b. Alexotas, Russia (present day Kaunas, Lithuania), 1864;
d. Göttingen, Germany, 1909
Number theory; geometry; relativity theory
In 1883, the Paris Academy of Sciences awarded its prestigious Grand Prix for mathematical science to the eighteen-year-old student Hermann Minkowski. The prize problem was to give the number of representations of an integer as a sum of five squares of integers. In a manuscript of 140 pages written in German, Minkowski developed a general theory of QUADRATIC CORMS [III.73] that contains the solution to this problem as a special case. Two years later, Minkowski obtained his Ph.D. in Königsberg, and in 1887 he received his habilitation in Bonn with further work on quadratic forms in n variables.
While a student in Königsberg, Minkowski became a close friend of Adolf Hurwitz and HILBERT [VI.63]. In 1894, after Hurwitz had moved to Zürich, Minkowski returned from Bonn to his alma mater, and soon became Hilbert’s successor after Hilbert left for Göttingen. In 1896, Minkowski moved on to Zurich to become Hurwitz’s colleague. In 1902, Hilbert negotiated for another chair of mathematics to be created for Minkowski in Göttingen. There he worked as Hilbert’s colleague and closest friend until he died, unexpectedly, of a ruptured appendix in early 1909.
Minkowski’s later work is characterized by an ingenious use of geometric intuition for the solution of number-theoretic problems. His starting point was a theorem of HERMITE [VI.47] on the smallest positive real that can be represented by a given positive-definite quadratic form of n integer-valued nonzero variables. By interpreting the quadratic forms in terms of geometric objects such as ellipses (for n = 2) or ellipsoids (for n = 3), and considering the integer values of the variables as the coordinates of the points of a regular lattice, Minkowski was able to employ the notion of volume to arrive at nontrivial number-theoretic results. His investigations were published in 1896 in a book entitled The Geometry of Numbers. Realizing that the geometric arguments based on ellipsoids used only the property of convexity, Minkowski further generalized his theory by introducing a general concept of convex point sets. A convex body, according to Minkowski, is one in which the straight line connecting any two interior points lies completely within the set. This notion allowed Minkowski to investigate a geometry in which the Euclidean axiom about the congruence of triangles is replaced by the weaker axiom that the sum of two sides of a triangle is always larger than the third one (which we would nowadays call the triangle inequality, the key notion in metric spaces). Theorems about this Minkowskian geometry also produced immediate nontrivial number-theoretic results. Further results were obtained in the theory of CONTINUED FRACTIONS [III.22]. In 1907, Minkowski published introductory lectures on number theory under the title Diophantine Approximations.
Minkowski always had a deep interest in physics. In 1906, he wrote the article on capillarity for the authoritative Encyclopedia of the Mathematical Sciences (edited by KLEIN [VI.57] and others). In Göttingen, Hilbert and Minkowski gave joint seminars in which they studied contemporary work in electrodynamics by POINCARÉ [VI.61], Einstein, and others. Minkowski soon realized the significance of the fact that the special theory of relativity was a consequence of the invariance of the Maxwell equations under the group of Lorentz transformations (see GENERAL RELATIVITY AND THE EINSTEIN EQUATIONS [IV.13 §1]). He reinterpreted Maxwell-Lorentz electrodynamics geometrically in a mathematical formulation in which no formal distinction between the space and time coordinates exists. This is expressed in the famous opening words of his address to the Cologne meeting of the Society of German Scientists and Physicians a few weeks before his death: “From this hour on, space by itself and time by itself are to sink fully into shadows and only a kind of union of the two should yet preserve autonomy.” Minkowski’s four-dimensional Lorentz-covariant formulation of special relativity was a prerequisite for Einstein’s later general theory of relativity.
Hilbert, D. 1910. Hermann Minkowski. Mathematische Annalen 68:445-71.
Walter, S. 1999. Minkowski, mathematicians, and the mathematical theory of relativity. In The Expanding Worlds of General Relativity, edited by H. Goenner et al., pp. 45-86. Boston: Birkhäuser.