Search Results for Minkowski


  1. Minkowski biography
    • Hermann Minkowski .
    • Hermann Minkowski's parents were Lewin Minkowski, a businessman, and Rachel Taubmann.
    • The second brother Oskar (1858-1931) was a physician, best known for his work on diabetes, and father of astrophysicist Rudolph Minkowski (1895-1976).
    • Apart from Max and Oskar, Minkowski also had an older sister, Fanny (1863-1954) and a younger brother, Toby (1873-1906).
    • Lewin and Rachel Minkowski were Germans although their son Hermann was born while they were living in Russia.
    • When Hermann was eight years old the family returned to Germany and settled in Konigsberg where Lewin Minkowski conducted his business.
    • Minkowski first showed his talent for mathematics while studying at the Gymnasium in Konigsberg.
    • His became close friends with Hilbert while at Konigsberg, for Hilbert was an undergraduate at the same time as Minkowski.
    • The student Minkowski soon became close friends with the newly appointed academic Hurwitz.
    • He received his doctorate in 1885 from Konigsberg for a thesis entitled Untersuchungen uber quadratische Formen, Bestimmung der Anzahl verschiedener Formen, welche ein gegebenes Genus enthalt Ⓣ Minkowski became interested in quadratic forms early in his university studies.
    • Minkowski, although only eighteen years old at the time, reconstructed Eisenstein's theory of quadratic forms and produced a beautiful solution to the Grand Prix problem.
    • The decision was that the prize be shared between Minkowski and Smith but this was a stunning beginning to Minkowski's mathematical career.
    • On 2 April 1883 the Academy granted the Grand Prize in Mathematics jointly to the young Minkowski at the start of his career and the elderly Smith at the end of his.
    • Minkowski's doctoral thesis, submitted in 1885, was a continuation of this prize winning work involving his natural definition of the genus of a form.
    • In 1887, a professorship became vacant at the University of Bonn, and Minkowski applied for that position; according to the regulations of German universities, he had to submit orally to the faculty an original paper, as an Habilitationsschrift.
    • Minkowski presented Raumliche Anschauung und Minima positiv definiter quadratischer Formen Ⓣ which was not published at the time but in 1991 the lecture was published in [',' J Schwermer, Raumliche Anschauung und Minima positiv definiter quadratischer Formen.
    • Zur Habilitation von Hermann Minkowski 1887 in Bonn, Jahresber.
    • This lecture is particularly interesting, for it contains the first example of the method which Minkowski would develop some years later in his famous "geometry of numbers".
    • Minkowski taught at Bonn from 1887, being promoted to assistant professor in 1892.
    • Minkowski married Auguste Adler in Strasbourg in 1897; they had two daughters, Lily born in 1898 and Ruth born in 1902.
    • The family left Zurich in the year that their second daughter was born for Minkowski accepted a chair at the University of Gottingen in 1902.
    • It was Hilbert who arranged for the chair to be created specially for Minkowski and he held it for the rest of his life.
    • Minkowski developed a new view of space and time and laid the mathematical foundation of the theory of relativity.
    • By 1907 Minkowski realised that the work of Lorentz and Einstein could be best understood in a non-euclidean space.
    • Minkowski worked out a four-dimensional treatment of electrodynamics.
    • Kline, reviewing [',' L Pyenson, Hermann Minkowski and Einstein’s Special Theory of Relativity : With an appendix of Minkowski’s ’Funktiontheorie’ manuscript, Arch.
    • In a paper published in 1908 Minkowski reformulated Einstein's 1905 paper by introducing the four-dimensional (space-time) non-Euclidean geometry, a step which Einstein did not think much of at the time.
    • But more important is the attitude or philosophy that Minkowski, Hilbert - with whom Minkowski worked for a few years - Felix Klein and Hermann Weyl pursued, namely, that purely mathematical considerations, including harmony and elegance of ideas, should dominate in embracing new physical facts.
    • In this view Minkowski followed Poincare whose philosophy was that mathematical physics, as opposed to theoretical physics, can furnish new physical principles.
    • In fact Minkowski had a major influence on Einstein as Corry points out in [',' L Corry, The influence of David Hilbert and Hermann Minkowski on Einstein’s views over the interrelation between physics and mathematics, Endeavor 22 (3) (1998), 95-97.','7]:- .
    • A main motive behind this change was the influence of two prominent German mathematicians: David Hilbert and Hermann Minkowski.
    • We have mentioned several times in this biography that Minkowski and Hilbert were close friends.
    • Less well known is the fact that Minkowski actually suggested to Hilbert what he should take as the theme for his famous 1900 lecture in Paris.
    • Minkowski, in a letter to Hilbert written on 5 January 1900, writes:- .
    • Time has certainly proved Minkowski correct! .
    • [',' S U Eminger, C F Geiser and R Rudio: the men behind the First International Congress of Mathematicians St Andrews PhD thesis (2014) 123-130.','8] Minkowski joined the organising committee in December 1896 -- he might not yet have been in Zurich for the preliminary meeting in July.
    • Minkowski also offered to give a talk himself in one of the section meetings, but for reasons that are not explained in the minutes he did not after all.
    • Minkowski acted as one of the secretaries at the 1900 ICM in Paris, and gave a talk in section I at the 1904 ICM in Heidelberg, entitled Zur Geometrie der Zahlen (On the Geometry of Numbers).
    • Minkowski's original mathematical interests were in pure mathematics and he spent much of his time investigating quadratic forms and continued fractions.
    • Minkowski published Diophantische Approximationen: Eine Einfuhrung in die Zahlentheorie Ⓣ in 1907.
    • At the young age of 44, Minkowski died suddenly from a ruptured appendix.
    • A Poster of Hermann Minkowski .
    • Honours awarded to Hermann Minkowski .
    • 1.nLunar featuresnCrater Minkowski .
    • .

