Hermann Minkowski

How to use this course

Physicists know Hermann Minkowski for one lecture: the 1908 address in Cologne in which space and time were fused into a single four-dimensional world. Mathematicians know him for a whole subject, the geometry of numbers, in which a question about integers is answered by drawing a convex shape and measuring its volume. Both sides of his work rest on one habit of mind. He looked at algebra and saw geometry.

Each technical lesson opens with an italic paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Lessons 04 to 08 cover number theory and convex geometry (1881–1907). Lessons 09 and 10 cover relativity (1907–1908). Readers interested only in spacetime can read Lessons 09 and 10 on their own.

Timeline

Kovno and Königsberg, 1864–1882

Hermann Minkowski was born on 22 June 1864 in Aleksotas, a small town across the Neman river from Kovno, today a district of Kaunas in Lithuania. The region belonged to the Russian Empire. His parents, Lewin Minkowski and Rachel Taubmann, were Jewish. Lewin was a merchant. Hermann was the third son. The eldest, Max, later ran the family business in Königsberg. The second, Oskar, became a physician and pathologist and in 1889, with Joseph von Mering, showed that removing the pancreas causes diabetes.

In 1872, when Hermann was eight, the family left the Russian Empire, where restrictions on Jews were tightening, and settled in Königsberg in East Prussia. He attended the Altstädtisches Gymnasium, moved quickly through the classes, and read Richard Dedekind, Peter Gustav Lejeune Dirichlet and Carl Friedrich Gauss on his own.

Two friends

He entered the University of Königsberg in April 1880, aged fifteen. Heinrich Weber, then professor there, wrote to Dedekind in 1881 about his exceptional talent. Among the students was David Hilbert, two years older and one semester behind him. The two became inseparable. When Adolf Hurwitz arrived as extraordinary professor in 1884, the three began a habit of daily walks in which, by Hilbert’s later account, they covered every corner of mathematics. Hilbert said that he learned much of his mathematics on those walks. Minkowski spent three semesters at the University of Berlin, from the winter of 1882–83, where his teachers included Leopold Kronecker and Karl Weierstrass.

The Paris prize and the doctorate, 1881–1885

In 1881 the Paris Academy of Sciences announced its Grand Prix des Sciences Mathématiques for 1883. The problem was to find the number of representations of an integer as a sum of five squares. Gotthold Eisenstein had stated formulas in 1847 without proof. Henry Smith of Oxford had sketched a proof in 1867 in a paper the Academy had overlooked.

Minkowski, seventeen when the problem was set, did far more than was asked. He rebuilt from scratch the theory of quadratic forms in any number of variables with integer coefficients, including the theory of genera, and derived the five-squares formulas as a consequence. Smith reworked his own proofs and submitted them too. On 2 April 1883 the Academy divided the prize between them. Smith had died in February. The award to an unknown eighteen-year-old caused controversy among English mathematicians, who felt Smith’s priority had been slighted. Charles Hermite, by contrast, praised Minkowski’s memoir warmly.

The thesis

Minkowski took his doctorate at Königsberg on 30 July 1885, with Ferdinand von Lindemann, who had proved the transcendence of \(\pi\) three years earlier, as his advisor. The dissertation, Untersuchungen über quadratische Formen, continued the prize memoir: it determined the number of distinct classes in a given genus of forms. This line of work, completed by Carl Ludwig Siegel in the 1930s, is now the Smith–Minkowski–Siegel mass formula. Lesson 04 sets out the ideas.

Bonn, Königsberg, Zürich, 1887–1902

Minkowski habilitated at Bonn in 1887. His habilitation lecture, on spatial intuition and minima of positive definite quadratic forms, already contains the geometric viewpoint of his later work; it was published only in 1991. At Bonn he also wrote a paper on hydrodynamics (1888), a first sign of interest in physics. He was promoted to extraordinary professor in 1892.

The decisive idea came in 1889–1891. Studying the minima of quadratic forms, Minkowski realised that a positive definite form is an ellipsoid and that the set of integer points is a lattice. Statements about forms then become statements about convex bodies and lattices, and volume becomes the key quantity. He developed this in a series of papers and in Geometrie der Zahlen, whose first part appeared in 1896 (Lessons 05–08).

