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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 15, 2026 Read
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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 14, 2026 Read
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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 15, 2026 Read
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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 14, 2026 Read
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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 14, 2026 Read