  2. Born biography
    • Back in Breslau he talked to his fellow students Toeplitz and Hellinger who told him of the great teachers of mathematics, Klein, Hilbert and Minkowski, at the University of Gottingen.
    • Born was soon in Gottingen attending lectures by Hilbert and Minkowski.
    • He became Hilbert's assistant in 1905, continuing to attend lectures by Klein and Runge on elasticity and a seminar by Hilbert and Minkowski on electrodynamics.
    • Perhaps the most benefit he derived from his famous teachers was during walks he would make in the woods with Hilbert and Minkowski where all manner of fascinating subjects were discussed in addition to mathematics, such as problems of philosophy, problems of politics, and social problems.
    • His work on combining ideas of Einstein and Minkowski led to an invitation to Gottingen in 1909 and he began a collaboration with Minkowski who died only weeks after the collaboration had begun [',' N Kemmer and R Schlapp, Max Born, Biographical Memoirs of Fellows of the Royal Society of London 17 (1971), 17-52.','8]:- .

  3. Hilbert biography
    • In the spring of 1882, Hermann Minkowski returned to Konigsberg after studying in Berlin.
    • Hilbert and Minkowski, who was also a doctoral student, soon became close friends and they were to strongly influence each others mathematical progress.
    • Minkowski, after reading the thesis, wrote to Hilbert (see [',' C Reid, Hilbert (Berlin- Heidelberg- New York, 1970).','8]):- .
    • Hilbert turned down the Berlin chair, but only after he had used the offer to bargain with Gottingen and persuade them to set up a new chair to bring his friend Minkowski to Gottingen.
    • Insofar as the creation of new ideas is concerned, I would place Minkowski higher, and of the classical great ones, Gauss, Galois, and Riemann.