In 1894 he returned to Königsberg. When Hilbert left for Göttingen in 1895, Minkowski succeeded to his chair. A year later he moved to the Eidgenössisches Polytechnikum (now ETH) in Zürich, where Hurwitz had been professor since 1892. One of the students in his classes from 1896 to 1900 was Albert Einstein, who attended irregularly. Minkowski later remembered him as a lazy student who did not bother with mathematics. Einstein later regretted not having learned more of it.

In 1897 Minkowski married Auguste Adler in Strasbourg. They had two daughters, Lily (born 1898) and Ruth (born 1902). The same year he served on the organizing committee of the first International Congress of Mathematicians, held in Zürich. In January 1900 he urged Hilbert to use his Paris Congress lecture to look forward, by surveying open problems. The result was Hilbert’s list of 23 problems.

Quadratic forms and the local–global idea

For the general reader. Which numbers are sums of two squares? Of five? Questions like these concern quadratic forms, polynomials in which every term has degree two. Minkowski’s first work organised such forms into families called genera and found that a form over the rationals is pinned down by a few simple invariants, one for each prime. That idea, testing a global question one prime at a time, became one of the main principles of modern number theory.

Forms, equivalence, genus

An integral quadratic form in \(n\) variables is \(Q(x)=x^{\mathsf T}Ax\) with \(A\) a symmetric matrix and \(Q\) taking integer values. Two forms are equivalent if \(A'=U^{\mathsf T}AU\) for some \(U\in\mathrm{GL}_n(\mathbb Z)\); equivalent forms represent the same integers. The determinant \(\det A\) is an invariant. Two forms are in the same genus if they are equivalent over the real numbers and over the \(p\)-adic integers \(\mathbb Z_p\) for every prime \(p\) (equivalently, modulo every integer). A genus contains finitely many classes.

Rational equivalence

In 1890 Minkowski found complete invariants for equivalence over \(\mathbb Q\): the dimension, the discriminant up to squares, the signature, and one sign-type invariant for each prime. In 1921 Helmut Hasse recast these results in the language of Kurt Hensel’s \(p\)-adic numbers as a local–global principle.

Over \(\mathbb Q_p\) the question is decided by finitely many congruences, by Hensel’s lemma, so the theorem turns a global question into a checkable list. Every form in five or more variables represents 0 over every \(\mathbb Q_p\). Hence an indefinite rational form in at least five variables always has a non-trivial rational zero (Meyer’s theorem, 1884).

The mass formula

For a positive definite form, the classes \(Q_1,\dots,Q_h\) in a genus have finite automorphism groups \(\mathrm O(Q_i)\). Smith and Minkowski gave a formula for the mass

as a product of local densities over all primes and a factor at infinity. Siegel proved the general version in 1935 and extended it to representation numbers. For the sum of five squares the genus has one class, so the mass formula gives the representation numbers directly, which is how the Paris problem is solved. Minkowski’s original formulas contained errors in the factor at the prime 2, corrected later by others.

Lattices and the convex body theorem

For the general reader. Plant a tree at every point with integer coordinates and stand at the origin. If you draw a large enough symmetric, bulging shape around yourself, it must contain another tree. Minkowski turned this simple picture into a precise theorem and used it to prove classical results about sums of squares and rational approximation in a few lines.

Lattices

A lattice \(\Lambda\subset\mathbb R^n\) is the set of integer combinations of a basis \(b_1,\dots,b_n\). Its covolume \(\det\Lambda=|\det(b_1,\dots,b_n)|\) is the volume of a fundamental parallelepiped and does not depend on the basis. A sublattice of index \(m\) in \(\mathbb Z^n\) has covolume \(m\).

Proof

Blichfeldt’s lemma. If \(S\) is measurable with \(\operatorname{vol}(S)>\det\Lambda\), there are distinct \(x,y\in S\) with \(x-y\in\Lambda\). Indeed, cut \(S\) along the translates of a fundamental domain \(P\) and slide every piece back into \(P\). The pieces have total volume greater than \(\operatorname{vol}(P)\), so two of them overlap.

