Langlands program – Quanta Magazine

Monumental Proof Settles Geometric Langlands Conjecture

Erica Klarreich — Fri, 19 Jul 2024 14:02:17 +0000

A group of nine mathematicians has proved the geometric Langlands conjecture, a key component of one of the most sweeping paradigms in modern mathematics. The proof represents the culmination of three decades of effort, said Peter Scholze, a prominent mathematician at the Max Planck Institute for Mathematics who was not involved in the proof. “It’s wonderful to see it resolved.

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What Are Sheaves?

Konstantin Kakaes — Fri, 19 Jul 2024 14:00:55 +0000

In 1940, the French mathematician and artillery officer Jean Leray was taken prisoner by the Germans. He told his captors that he was a topologist, fearful that if they discovered his true area of expertise, hydrodynamics, they would force him to aid the German war effort. For the nearly five years of his imprisonment, Leray kept up this subterfuge by carrying out research in topology...

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A Rosetta Stone for Mathematics

Kevin Hartnett — Mon, 06 May 2024 14:23:30 +0000

In 1940, from a jailhouse in Rouen, France, André Weil wrote one of the most consequential letters of 20th-century mathematics. He was serving time for refusing to join the French army, and he filled his days in part by writing letters to his sister, Simone, an accomplished philosopher living in London. In a previous letter, Simone had asked André to tell her about his work. With war all around...

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Elliptic Curve ‘Murmurations’ Found With AI Take Flight

Lyndie Chiou — Tue, 05 Mar 2024 15:12:30 +0000

Elliptic curves are among the more beguiling objects in modern mathematics. They don’t seem complicated, but they form an expressway between the math that many people learn in high school and research mathematics at its most abstruse. They were central to Andrew Wiles’ celebrated proof of Fermat’s Last Theorem in the 1990s. They are key tools in modern cryptography. And in 2000...

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The Year in Math

Konstantin Kakaes — Sat, 23 Dec 2023 01:33:58 +0000

Mathematical truths are often born of the conflict between order and disorder. Mathematicians discover patterns, and, to better understand the mysterious forces at play, they look for countervailing impulses that disrupt those patterns. That tension came up repeatedly in our coverage this past year. We covered breakthroughs in graph theory, combinatorics, number theory and geometry — areas where...

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