  4. Speiser Andreas biography
    • Von der Muhll advised Speiser to study at the University of Gottingen and, having entered in 1904, he was taught by several leading mathematicians including Felix Klein, David Hilbert and Hermann Minkowski.
    • While he was studying there he became friends with Constantin Caratheodory who had been a student of Minkowski's and was a docent at Gottingen for most of the time Speiser was studying there.
    • Speiser undertook research on quadratic forms advised mainly by Minkowski but also helped by Hilbert.
    • He had completed the work for his thesis Theorie der binaren quadratischen Formen mit Koeffizienten und Unbestimmten in einem beliebigen Zahlkorper Ⓣ when, in January 1909, Minkowski died suddenly from a ruptured appendix.

  5. Konig Denes biography
    • While in Gottingen, he attended Hermann Minkowski's lectures on topology (called Analysis Situs at this time) in session 1904-05.
    • At Gottingen, Konig had been influenced by Minkowski's lectures on the four colour problem in the 1904-05 session [',' T Gallai, The life and scientific work of Denes Konig (1884-1944), Linear Algebra Appl.
    • In these, in addition to the characterisation of the topological properties of two-dimensional surfaces and the generation of various normal types, Minkowski intended to present a proof, given by Wemicke, of the four-colour conjecture.

  6. Hurwitz biography
    • Here he taught Hilbert and Minkowski, becoming a life long friend of Hilbert.
    • Even after Minkowski left the University of Konigsberg and went to Bonn, he still returned to Konigsberg for every vacation and joined Hurwitz and Hilbert in their almost daily walks [',' W H Young, Adolf Hurwitz, Proc.
    • Together with Geiser and Minkowski, Hurwitz was responsible for choosing the plenary speakers (on Rudio's suggestion).

  7. Weber Heinrich biography
    • However, one of the most significant things to happen during his time teaching at Konigsberg was in 1880 when two young students, David Hilbert and Hermann Minkowski, enrolled there.
    • All of these courses were attended by the young Hilbert (and probably also by Minkowski).
    • The third volume of Lehrbuch der Algebra Ⓣ, published in 1903, was dedicated by Weber to Dedekind, Hilbert and Minkowski:- .

  8. Bieberbach biography
    • Bieberbach decided to look at the list of announcements of mathematics courses given in different universities which appeared in the Jahresbericht der Deutschen Mathematiker-Vereinigung and, after studying the possibilities, decided that Hermann Minkowski's course on 'Invariant theory' at the University of Gottingen looked the most attractive.
    • He attended the algebra course by Minkowski which had brought him there, but he was influenced even more strongly by Felix Klein and his lectures on elliptic functions.

  9. Nirenberg biography
    • He published the results of his thesis in 1953 in the paper The Weyl and Minkowski problems in differential geometry in the large.
    • Caffarelli mentions Nirenberg's areas of interest in partial differential equations: Regularity and solvability of elliptic equations of order 2n; the Minkowski problem and fully nonlinear equations; the theory of higher regularity for free boundary problems; and symmetry properties of solutions to invariant equations.

  10. Kelly biography
    • This paper was published in the Bulletin of the American Mathematical Society, as was his papers On isometries of product sets of 1948 and On Minkowski bodies of constant width of 1949.
    • Kelly showed that this theorem went over to entire subsets of Minkowski n-space.

  11. Golab biography
    • He was awarded the degree based on his paper Quelques problemes metrique de la geometrie de Minkowski Ⓣ in 1932.
    • One can divide them into three almost equal parts; papers on the theory of geometric objects (40), papers on classical differential geometry under weak regularity assumptions (43) and papers belonging to various other domains in geometry (50) mainly connected with some special spaces such as spaces with linear or projective connection, Riemann, Minkowski and Finsler spaces, general metric spaces, etc.

  12. Szele biography
    • In 1896 Hermann Minkowski had conjectured that if n-dimensional Euclidean space is filled by n-dimensional cubes so that every point is covered by a cube and no two cubes have interior points in common then there are cubes sharing n-1 dimensional faces.
    • No progress was made until 1942 when Gyorgy Hajos translated Minkowski's conjecture into a problem in abelian group theory and so proved the conjecture true.