The theorem. Apply the lemma to \(S=\tfrac12K\), of volume \(2^{-n}\operatorname{vol}(K)>\det\Lambda\). We get \(x\ne y\) in \(\tfrac12K\) with \(x-y\in\Lambda\). Now \(2x\in K\) and \(-2y\in K\) by symmetry, so their midpoint \(x-y\) lies in \(K\) by convexity. For the compact case apply the strict version to \((1+\varepsilon)K\) and let \(\varepsilon\to0\); a compact set meets the discrete set \(\Lambda\setminus\{0\}\) in finitely many points, so one point works for all \(\varepsilon\).

The constant \(2^n\) is sharp: the open cube \((-1,1)^n\) has volume \(2^n\) and contains no non-zero point of \(\mathbb Z^n\).

The region \(\{|L_i(x)|\le c_i\}\) is a compact symmetric parallelepiped of volume \(2^nc_1\cdots c_n/|\Delta|\ge 2^n\), so the compact theorem applies.

Two squares in one picture

Let \(p\equiv1\pmod4\) be prime, so there is \(u\) with \(u^2\equiv-1\pmod p\). The lattice \(\Lambda=\{(a,b)\in\mathbb Z^2: b\equiv ua \pmod p\}\) has index \(p\), so \(\det\Lambda=p\). The open disk \(a^2+b^2<2p\) has area \(2\pi p>4p\). So it contains \((a,b)\ne0\) in \(\Lambda\). Then \(a^2+b^2\equiv a^2(1+u^2)\equiv0\pmod p\) and \(0<a^2+b^2<2p\), hence \(a^2+b^2=p\).

Geometry of numbers in number fields

For the general reader. In a number field, numbers built from roots of polynomials, factorization can fail to be unique, and the class number measures how badly. Minkowski placed the integers of such a field as a lattice in ordinary space. His convex body theorem then shows that the class number is finite and computable, and that every number field other than the rationals has primes that ramify.

The Minkowski embedding

Let \(K\) be a number field of degree \(n\) with \(r_1\) real embeddings and \(r_2\) pairs of complex embeddings, so \(n=r_1+2r_2\). The map \(j:K\to\mathbb R^{r_1}\times\mathbb C^{r_2}\cong\mathbb R^n\) sends \(\alpha\) to its real embeddings and one embedding from each complex pair. The image of the ring of integers \(\mathcal O_K\) is a lattice with

for a non-zero ideal \(\mathfrak a\), where \(d_K\) is the discriminant and \(N(\mathfrak a)=|\mathcal O_K/\mathfrak a|\).

Proof sketch

Apply the compact convex body theorem to the lattice \(j(\mathfrak a)\) and the body \(K_t=\{\sum_{\text{real}}|x_i|+2\sum_{\text{complex}}|z_k|\le t\}\), whose volume is \(2^{r_1}(\pi/2)^{r_2}t^n/n!\). Choosing \(t\) so this equals \(2^n\det j(\mathfrak a)\) produces a non-zero \(\alpha\in\mathfrak a\) whose embeddings satisfy the bound. The inequality of arithmetic and geometric means gives \(|N(\alpha)|\le(t/n)^n\), that is, \(|N(\alpha)|\le M_K\,N(\mathfrak a)\). Then \(\mathfrak b=\alpha\mathfrak a^{-1}\) is integral, lies in the class of \(\mathfrak a^{-1}\), and has norm at most \(M_K\). Since \(\mathfrak a\) was arbitrary, every class is reached.

Since \(N(\mathfrak b)\ge1\), the bound gives \(\sqrt{|d_K|}\ge\frac{n^n}{n!}(\pi/4)^{r_2}\ge\frac{n^n}{n!}(\pi/4)^{n/2}\). For \(n=2\) the right side squared is \(\pi^2/4>1\), and it increases with \(n\). By Dedekind’s theorem a prime ramifies in \(K\) exactly when it divides \(d_K\), so every proper extension of \(\mathbb Q\) is ramified at some prime. Combined with Charles Hermite’s theorem, the bound also shows there are only finitely many number fields of bounded discriminant, the Hermite–Minkowski theorem.

Minima, reduction, packing, continued fractions

For the general reader. How tightly can identical balls be packed in a regular grid? How do you pick the “best” basis of a lattice? Minkowski answered the second question completely and gave lasting bounds for the first. Along the way he invented a strange function that turns the continued fraction of a number into its binary expansion.

