Analysis

The life of Niels Henrik Abel

Timeline 1802Born on 5 August near Stavanger, on the west coast of Norway. 1804The family moves to Gjerstad, where his father is the pastor. 1815Enters the Cathedral School of Christiania, today’s Oslo. 1818Bernt Holmboe becomes his mathematics teacher. 1820His father dies. Abel begins to support part of the family. 1821Enters the university. Believes he has solved the quintic and finds the mistake. 1824Publishes, at his own expense, the proof that there is no formula for the fifth degree. 1825Travels to Berlin, where he meets August Leopold Crelle. 1826In Paris, hands the Academy his great memoir on integrals. Cauchy loses it. 1827Returns to Norway, jobless and in debt. Publishes elliptic functions. 1829Dies of tuberculosis at Froland on 6 April, aged 26. 1830Receives, posthumously, the Grand Prize of the Paris Academy, with Jacobi. ◀ Previous Follow the journey Next ▶ 1802 near StavangerHe is born on the west coast of Norway, the second of seven children of a pastor. 1804–1815 GjerstadHe grows up in his father’s parish in the south of the country. 1815–1825 ChristianiaAt the Cathedral School his teacher Holmboe discovers his talent. In 1824 he publishes the six-page pamphlet on the quintic. 1825–1826 BerlinHe meets August Leopold Crelle, who publishes seven of his papers in the first volume of his journal. 1826 ParisHe hands the Academy his great memoir on integrals. Nobody pays attention to him, and Cauchy loses the manuscript. 1827–1828 ChristianiaHe returns jobless and in debt. He publishes elliptic functions and races Jacobi. 1828–1829 FrolandHe travels by sleigh at Christmas to see his fiancée, falls ill and dies of tuberculosis on 6 April. Two days later the appointment from Berlin arrives. Enable JavaScript to see the map. The stages are listed below. Abel’s great journey, from Norway to Berlin and Paris, and the return to die at Froland. Map: © OpenStreetMap contributors

October 15, 2026 Read

The life of John Nash

Timeline 1928Born on 13 June in Bluefield, West Virginia, in the United States. 1945Enters the Carnegie Institute of Technology, in Pittsburgh, to study chemical engineering. Switches to mathematics. 1948Arrives at Princeton. Invents the game his colleagues call “Nash”, known today as Hex. 1950Defends the thesis “Non-cooperative games”, with the Nash equilibrium. Publishes “The bargaining problem”. 1951Becomes an instructor at MIT. 1956Publishes the isometric embedding theorem. 1957Marries Alicia Lardé, a physics student at MIT. 1958Publishes the result on the regularity of parabolic equations. 1959First psychiatric hospitalisation. Three decades of illness begin. 1994Receives the Nobel Prize in Economics, with John Harsanyi and Reinhard Selten. 2001Remarries Alicia. The film A Beautiful Mind is released. 2015Receives the Abel Prize, with Louis Nirenberg. Dies in a car crash on 23 May, together with Alicia. ◀ Previous Follow the journey Next ▶ 1928–1945 BluefieldHe grows up in a railway and coal town. He reads Men of Mathematics and proves Fermat’s little theorem by himself. 1945–1948 PittsburghAt the Carnegie Institute of Technology he swaps chemical engineering for mathematics and has the bargaining idea. 1948–1951 PrincetonHe invents Hex and defends his 28-page thesis with the Nash equilibrium. 1951–1959 MITA professor at MIT. He proves the embedding theorem and the result on parabolic equations. In 1959 his illness begins. 1970–2015 PrincetonHe wanders the corridors as “the phantom of Fine Hall” and recovers at the end of the 1980s. 1994 StockholmHe receives the Nobel Prize in Economics, with John Harsanyi and Reinhard Selten. 2015 OsloHe receives the Abel Prize, with Louis Nirenberg. On the way home, on 23 May, he and Alicia die in a taxi crash in New Jersey. Enable JavaScript to see the map. The stages are listed below. John Nash’s places, from Bluefield to Princeton, and the journeys to receive the Nobel and the Abel. Map: © OpenStreetMap contributors

October 14, 2026 Read

The fifth-degree equation: the formula that does not exist

Geometers have been much occupied with the general solution of algebraic equations, and several of them have sought to prove its impossibility. But, if I am not mistaken, nobody has succeeded so far. — Niels Henrik Abel, Memoir on algebraic equations (1824) Everyone who has been to school knows one formula for solving equations: the quadratic formula. The equation $ax^2+bx+c=0$ has the roots $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$$The recipe uses only the four operations and a square root. The Babylonians could already solve problems of this kind almost four thousand years ago, and Euclid solved them with squares and rectangles. The natural question is: are there recipes like this for equations of higher degree?

October 15, 2026 Read

The game of Hex: why the first player wins

In 1948, in the common room of the Princeton mathematics building, where professors and students played chess and Go in the afternoons, John Nash presented a new game. The board is a rhombus covered with hexagonal cells, like a honeycomb. Two players, red and blue, take turns placing pieces, one at a time, on empty cells. The pieces never move. Red wants to form a chain of red pieces joining the top edge to the bottom one; blue, a blue chain joining the left edge to the right one.

