon. The rivalry between the two men was carried on at a level that few recognized; the differences between their derivations were subtle. Ultimately, Einstein used Minkowski's ideas to develop his general theory of relativity, which was published seven years after Minkowski's death.