Analysis

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 15, 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 16, 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 15, 2026 Read

The legacy of Niels Henrik Abel

Abel has left mathematicians enough work for a hundred and fifty years. — Saying attributed to Charles Hermite Abel worked as a mathematician for less than eight years. In that time he settled the oldest question in algebra, brought rigour to series, created elliptic functions and wrote a memoir that was only fully understood decades after his death. His name is now found in more places in mathematics than that of almost anyone else: abelian groups, abelian integrals, abelian functions, abelian varieties, Abel’s theorem, Abel summation, Abel’s test.

October 16, 2026 Read

The heat equation and the regularity of De Giorgi and Nash

In 1900, at the International Congress of Mathematicians in Paris, David Hilbert presented a list of 23 problems that, in his view, would guide twentieth-century mathematics. Problem 19 asked, in simple terms: are the solutions of the “well-behaved” problems of the calculus of variations always smooth functions? Half a century later, the question was still open for an important class of equations. In 1956 Nash heard about it from a colleague and decided to attack it.

October 15, 2026 Read

Image credits

Historical images and photographs All come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. Portrait of Abel, drawing by Johan Gørbitz, Paris, 1826: source page · public domain. First volume of Crelle’s Journal, Berlin, 1826: source page · public domain. Abel’s grave at Froland, photo by Karl Ragnar Gjertsen: source page · CC BY 4.0. Abel monument in Oslo, sculpture by Gustav Vigeland, photo by Wolfmann: source page · CC BY-SA 4.0. Blog figures The figure of the lemniscate divided into eight equal arcs is an original illustration. The points were computed with the formula $r=\sqrt{\sqrt2-1}$ derived in the post. The figure has no text and serves every language of the site.

October 16, 2026 Read

The legacy of John Nash

Statistically, it would seem improbable that any mathematician or scientist, at the age of 66, would be able through continued research efforts, to add much to his or her previous achievements. However I am still making the effort. — John Nash, autobiographical note for the Nobel Foundation (1994) John Nash is the only person to have received both the Nobel Prize and the Abel Prize, the two highest honours in economics and in mathematics. The two halves of his work seem to have nothing in common: one is about human decisions, the other about surfaces and equations. What unites them is the style: attacking problems everyone thought impossible with ideas nobody had had.

October 15, 2026 Read

Image credits

Photographs and images All come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. Portrait of John Nash, photo by Peter Badge: source page · CC BY-SA 3.0. John Nash in 2006, in Cologne, photo by Elke Wetzig: source page · CC BY-SA 3.0. Isometric embedding of the flat torus, Hevea project, uploaded by Romainbehar: source page · CC0. Blog figures The cover, the Hex board and the graph of the product of gains in bargaining are original illustrations, with versions in each language of the site. The board shows an invented game, in which red has completed its chain.

October 15, 2026 Read

The legacy of Augustin-Louis Cauchy

Cauchy is mad, and there is nothing to be done with him; and yet he is, at present, the one who knows how mathematics ought to be done. — Niels Henrik Abel, letter to Holmboe (1826) Cauchy published about 800 papers and seven books. His Œuvres complètes, edited by the Academy of Sciences, came out between 1882 and 1974 in 27 thick volumes. Among mathematicians, only Euler and very few others left a body of work comparable in size.

October 31, 2026 Read