Successive minima

Let \(K\) be a compact convex symmetric body with non-empty interior and \(\Lambda\) a lattice. The \(i\)-th successive minimum \(\lambda_i\) is the smallest \(\lambda\) such that \(\lambda K\) contains \(i\) linearly independent lattice vectors. So \(\lambda_1\le\cdots\le\lambda_n\), and the convex body theorem says \(\lambda_1^n\operatorname{vol}(K)\le2^n\det\Lambda\).

The lower bound is easy: the cross-polytope spanned by \(\pm v_i/\lambda_i\), for independent \(v_i\) realizing the minima, lies in \(K\) and has volume \(\frac{2^n}{n!}|\det(v_i)|/\prod\lambda_i\ge\frac{2^n}{n!}\det\Lambda/\prod\lambda_i\). The upper bound is much harder; Minkowski’s proof was long, and shorter ones were found only decades later.

Packing and Hermite’s constant

Balls of radius \(r=\lambda_1/2\) (with \(K\) the unit ball) centred at the points of \(\Lambda\) do not overlap. The packing density is

In the language of forms, Hermite’s constant \(\gamma_n\) is the maximum over lattices of \(\lambda_1^2/(\det\Lambda)^{2/n}\), and maximizing density is the same problem. The convex body theorem gives \(\gamma_n\le4\operatorname{vol}(B_n)^{-2/n}\), roughly linear in \(n\). Minkowski also connected the densest lattice packing of a convex body with the minimum of the associated distance function, opening a line of work continued to the present.

The proof averages over all lattices of covolume 1: the average number of non-zero lattice points in a set of volume \(V\) is \(V\), so some lattice does no worse. No explicit family of lattices is known that beats this bound by more than a polynomial factor in high dimension.

Reduction

A basis \(b_1,\dots,b_n\) of \(\Lambda\) is Minkowski reduced if, for each \(k\), \(b_k\) is a shortest lattice vector that can be extended to a basis together with \(b_1,\dots,b_{k-1}\). In a 1905 paper Minkowski showed that the reduced positive definite forms fill a convex polyhedral cone defined by finitely many inequalities, a fundamental domain for \(\mathrm{GL}_n(\mathbb Z)\) acting on positive definite forms. He used its volume to obtain asymptotics for class numbers of forms. Reduction theory later passed to arithmetic groups through Siegel, Armand Borel and Harish-Chandra; the LLL algorithm of 1982 is a computable weakening of Minkowski reduction.

The question-mark function

In his 1904 Heidelberg lecture Minkowski defined, for \(x=[a_0;a_1,a_2,\dots]\) in continued fraction form,

It is a strictly increasing homeomorphism of \([0,1]\), maps rationals to dyadic rationals and quadratic irrationals to rationals, and has derivative zero almost everywhere. Minkowski introduced it to illustrate an analogue of Lagrange’s theorem that quadratic irrationals have periodic continued fractions.

Convex bodies and mixed volumes

For the general reader. To use convex shapes in number theory, Minkowski needed a theory of the shapes themselves. He created one. He added shapes together, measured how volume grows, and showed among other things that of all shapes with a given area the circle has the shortest boundary. Modern convex geometry starts with this work.

Sums and gauges

The Minkowski sum of sets is \(A+B=\{a+b: a\in A, b\in B\}\). A convex body \(K\) is described by its support function \(h_K(u)=\max_{x\in K}\langle x,u\rangle\), and \(h_{A+B}=h_A+h_B\). A symmetric convex body with the origin inside defines a gauge \(\|x\|_K=\min\{\lambda\ge0: x\in\lambda K\}\), a norm whose unit ball is \(K\). Minkowski introduced such general distance functions, anticipating normed spaces; the triangle inequality for \(\ell^p\) norms is still called Minkowski’s inequality.

Mixed volumes

For convex bodies \(K_1,\dots,K_m\) and \(t_i\ge0\), Minkowski showed that \(\operatorname{vol}(t_1K_1+\cdots+t_mK_m)\) is a homogeneous polynomial of degree \(n\) in the \(t_i\); its coefficients are the mixed volumes \(V(K_{i_1},\dots,K_{i_n})\). In the plane, for the unit disk \(B\),

so the mixed area \(V(K,B)\) equals half the perimeter.

