
Credit: Wikimedia
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Introduction
Most of us learn the basic scheme for matrix multiplication in high school. What could be more simple and straightforward? Thus it may come as a surprise to some the basic scheme is not the most efficient.
German mathematician Volker Strassen was the first to show this, in 1969, by exhibiting a scheme that yields practical speedups even for moderately sized matrices. In the years since Strassen’s discovery, numerous other researchers have found even better schemes for certain specific matrix size classes. For a good overview of these methods, together with some
Continue reading Machine learning program finds new matrix multiplication algorithms
Highlevel diagram of Shor’s algorithm for factoring integers. Credit: ResearchGate; Archimedes Pavlidis
The Breakthrough Prizes
The Breakthrough Prizes are awarded annually for scientists who do groundbreaking work addressing fundamental questions in physics, mathematics, computer science and life sciences. The prizes include a stipend of US$3 million (some earlycareer awards have smaller stipends, typically US$50,000). The Breakthrough Prizes were founded by Sergey Brin (cofounder of Google, now Alphabet), Pricilla Chan, Mark Zuckerberg (cofounder of Facebook, now Meta), Yuri Milner (founder of DST Global, a global technology investor), Julia Milner, and Anne Wojcicki (founder of 23andMe, a genomics firm).
The latest
Continue reading Breakthrough Prizes honor AlphaFold and quantum computing pioneers
The 2022 recipients of the Fields Medal, arguably the highest honor in the field of research mathematics, have been announced by the International Mathematical Union, as part of the quadrennial International Congress of Mathematicians, which this year is being held in Helsinki, Finland.
This year’s award recipients are interestingly diverse. One was raised in Ukraine, and grieves over her childhood city being bombed in the current military activity; one is known for his passionately independent approach to both life and mathematics; one is very active athletically, and has often found key insights while engaged in these activities; and one
Continue reading 2022 Fields Medalists: Diverse backgrounds, breakthrough mathematics
Artificial intelligence in technology; credit: iStockmetamorworks
A brief history
The modern field of artificial intelligence (AI) arguably dates to 1950, when Alan Turing outlined the basics of AI in his paper “Computing machinery and intelligence” [Paper]. He even proposed a test, now known as the Turing test, for establishing whether true AI had been achieved. Early computer scientists were confident that true AI system would soon be a reality. In 1965 Herbert Simon wrote that “machines will be capable, within twenty years, of doing any work a man can do.” In 1970 Marvin Minsky declared, “In from three to
Continue reading Advances in artificial intelligence raise major questions
Delicate Arch at night; credit: Astronomy.com
Introduction
In 1950, while having lunch with colleagues Edward Teller and Herbert York, who were chatting about a recent cartoon in the New Yorker depicting aliens, physicist Enrico Fermi suddenly blurted out, “Where is everybody?,” a question now known as Fermi’s paradox. This article presents background on Fermi’s paradox, explains why many of the proposed solutions are not viable, and mentions a few promising new results and directions.
Behind Fermi’s question was this line of reasoning: (a) Given the vast number of stars in the Milky Way (not to mention the larger
Continue reading Where are the extraterrestrials? Fermi’s paradox, diversity and the origin of life
So soon? Yes, it is that time of year again — PiDay, namely Mar 14 (from 3/14 in North American date notation) is less than two weeks away. Continuing a long tradition on the Math Scholar blog, we present a customconstructed crossword puzzle to commemorate the occasion.
This year’s puzzle commemorates a wellknown pirelated theorem, one of the most beautiful facts in mathematics, which was originally discovered in the 18th century. The theorem is stated, in full, in the completed puzzle (see clues 20A, 30A and 44A).
My spouse and one daughter, who solved the puzzle, agreed that in terms
Continue reading PiDay 2022 crossword puzzle
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The standard model and the LambdaCDM model
The standard model of physics, namely the framework of mathematical laws at the foundation of modern physics, has reigned supreme since the 1970, having been confirmed in countless exacting experimental tests. Perhaps its greatest success was the prediction of the Higgs boson, which was experimentally discovered in 2012, nearly 50 years after it was first predicted.
One application of the standard model, together with general relativity, is the Lambda Cold Dark Matter model (often abbreviated LambdaCDM or ΛCDM), which governs the evolution of the entire universe from the
Continue reading Latest experimental data compounds the Hubble constant discrepancy
Block function approximation to normal distribution
Introduction
Today, arguably more than ever before, the world is governed by the science of probability and statistics. “Big data” is now the norm in scientific research, with terabytes of data streaming into research centers from satellites and experimental facilities, analyzed by supercomputers. “Data mining” is now an essential part of mathematical finance and business management. Numerous public opinion polls, expertly analyzed, guide the political arena. Covid19 infection rates, immunization levels and r0 factors are a staple of nightly newscasts.
Yet the public at large remains mostly ignorant of the basic principles
Continue reading The brave new world of probability and statistics
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 ZermeloFraenkel 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
Continue reading Computer theorem prover verifies sophisticated new result
Johann Sebastian Bach; credit Wikimedia
OK. Johann Sebastian Bach (16851750) 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
Continue reading Bach as mathematician

