algebra – Quanta Magazine

‘Groups’ Underpin Modern Math. Here’s How They Work.

Leila Sloman — Fri, 06 Sep 2024 15:05:01 +0000

Mathematics started with numbers — clear, concrete, intuitive. Over the last two centuries, however, it has become a far more abstract enterprise. One of the first major steps down this road was taken in the late 18th and early 19th centuries. It involved a field called group theory, and it changed math — theoretical and applied — as we know it. Groups generalize essential properties of the whole...

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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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What Makes for ‘Good’ Mathematics?

Steven Strogatz — Thu, 01 Feb 2024 14:03:12 +0000

We tend to think of mathematics as purely logical, but the teaching of math, its values, its usefulness and its workings are packed with nuance. So what is “good” mathematics? In 2007, the mathematician Terence Tao wrote an essay for the Bulletin of the American Mathematical Society that sought to answer this question. Today, as the recipient of a Fields Medal, a Breakthrough Prize in Mathematics...

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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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‘A-Team’ of Math Proves a Critical Link Between Addition and Sets

Leila Sloman — Wed, 06 Dec 2023 15:40:14 +0000

In a randomly chosen set of numbers, addition can go wild. Add together every pair from such a set, and you’ll end up with a new list that’s likely to have a lot more numbers than what you started with. Start with 10 random numbers, and this new list (called the sumset) will have about 50 elements. Start with 100 and the sumset will probably have around 5,000; 1,000 random initial numbers will...

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