Isoperimetric inequality. In the plane, put \(A=K\) and \(B=tB_1\): \(\operatorname{area}(K+tB_1)\ge(\sqrt{\operatorname{area}K}+t\sqrt\pi)^2\). Compare the coefficients of \(t\) with Steiner’s formula as \(t\to0\): \(\operatorname{perim}(K)\ge2\sqrt{\pi\operatorname{area}K}\), that is, \(L^2\ge4\pi A\), with equality only for disks.

The condition \(\sum a_iu_i=0\) is necessary: it is the divergence theorem applied to a constant vector field. Minkowski extended the result to smooth convex surfaces with prescribed Gauss curvature as a function of the normal, the Minkowski problem, whose regularity theory occupied geometric analysis for much of the twentieth century. He also studied bodies of constant width.

Space and time become spacetime

For the general reader. Einstein’s 1905 theory said that moving clocks run slow and moving rods shrink. Minkowski saw a simpler way to say it. There is one four-dimensional world, and different observers slice it into “space” and “time” in different ways, much as different people looking at a room choose different directions as “forward”. What all observers agree on is a kind of distance in this world, with a minus sign where Euclid had a plus.

The interval

Label an event by \(x=(x^0,x^1,x^2,x^3)=(ct,x,y,z)\). Minkowski singled out the quadratic form

(Minkowski himself wrote \(x_4=ict\) to make the form look Euclidean; the real form with signature is now standard.) A vector is timelike if \(s^2>0\), lightlike if \(s^2=0\), and spacelike if \(s^2<0\). The lightlike vectors form the light cone. Minkowski called points of spacetime world points and the history of a particle its world line, and he named the whole structure the world.

Boosts are hyperbolic rotations

In the \((x^0,x^1)\)-plane a boost with velocity \(v=c\tanh\phi\) is

It preserves \((x^0)^2-(x^1)^2\) because \(\cosh^2\phi-\sinh^2\phi=1\). The rapidity \(\phi\) plays the role of an angle, and boosts in one direction compose by adding rapidities. On a diagram (see the hero figure) the moving time axis \(x'^1=0\) and space axis \(x'^0=0\) tilt toward the light cone by equal amounts, and the unit marks lie on the hyperbolas \(s^2=\pm1\).

Proper time

A clock moving along a timelike world line \(x(t)\) measures the proper time

Among timelike curves between two events, the straight one has the greatest proper time: the reverse of the Euclidean triangle inequality. Time dilation, the twin paradox and the relativity of simultaneity all follow from this geometry. Minkowski presented it in Göttingen on 5 November 1907 and to a wide audience in Cologne on 21 September 1908. His opening sentence there announced that space by itself and time by itself were to sink into shadows (“zu Schatten herabsinken”), with only a union of the two preserving independent reality.

Four-vectors and electrodynamics

For the general reader. In Minkowski’s 1908 paper on electrodynamics, the electric and magnetic fields stop being two separate things. They are two faces of one object in four dimensions, and Maxwell’s equations shrink to two short lines. The same geometry turns energy and momentum into one four-dimensional arrow whose length is the mass.

Kinematics

For a world line parametrized by proper time, the four-velocity is \(u^\mu=dx^\mu/d\tau=\gamma(c,\mathbf v)\), with \(\eta_{\mu\nu}u^\mu u^\nu=c^2\). The four-momentum is \(p^\mu=mu^\mu=(E/c,\mathbf p)\). Then

and at rest \(E=mc^2\). Minkowski’s 1908 paper gave the four-dimensional form of the equation of motion of a point mass, \(m\,du^\mu/d\tau=K^\mu\), with the four-force \(K^\mu\) orthogonal to \(u^\mu\).

The field tensor

In units with \(c=1\) (Heaviside–Lorentz), let \(A^\mu=(\varphi,\mathbf A)\) and \(F^{\mu\nu}=\partial^\mu A^\nu-\partial^\nu A^\mu\). Then \(F^{i0}=E^i\) and \(F^{ij}=-\varepsilon^{ijk}B^k\). Minkowski called such an antisymmetric object a “space-time vector of the second kind”, now a 2-form.

