Computer theorem prover verifies sophisticated new result

Courtesy Maria Nguyen, Quanta Magazine

Computer discovery of mathematical theorems

In 1983 the present author recalls discussing the future of mathematics with Paul Cohen, who in 1963 proved that the continuum hypothesis is independent from the axioms of Zermelo-Fraenkel set theory. Cohen was convinced that the future of mathematics, and much more, lies in artificial intelligence. Reuben Hersch recalls Cohen saying specifically that at some point in the future mathematicians would be replaced by computers. So how close are we to Cohen’s vision?

In fact, computer programs that discover new mathematical identities and theorems are already a staple

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Bach as mathematician

Johann Sebastian Bach; credit Wikimedia

OK. Johann Sebastian Bach (1685-1750) was not a mathematician in a strict sense of the word. There is no “Bach convergence theorem” in real analysis, nor is there a “Bach isomorphism theorem” in algebra. Bach had no formal training in mathematics beyond elementary arithmetic.

But, as we will see, Bach was definitely a mathematician in a more general sense, as a composer whose works are replete with patterns, structures, recursions and other precisely crafted features. There are even hints of Fibonacci numbers and the golden ratio in Bach’s music (see below). Indeed, in this larger

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AI system finds counterexamples to graph theory conjectures

Courtesy: towardsdatascience.com

AI comes of age

After several decades of disappointment, effective artificial intelligence (AI) systems emerged in the late 1990s and early 2000s, with the emergence of Bayes-theorem-based methods, combined with steadily advancing computer technology.

One notable milestone came in March 2016, when a computer program named “AlphaGo,” developed by researchers at DeepMind, a subsidiary of Alphabet (Google’s parent company), defeated champion Go master Lee Se-dol, an achievement that many observers had not expected to occur for decades. Then in October 2017, Deep Mind researchers announced a new program, AlphaGo Zero, which was programmed only with the rules

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Muon result may rewrite standard model of physics

Muon g-2 experiment (courtesy Fermilab)

A new measurement of the magnetic moment of the muon may draw into question the standard model of physics, the reigning theoretical construct describing all known fundamental forces and particles of physics. The new result was released in a paper dated today (7 April 2021) with 240 authors, led by researchers at Fermilab in the U.S., but also including researchers from Italy, Germany, United Kingdom, Russia, South Korea, China and Croatia.

The standard model of physics is arguably is the most successful physical theory ever devised, explaining all known fundamental particles and all known forces

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Aho and Ullman receive the ACM Turing Award

Alfred V. Aho (courtesy ACM)

Jeffrey D. Ullman (courtesy ACM)

The 2020 Alan M. Turing Award, bestowed by the Association for Computing Machinery, has been granted to Alfred V. Aho, Professor Emeritus at Columbia University, and Jeffrey D. Ullman, Professor Emeritus at Stanford University. The ACM Turing Award, which is named after computing pioneer Alan Turing, is widely considered to be the most prestigious award in the field of computer science. Past recipients include many of the most accomplished figures in the field, including Richard Hamming, Donald Knuth, William Kahan, Edward Feigenbaum, Jim Gray, Tim Berners-Lee, John Hennessy and David

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Two researchers share Abel prize for work in discrete mathematics and computer science

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The Abel Prize

The 2021 Abel Prize, arguably the closest equivalent in mathematics to the Nobel prize, has been awarded jointly to Avi Widgerson of the Institute for Advanced Study in Princeton, and László Lovász of the Eötvös Loránd University in Budapest, for their research linking discrete mathematics and computer science. The recipients will split the award, which is approximately USD$880,000.

According to Hanz Munthe-Kaas of the University of Bergen in Norway, who chaired the Abel Prize committee, Widgerson and Lovász “really opened up the landscape

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Pandemics, misinformation and pseudoscience

Credit: U.S. Center for Disease Control and Prevention

Covid-19 and the misinformation pandemic

The Covid-19 pandemic has disrupted human life like no other event of modern history. As of the present date (1 Mar 2021), over 114,000,000 confirmed cases and 2.5 million deaths have been recorded worldwide, according to the Johns Hopkins University database. The U.S. has recorded over 514,000 deaths, and the U.K. has recorded over 123,000. The U.S. death toll, for instance, exceeds the combined combat death toll of all wars fought in its 245-year history (save only the civil war). And for every death there are

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PiDay 2021 crossword puzzle

Yes, it is that time of year — Pi Day (March 14, or 3/14 in North American month/day date notation) is approaching. So in honor of the occasion, I have constructed a new crossword puzzle — see below. This puzzle employs a certain pi-related feature that will become evident as you solve it.

This puzzle conforms to the New York Times crossword conventions. As far as difficulty level, it would be comparable to the NYT Tuesday or Wednesday puzzles (the NYT puzzles are graded each week from Monday [easiest] to Saturday [most difficult]).

If you would like a full-page version

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Machine-learning breakthrough in protein folding

Structure of the Nsp 15 hexamer, a component of Covid-19 (Courtesy Argonne Natl Lab)

Protein folding

Proteins are the workhorses of biology. A few examples in human biology include actin and myosin, the proteins that enable muscles to work; keratin, which is the basis of skin and hair; hemoglobin, the basis of red blood that carries oxygen to cells throughout the body; pepsin, an enzyme that breaks down food for digestion; and insulin, which controls metabolism. A protein known as “spike” is the key for the coronavirus to invade healthy cells. And for every protein in human biology, there

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Do odd perfect numbers exist? New results on an old problem

Euclid, from Rafael’s “School of Athens”, Vatican Museum, Rome, photo courtesy Clay Mathematics Institute

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Perfect numbers

A perfect number is a positive integer whose divisors (not including itself) add up to the integer. The smallest perfect number is $6$, since $6 = 1 + 2 + 3$. The next is $28 = 1 + 2 + 4 + 7 + 14$, followed by $496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248$ and $8128 = 1 + 2 + 4 + 8 + 16 + 32

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