  13. Segal biography
    • The axioms are properties of Minkowski spacetime M' and admit only one other model M which can briefly be described as the supposedly discredited cosmological model known as the Einstein universe first proposed by Einstein in 1917.
    • The key point is that time and its conjugate variable, energy, are fundamentally different in the Einstein Universe from the conventional time and energy in the local flat Minkowski space that approximates the Einstein Universe at the point of observation.

  14. Lewy biography
    • Among the first papers he published after emigrating to the United States were A priori limitations for solutions of Monge-Ampere equations (two papers, the first in 1935, the second two years later), and On differential geometry in the large : Minkowski's problem (1938).
    • Minkowski's problem is to construct a convex surface in three dimensional space that realises a given curvature as a function of the direction of the normal.

  15. Wien biography
    • Arnold Sommerfeld and Hermann Minkowski were both pupils at this Gymnasium at the time and, perhaps for the first time in his life, Wien made good academic progress.

  16. Schwarzschild biography
    • In Gottingen he collaborated with Klein, Hilbert and Minkowski.

  17. Christoffel biography
    • This second group, which may partly overlap with the former, would include such illustrious names as Mobius, von Staudt, Plucker, Heine, Du Bois-Reymond, Carl Neumann, Lipschitz, Fuchs, Schwarz, Hurwitz and Minkowski.

  18. Fejer biography
    • Fejer spent the winter of 1902-3 on a visit to Gottingen, attending lectures by David Hilbert and Hermann Minkowski, and the summer of 1903 in Paris where he attended lectures by Emile Picard and Jacques Hadamard.

  19. Ehrenfest biography
    • Klein, Hilbert, Minkowski and Caratheodory were all working in Gottingen at this time and it was an important period for Ehrenfest's research.

  20. Wiltheiss biography
    • Wiltheiss was a founder member of the German Mathematical Society along with his colleague at Halle Hermann Wiener, as were Cantor, Gordan, Hilbert, Klein, Minkowski, Study and Heinrich Weber who all gave lectures at the Bremen meeting.

  21. Noether Emmy biography
    • During 1903-04 she attended lectures by Karl Schwarzschild, Otto Blumenthal, David Hilbert, Felix Klein and Hermann Minkowski.

  22. Mahler biography
    • Other major themes of his work were rational approximations of algebraic numbers, p-adic numbers, p-adic Diophantine approximation, geometry of numbers (a term coined by Minkowski to describe the mathematics of packings and coverings) and measure on polynomials.

  23. Jarnik biography
    • During the decade 1939-49 he wrote a series of papers dealing with the geometry of numbers, in particular dealing with Minkowski's inequality for convex bodies.

  24. Helly biography
    • At Gottingen Helly studied under Hilbert, Klein, Minkowski and Runge in 1907-8.

  25. Plucker biography
    • Strangely enough, in the period when four-dimensional manifolds appeared in relativity theory and became fashionable, nobody compared the Minkowski fourfold and the Plucker fourfold which appeared 50 years earlier.

  26. Reinhardt biography
    • In this thesis Schmidt proved a conjecture of Hermann Minkowski in dimension n, where n < 8.

  27. Voronoy biography
    • There he met Minkowski and they discovered that they were each working on similar topics.

  28. Wiener Hermann biography
    • Wiener was a founder member of the German Mathematical Society, as were Cantor, Gordan, Hilbert, Klein, Minkowski, Study and Heinrich Weber who all gave lectures at the Bremen meeting.

  29. Smith biography
    • After his death the Academy awarded two full prizes, one to Smith and one to Minkowski.

  30. Vladimirov biography
    • Hermann Minkowski had initiated a study of the geometry of numbers in 1890 and, over the next twenty years, he studied many problems including packing problems for convex bodies.

  31. Dubreil biography
    • In his first year of study, he also attended some sessions of Vessiot's seminar on the geometry of numbers according to Minkowski and some sessions of Hadamard's seminars but Dubriel writes in [',' P Dubreil, L’algebre, en France, de 1900 a 1935, Cahiers du Seminaire d’Histoire des Mathematiques 3 (Inst.