Check one component: for \(\nu=0\), \(\partial_iF^{i0}=\nabla\cdot\mathbf E=\rho\). For the force with \(\mu=i\): \(F^{i0}u_0=\gamma E^i\) and \(F^{ij}u_j=\gamma(\mathbf v\times\mathbf B)^i\), so \(d\mathbf p/d\tau=q\gamma(\mathbf E+\mathbf v\times\mathbf B)\).

Matter

The larger part of Minkowski’s paper treated fields in moving media, with \(\mathbf D,\mathbf H\) joined into a second tensor and constitutive relations written covariantly. His expression for the energy–momentum tensor in media differs from the one Max Abraham proposed in 1909. The resulting Abraham–Minkowski debate over the momentum of light in a medium continued for a century; the modern view is that both are correct once the momentum of the medium is accounted for.

Göttingen, 1902–1909

In 1902 Hilbert persuaded the Prussian ministry to create a third chair of mathematics at Göttingen, beside Felix Klein’s and his own, and Minkowski took it. For six years the two friends led the most active mathematics department in the world. They taught joint seminars, walked daily, and argued about everything. Minkowski’s doctoral students there included Constantin Carathéodory (1904), Paul Wernicke (1904) and Elijah Swift (1907); Louis Kollros (Zürich, 1905) and Dénes Kőnig (1907) also count him as advisor.

At the 1904 International Congress in Heidelberg he lectured on the geometry of numbers, and in 1907 he published Diophantische Approximationen, an introduction to number theory through his geometric methods. His attention, however, was moving toward physics. In the summer of 1905 he and Hilbert ran a seminar on the electron theory of Hendrik Lorentz, Henri Poincaré and others. After Einstein’s paper of 1905 he studied the new kinematics and saw its geometric content.

Three texts on relativity

• 5 November 1907: Das Relativitätsprinzip, a lecture to the Göttingen Mathematical Society, published posthumously in 1915 in a version edited by Arnold Sommerfeld.
• 21 December 1907: Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern, presented to the Göttingen Society of Sciences and printed in its Nachrichten in 1908 (Lesson 10).
• 21 September 1908: Raum und Zeit, the Cologne lecture, printed in 1909 in the Physikalische Zeitschrift and the Jahresbericht of the German Mathematical Society.

Minkowski presented relativity as the discovery of a new invariant geometry. He regarded Einstein’s 1905 paper as an important step but felt its mathematical structure had not been fully seen. Einstein at first thought the four-dimensional formalism added learned complication without new physics. Sommerfeld reported that Einstein joked he no longer understood his own theory once the mathematicians had taken it up. After 1912, working on gravitation, Einstein adopted Minkowski’s geometry fully; general relativity is the theory in which Minkowski’s flat world becomes curved.

Max Born, who had heard the Cologne lecture, joined Minkowski as a collaborator in December 1908 to work on relativistic electrodynamics.

Sudden death and long legacy

Shortly after Christmas 1908 Minkowski fell ill with appendicitis. Peritonitis set in, an operation came too late, and he died in Göttingen on 12 January 1909, aged 44. Hilbert, who lost his closest friend, gave the memorial address and edited the Gesammelte Abhandlungen (1911), with Andreas Speiser and Hermann Weyl. Born and Speiser sorted his papers, and Born completed his unfinished work on the electron. The second part of Geometrie der Zahlen appeared in 1910. Minkowski was buried in Göttingen. In 1932 his remains were moved to the Heerstraße cemetery in Berlin, beside those of his brother Oskar.

Legacy

• Physics: Minkowski space is the arena of special relativity and quantum field theory, and the local model of every spacetime in general relativity. Words he coined, world line and world point, remain in use.
• Number theory: the geometry of numbers underlies the finiteness theorems of algebraic number theory, Diophantine approximation, the subspace theorem, reduction theory of arithmetic groups and Arakelov geometry.
• Convex geometry and analysis: mixed volumes, the Brunn–Minkowski inequality and the Minkowski problem are central in convex and geometric analysis; Minkowski sums and gauges are basic tools in optimization.
• Computation and cryptography: lattice reduction and the hardness of short-vector problems, both descended from his questions, underlie modern lattice-based cryptography.
• Family: his nephew Rudolph Minkowski, son of Oskar, became an astronomer known for work on supernovae and the Palomar Sky Survey.
• Honours: the lunar crater Minkowski and asteroid 12493 Minkowski bear his name.

