Hermann Minkowski | Biography

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. He also developed the mathematical theory known as the geometry of numbers and laid the mathematical foundation for Albert Einstein' s theory of relativity.

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.

Even as a student at the University of Königsberg, Minkowski demonstrated a rare mathematical talent. In 1881, the Paris Academy of 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 Paris Academy, the British mathematician J. H. Smith had in fact already outlined at the time. Minkowski produced the proof independently while Smith sent in his own work. In 1883 both Smith and Minkowski received the prize. At that time, the nineteen-year-old Minkowski was two years away from receiving his doctorate from the University of Königsberg. The work contained in the 140-page manuscript he submitted to the Academy was, in fact, considered a better formulation than Smith's because the young Minkowski used more natural and more general definitions in arriving at his proof.

After receiving his doctorate from the University of Königsberg, Minkowski taught at the University of Bonn until 1894. Returning to teach at the University of Königsberg for two years, he then taught until 1902 at the University of Zurich. 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.

After working on the arithmetic of quadratic forms for several years and making contributions particularly to work in n variables, Minkowski extended his work to what is most commonly known as the geometry of numbers. In 1889, he introduced what has been characterized as his most original achievement, when he included volume in his work with ternary quadratic forms. With this extension it became possible to give mathematical descriptions of the properties of, for example, convex bodies in both two and three dimensions. Common examples of convex regions are those bounded by circles, ellipses, and parallelograms. Using the two- and three-dimensional versions of Minkowski's convex body theorem, mathematicians are able to prove some fundamental facts about algebraic number theory and can derive new proofs of some theorems from elementary number theory. Minkowski extended these ideas to investigations of the geometrical properties of convex sets in n-dimensional space, and his observations ultimately formed the basis for modern functional analysis.

At the urging of a former classmate, the great mathematician David Hilbert, 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. Albert Einstein had been one of Minkowski's pupils, and Minkowski was very interested in the special theory of relativity formulated by Einstein, which at the time competed with the more widely accepted electron theory of the Dutch physicist Hendrik Lorentz as an explanation of subatomic phenomena. Minkowski was the first to recognize the consequences of the relativity theory in consideration of time and space. "From now on," he said, "space by itself and time by itself are mere shadows and only a blend of the two exists in its own right." By 1907, Minkowski had placed a formal geometric interpretation upon relativity. He believed that in the universe, time and space exist as a fused "time-space." In his book Raum und Zeit (translated into English as Time and Space ), he demonstrated that relativity made it necessary mathematically to take time into account as a fourth dimension besides the spatial dimensions of length, width, and depth. Einstein used Minkowski's ideas to develop his general theory of relativity, published nine years later, several years after Minkowski's death in Göttingen, Germany, on January 12, 1909.