  32. Janiszewski biography
    • Having chosen excellent centres of mathematical research at which to study he was taught by many outstanding mathematicians including Burkhardt, Hilbert, Minkowski, and Zermelo .

  33. Stackel biography
    • Stackel's stay in Halle lasted until 1895 when he was called to take up the post of associate professor at the University of Konigsberg as a successor to Minkowski, who had himself recommended Stackel for the post, having been impressed by his work at Halle.

  34. Marcolongo biography
    • Lorentz, Minkowski, and Poincare were the authors he cited most frequently; Einstein's name never appeared.

  35. Egorov biography
    • He returned to Germany to spend the summer of 1903 in Gottingen attending lectures by Klein, Hilbert and Minkowski.

  36. Herzog biography
    • In special, highly popular seminars he introduced his students to more advanced work by Maxwell and Minkowski, amongst others.

  37. Rogers James biography
    • Wiley, New York 1984).','3], [',' U Dudley, Real Analysis and Probability (Wadsworth, 1989).','5], [',' L Maligranda, Equivalence of the Holder-Rogers and Minkowski inequalities, Math.

  38. Rapcsak biography
    • Continuing with his teaching career, Rapcsak undertook research on differential geometry advised by Varga and submitted his thesis The theory of surfaces in Minkowski space (Hungarian) in 1947.

  39. Fischer biography
    • He spent 1899 at the University of Berlin, then studied at Zurich and Gottingen with Minkowski.

  40. Rogers biography
    • He was a plenary speaker at the British Mathematical Colloquium in 1964 giving the lecture The Brunn-Minkowski theorem and related inequalities.

  41. Hirsch Arthur biography
    • Hirsch became Titularprofessor in 1897, and in 1903 he was appointed to an ordinary professorship, succeeding Minkowski in his chair for higher mathematics [',' G Frei and U Stammbach, Hermann Weyl und die Mathematik an der ETH Zurich 1913-1930, Birkhauser, Basel, 1992 ','3].

  42. Landau biography
    • In 1909 he was appointed to an ordinary professorship at Gottingen as successor to Minkowski.

  43. Sommerfeld biography
    • Two slightly older pupils at the same school were Minkowski and Wien.

  44. Beckenbach biography
    • The book begins with a study of axiomatics, then examines several classical inequalities of analysis such as the relationship between the arithmetic mean and geometric mean, the Cauchy, Holder, and Minkowski inequalities, and the triangle inequality.

  45. Finsler biography
    • A Finsler space is a generalisation of a Riemannian space where the length function is defined differently and Minkowski's geometry holds locally.

  46. Bliss biography
    • He left Minnesota in 1902 to spend a year in Gottingen where he interacted with Klein, Hilbert, Minkowski, Zermelo, Schmidt, Max Abraham and Caratheodory.

  47. Dvoretzky biography
    • Among many doctoral students that he advised at the Hebrew University we mention three who went on to a highly successful academic career; Branko Grunbaum, who wrote the thesis On Some Properties of Minkowski Spaces (1957), Joram Lindenstrauss, who wrote the thesis Extension of Compact Operators (1962), and Aldo Joram Lazar, who wrote the thesis Spaces of Affine Functions on Simplexes (1968).

  48. Cooper William biography
    • His first few papers on mathematics (all written with are Abraham Charnes) are: The stepping stone method of explaining linear programming calculations in transportation problems (1954); A model for optimizing production by reference to cost surrogates (1955); Optimal estimation of executive compensation by linear programming (1955); Nonlinear power of adjacent extreme point methods in linear programming (1957); Management models and industrial applications of linear programming (1957); The theory of search: optimum distribution of search effort (1958); Nonlinear network flows and convex programming over incidence matrices (1958); and The strong Minkowski-Farkas-Weyl theorem for vector spaces over ordered fields (1958).