October 14, 2026 Read

The binomial series and the devil's series

Divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever. — Niels Henrik Abel, letter to Holmboe (January 1826) In 1665 the young Newton discovered that the binomial formula also held for exponents that are not whole numbers: $$(1+x)^m=1+mx+\frac{m(m-1)}{2}x^2+\frac{m(m-1)(m-2)}{6}x^3+\cdots$$For a positive integer $m$ the sum stops after $m+1$ terms. For $m=\frac12$ or $m=-1$ it never stops: it is a series, a sum with infinitely many terms. Newton, Euler and all the mathematicians of the eighteenth century used this series with enormous success. Almost nobody asked when it works.

October 15, 2026 Read

The Nash equilibrium: nobody does better alone

An equilibrium point is an n-tuple such that each player’s mixed strategy maximizes his payoff if the strategies of the others are held fixed. — John Nash, Proceedings of the National Academy of Sciences (1950) In 1944 John von Neumann and the economist Oskar Morgenstern published Theory of Games and Economic Behavior, a huge book that turned strategic decisions into mathematics: card games, wars, markets. Von Neumann’s great result applied to two-player zero-sum games, in which what one wins the other loses, like chess or poker. But most of life is not like that: in a trade deal, a negotiation or traffic, both sides can win together, or lose together.

October 14, 2026 Read

Inverting the integral: elliptic functions

Start with a familiar function, seen in an odd way. The integral $$\arcsin x=\int_0^x\frac{dt}{\sqrt{1-t^2}}$$gives the length of an arc of a circle of radius 1, measured from the lowest point, as a function of the height $x$. It is a complicated function. But its inverse, which gives the height as a function of the arc, is the sine, the simplest function in trigonometry. And the sine has a property the integral hides: it is periodic. Since a quarter turn measures

October 15, 2026 Read

Mixed strategies: the penalty kick and odds and evens

Every finite game has an equilibrium point. — John Nash, Non-Cooperative Games, doctoral thesis (1950) Some games have no equilibrium at all if each player picks a fixed move. The simplest is odds and evens. Ana calls “evens”, Bruno calls “odds”, and each shows one or two fingers. If the sum is even, Ana wins a point; if it is odd, Bruno wins. Reducing each choice to “one” or “two” fingers, the payoffs are:

October 14, 2026 Read

Dividing the lemniscate with ruler and compass

In 1694 Jakob Bernoulli studied a curve shaped like a figure eight lying on its side, which he called lemniscus, a ribbon bow. With its foci at the points $(\pm\frac1{\sqrt2},0)$, it is the set of points whose distances to the two foci have product $\frac12$. Its equation is $$(x^2+y^2)^2=x^2-y^2,$$or, in polar coordinates, $r^2=\cos2\theta$. A century and a half later Abel proved about it a theorem that seems to have come out of Euclid’s book: for which numbers $n$ can the lemniscate be divided into $n$ arcs of equal length using only ruler and compass?

October 15, 2026 Read

The Nash bargaining solution

A new treatment is presented of a classical economic problem, one which occurs in many forms, as bargaining, bilateral monopoly, etc. — John Nash, The Bargaining Problem (1950) In 1948, still an undergraduate at the Carnegie Institute of Technology, Nash took a course in international economics, the only economics course of his life. He came out of it with a question: when two people negotiate, how do they share the gains from a deal? Economists thought the answer depended on each side’s psychology, skill and bluffing, and that there was nothing to say in general. Nash showed that, if the deal must satisfy a few reasonable requirements, there is only one possible answer. The paper, written before he arrived at Princeton, was published in 1950 in the journal Econometrica.

October 14, 2026 Read

Abelian groups: when order does not matter

Putting on your sock and then your shoe does not give the same result as putting on your shoe and then your sock. Adding 3 and then 5 gives the same as adding 5 and then 3. When the order of two operations does not change the result, we say they commute. The word mathematicians use for structures in which everything commutes is a tribute to Abel: abelian. The last memoir After proving that the general equation of the fifth degree has no formula, Abel wanted to know which equations do have one. In 1829, in one of the last papers he published in Crelle’s Journal, he studied a special family: equations in which all the roots can be obtained from one of them, $x$, by rational functions $\theta_1(x), \theta_2(x),\dots$ that commute with each other:

October 15, 2026 Read

Crumpling without stretching: the embedding theorem

Take a sheet of paper. You can roll it into a cylinder without tearing or stretching it: distances measured on the sheet, with a flexible ruler, stay the same. But you cannot wrap a ball with it without making folds. The sheet and the sphere have different intrinsic geometries, and Gauss proved in 1827 that this difference, the curvature, does not change when the surface is bent without stretching. In the nineteenth century Bernhard Riemann had an even bolder idea: to describe a surface, or a space of any dimension, by its internal measurements alone, without drawing it anywhere. All it takes is a rule, the metric, that says how to measure the length of any curve. This is how general relativity describes spacetime: from inside, without a larger space around it.

October 14, 2026 Read