Connections

People linked to Hermann Minkowski across this collection. A name without a link has no page here yet. See Hermann Minkowski in the network of all 337 lives →

Teachers and advisors

• Ferdinand von Lindemann 1885

Students

• Albert Einstein 1896-1900
• Constantin Carathéodory 1904
• Dénes Kőnig 1907
• Elijah Swift 1907
• Louis Kollros 1905
• Paul Wernicke 1904

Influenced by

• Charles Hermite
• Hendrik Lorentz
• Henri Poincaré
• Karl Weierstrass
• Leopold Kronecker
• Peter Gustav Lejeune Dirichlet
• Wilhelm Weber
• Gotthold Eisenstein

Influenced

• Albert Einstein 1896-1909
• Carl Ludwig Siegel
• Helmut Hasse
• Hermann Weyl
• Max von Laue

Colleagues and collaborators

• Arnold Sommerfeld
• Felix Klein 1902-1909
• Max Born 1908-1909

Rivals

• Henry John Stephen Smith 1883

Friends

• Adolf Hurwitz 1884-1909
• David Hilbert 1880-1909

Family

• Oskar Minkowski
• Rudolph Minkowski

Also mentioned in this life

• André-Marie Ampère
• Arthur Compton
• Richard Dedekind
• Euclid
• Leonhard Euler
• Michael Faraday
• Pierre de Fermat
• Carl Friedrich Gauss
• Harish-Chandra
• Oliver Heaviside
• Joseph-Louis Lagrange
• Tullio Levi-Civita
• James Clerk Maxwell
• Jean-Pierre Serre
• Jakob Steiner

Fields

• number theory 45 lives
• mathematical physics 45 lives
• special relativity 3 lives

On CoursesHub

• Minkowski Spacetime · Special Relativity

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Hermann Minkowski, MacTutor History of Mathematics.
2. Hermann Minkowski, Wikipedia.
3. Minkowski, Hermann, Neue Deutsche Biographie 17 (1994).
4. Minkowski, Hermann, Dictionary of Scientific Biography.
5. D. Hilbert, Hermann Minkowski, Math. Ann. 68 (1910), 445–471.
6. H. Minkowski, Briefe an David Hilbert, ed. L. Rüdenberg and H. Zassenhaus, Springer, 1973.
7. D. Rowe, A look back at Hermann Minkowski’s Cologne lecture “Raum und Zeit”, Math. Intelligencer 31 (2009), 27–39.
8. J. Schwermer, Räumliche Anschauung und Minima positiv definiter quadratischer Formen, Jahresber. DMV 93 (1991), 49–105.

Mathematics and physics

1. H. Minkowski, Geometrie der Zahlen, Teubner, 1896 and 1910.
2. H. Minkowski, Diophantische Approximationen, Teubner, 1907.
3. H. Minkowski, Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern, Nachr. Ges. Wiss. Göttingen (1908), 53–111.
4. H. Minkowski, Raum und Zeit, Physikalische Zeitschrift 10 (1909), 104–111; Jahresber. DMV 18 (1909), 75–88. English translation on Wikisource.
5. H. Minkowski, Gesammelte Abhandlungen, ed. D. Hilbert, 2 vols., Teubner, 1911.
6. J. W. S. Cassels, An Introduction to the Geometry of Numbers, Springer, 1959.
7. J.-P. Serre, A Course in Arithmetic, Springer, 1973 (Hasse–Minkowski theorem).
8. J. Neukirch, Algebraic Number Theory, Springer, 1999, ch. I (Minkowski theory).
9. R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, 2nd ed., Cambridge, 2014.
10. R. J. Gardner, The Brunn–Minkowski inequality, Bull. AMS 39 (2002), 355–405.
11. S. Walter, Minkowski, mathematicians, and the mathematical theory of relativity, in The Expanding Worlds of General Relativity, Birkhäuser, 1999.
12. G. Weinstein, Max Born, Albert Einstein and Hermann Minkowski’s space-time formalism of special relativity, arXiv:1210.6929.