  49. Geiser biography
    • For ten years he was its director (1881-1887 and 1891-1895) and the mathematicians who taught there during this period indicate the high status it was achieving: Richard Dedekind, Heinrich Durege, Elwin Christoffel, Hermann Schwarz, Heinrich Weber, Theodor Reye, Wilhelm Fiedler, Georg Frobenius, Friedrich Schottky, Adolf Hurwitz, Hermann Minkowski and Ernst Zermelo.

  50. Amberg biography
    • This sub-committee already consisted of Geiser, Hurwitz and Minkowski, and Franel joined together with Amberg.

  51. Wheeler biography
    • She went to Gottingen University where she attended lectures by David Hilbert, Felix Klein, Hermann Minkowski, Gustav Herglotz and Karl Schwarzschild.

  52. Caratheodory biography
    • He received his doctorate in 1904 from Gottingen University for his thesis Uber die diskontinuierlichen Losungen in der Variationsrechnung Ⓣ which he submitted to Hermann Minkowski.

  53. Miranda biography
    • Examples of his work around this time are: Su un problema di Minkowski Ⓣ (1939) which considers the problem of determining a convex surface of given Gaussian curvature; Su alcuni sviluppi in serie procedenti per funzioni non necessariamente ortogonali Ⓣ (1939) which examines expansion theorems in terms of the characteristic solutions of an integral equation whose kernel, although symmetric, involves the characteristic parameter; Nuovi contributi alla teoria delle equazioni integrali lineari con nucleo dipendente dal parametro Ⓣ (1940) which examines the development of the Hilbert-Schmidt theory for a particular type of linear integral equation; and Observations on a theorem of Brouwer (1940) which gave an elementary proof of the equivalence of Brouwer's fixed point theorem and a special case of Kronecker's index theorem.

  54. Hsiung biography
    • The central themes are the Gauss-Bonnet formula and uniqueness theorems for the Minkowski and Christoffel problems.

  55. Rohrbach biography
    • Before submitting his thesis, he published the paper Bemerkungen zu einem Determinantensatz von Minkowski Ⓣ in 1931.

  56. Szekeres biography
    • He continued to publish on relativity with work such as Kinematic geometry: An axiomatic system for Minkowski space-time (1968).

  57. Courant biography
    • At Gottingen Courant began by attending courses by Hilbert and Minkowski and he was also allowed to attend the joint seminar of the two mathematicians on mathematical physics.

  58. Koksma biography
    • One then finds a discussion of Minkowski's analysis, his 'Geometry of Numbers' and applications to homogeneous and non-homogeneous linear forms.

  59. Weierstrass biography
    • We name a few who are mentioned elsewhere in our archive: Bachmann, Bolza, Cantor, Engel, Frobenius, Gegenbauer, Hensel, Holder, Hurwitz, Killing, Klein, Kneser, Konigsberger, Lerch, Lie, Luroth, Mertens, Minkowski, Mittag-Leffler, Netto, Schottky, Schwarz and Stolz.

History Topics

  1. 20th century time
    • On 21 September 1908 Minkowski began his famous lecture at the University of Cologne with these words:- .
    • Weyl quickly understood the new notion that Minkowski put forward.
    • Before we move on from special relativity, we must consider one aspect which seems particularly difficult in Minkowski's 4-dimensional space-time, and indeed in any version of relativity.

  2. Special relativity
    • Also in 1908 Minkowski published an important paper on relativity, presenting the Maxwell-Lorentz equations in tensor form.
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  3. ETH history
    • Among the mathematicians who taught at the Polytechnic in the 19th and early 20th centuries we find, in chronological order: Joseph Ludwig Raabe, Richard Dedekind, Elwin Bruno Christoffel, Carl Theodor Reye, Hermann Amandus Schwarz, Heinrich Weber, Georg Ferdinand Frobenius, Friedrich Schottky, Adolf Hurwitz, Hermann Minkowski, Hermann Weyl, and George Polya.

Societies etc

  1. German Mathematical Society
    • Few societies can have come into existence at a meeting at which so many leading mathematicians spoke, for at this meeting Cantor, Gordan, Hilbert, Klein, Minkowski, Study and Heinrich Weber all gave lectures.


  1. Lunar features
    • (W) (L) Minkowski .

  2. International Congress Speaker
    • Louis Joel Mordell, Minkowski's Theorems and Hypotheses on Linear Forms.

  3. Lunar features
    • Minkowski .

  4. Young Mathematician prize
    • for works on the geometry of Minkowski spaces.


  1. References for Minkowski
    • References for Hermann Minkowski .
    • .
    • H Hancock, Development of the Minkowski Geometry of Numbers (New York, 1939).
    • F W Lanchester, Relativity : an elementary explanation of the space-time relations as established by Minkowski, and a discusson of gravitational theory based thereon (London, 1935).
    • W Benz, Lorentz-Minkowski geometry, De Sitter's world and Einstein's cylinder universe, in Charlemagne and his heritage.
    • L Corry, Hermann Minkowski and the postulate of relativity, Arch.
    • L Corry, The influence of David Hilbert and Hermann Minkowski on Einstein's views over the interrelation between physics and mathematics, Endeavor 22 (3) (1998), 95-97.
    • L Pyenson, Hermann Minkowski and Einstein's Special Theory of Relativity : With an appendix of Minkowski's 'Funktiontheorie' manuscript, Arch.
    • M F Ranada, David Hilbert, Hermann Minkowski, the axiomatization of physics and the Sixth Problem (Spanish), Gac.
    • Zur Habilitation von Hermann Minkowski 1887 in Bonn, Jahresber.
    • J-P Serre, Smith, Minkowski et l'Academie des Sciences, Gaz.
    • T M Tonietti, Arithmetic and Arithmetisierung : Felix Klein and Hermann Minkowski (Italian), in Epistemology of mathematics.
    • H J Zassenhaus, On the Minkowski- Hilbert dialogue on mathematization, Canad.
    • .

  2. References for Hilbert
    • L Corry, The influence of David Hilbert and Hermann Minkowski on Einstein's views over the interrelation between physics and mathematics, Endeavour 22 (3) (1998), 95-97.
    • M F Ranada, David Hilbert, Hermann Minkowski, the axiomatization of physics and the Sixth Problem (Spanish), Gac.
    • I Smadja, Local axioms in disguise: Hilbert on Minkowski diagrams, Synthese 186 (1) (2012), 315-370.
    • I Smadja, Erratum to: Local axioms in disguise: Hilbert on Minkowski diagrams, Synthese 186 (1) (2012), 441-442.
    • H J Zassenhaus, On the Minkowski-Hilbert dialogue on mathematization, Canad.

  3. References for Smith
    • J-P Serre, Smith, Minkowski et l'Academie des Sciences, Gaz.

  4. References for Rogers James
    • L Maligranda, Equivalence of the Holder-Rogers and Minkowski inequalities, Math.

Additional material

  1. Berge books
    • In fact, he supplies the preliminary mathematical concepts, such as sets, vector space, convexity of sets and of functions, and proves the Minimax Theorem of John von Neumann and its generalization by Maurice Sion (who replaces convexity in von Neumann's theorem by quasi-convexity), as well as the Farkas-Minkowski theorem [named after Gyula Farkas and Hermann Minkowski].
    • Chapter 3, Properties of Convex Sets and Functions in the Space Rn (18 pages), discusses (in Rn) the following topics: Separation of convex sets, supporting hyperplanes, the Krein-Milman theorem [named after Mark Grigorievich Krein and David Pinhusovich Milman (1912-1982)], intersections of convex sets, Helly's theorem [named after Eduard Helly], minimax theorems and several generalizations of the Farkas-Minkowski theorem [named after Gyula Farkas and Hermann Minkowski].
    • Their contents are respectively: separation theorems for convex sets, the Farkas-Minkowski [named after Gyula Farkas and Hermann Minkowski] and von Neumann minimax theorems, with divers extensions and corollaries; the various forms of the minimization problem (with equivalence theorems); algorithms of "simplex" type for the solution of convex and quadratic programming problems.

  2. Weyl on Hilbert
    • His speech was fairly fluent, not as hesitant as Minkowski's, and far from monotonous.
    • First, comments by Hilbert on Hermann Minkowski delivered in a memorial address in 1909 to the Gottingen Gesellschaft der Wissenschaften: .
    • Hilbert on Minkowski .

  3. Born Inaugural
    • At that time Felix Klein was the leading figure in a group of outstanding mathematicians at Gottingen, amongst them Hilbert and Minkowski.
    • Minkowski has shown that it is possible to get a description of the connection of all events which is independent of the observer, or invariant, as the mathematicians say, by considering them as points in a four-dimensional continuum with a quasi-Euclidean geometry.

  4. Perron books
    • The work of Minkowski and others on criteria for algebraic numbers is mentioned but not discussed and the chapter concludes with proofs of the transcendence of e and π.

  5. Einstein: 'Ether and Relativity
    • In Minkowski's idiom this is expressed as follows:- Not every extended conformation in the four-dimensional world can be regarded as composed of world-threads.

  6. Reviews of Shafarevich's books
    • Chapter 2 contains a discussion of norm-forms, Minkowski's Lemma, Dirichlet's Theorem on the structure of the group of units, and the conditions for a reduced binary quadratic form of negative discriminant.

  7. Serre reviews
    • Its objective is the classification of quadratic forms over the field of rational numbers (Hasse-Minkowski theorem).

  8. Brusotti publications
    • Luigi Brusotti, L'area di una superficie curva nella definizione di Minkowski e nell'insegnamento della geometria elementare, Periodico di Mat.

  9. Élie Cartan reviews
    • The second part is devoted to the theory of spinors in spaces of any number of dimensions, and particularly in the space of special relativity (Minkowski space).

  10. Who was who 1852
    • Jacobi is the founder of the Konigsberg school to which belonged men like L O Hesse (1811-1874) and R F A Clebsch (1833-1872), followed in later years by A Hurwitz (1859-1919), D Hilbert (1862-1942) and H Minkowski (1864-1909).

  11. David Hilbert: 'Mathematical Problems
    • As an example of an arithmetical theory operating rigorously with geometrical ideas and signs, I may mention Minkowski's work, Die Geometrie der Zahlen.

  12. Milnor's books
    • The theory of quadratic forms and the intimately related theory of symmetric bilinear forms have a long and rich history, highlighted by the work of Legendre, Gauss, Minkowski, and Hasse.

  13. Hardy Inaugural Lecture
    • The name of Minkowski is familiar today to many, even in Oxford, who have certainly never read a line of Smith.

  14. Hormander books
    • Later, it was used by Fermat, Cauchy, Minkowski, and others as a tool ancillary to other studies.

  15. G H Hardy addresses the British Association in 1922
    • The old-fashioned geometry of Euclid, the entertaining seven-point geometry of Veblen, the space-times of Minkowski and Einstein, are all absolutely and equally real.

  16. Rohrbach publications
    • Hans Rohrbach, Bemerkungen zu einem Determinantensatz von Minkowski, Jahresber.

  17. What to solve
    • Section 3 contains problems on lattice points; the latter have been used since Minkowski in number theory and are encountered in a wide range of mathematical topics.


  1. Quotations by Minkowski
    • Quotations by Hermann Minkowski .
    • .

  2. Quotations by Einstein
    • This has been done elegantly by Minkowski; but chalk is cheaper than grey matter, and we will do it as it comes.

Famous Curves

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EMS Archive

  1. EMS 1913 Colloquium
    • Without it to bring us back to the obvious world of apparent realities we should have been floundering hopelessly in the Absolute or in Minkowski's Welt.

BMC Archive

  1. BMC 1953
    • Rankin, R AThe Minkowski-Hajos theorem on linear forms and the factorisation of abelian groups .

  2. BMC 1964
    • Rogers, C AThe Brunn-Minkowski theorem and related inequalities .

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