Peter Borwein: A visionary mathematician

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Peter Borwein

Peter Borwein, former professor of mathematics at Simon Fraser University and director of the university’s Centre for Interdisciplinary Research in the Mathematical and Computational Sciences (IRMACS), died on August 23, 2020, at the age of 67, of pneumonia, after courageously battling multiple sclerosis for over 20 years.

The Notices of the American Mathematical Society has just published a memorial tribute, written by the present author, that summarizes Peter’s life and career. Here are a few highlights:

Peter Borwein is perhaps best known for discovering (often but not always with his brother Jonathan) new

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PiDay 2023 crossword puzzle

So soon? Yes, it is that time of year again — PiDay (March 14) is just a few days away. In honor of this mathematical holy day, we present once again a crossword puzzle with an appropriate pi-related theme.

This year’s puzzle implements a new design for a mathematical crossword, which to the present author’s knowledge has never before been employed. See, for example, clue 40 Across below. In all respects, though, the puzzle conforms to the standards of New York Times crosswords. In terms of overall difficulty (Monday = easiest; Saturday = most difficult), this puzzle most likely would

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Can ChatGPT prove math theorems?

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ChatGPT: A milestone in artificial intelligence-based language models

Many readers have doubtless heard of ChatGPT, the latest instance of a language generation tool developed by the technology startup OpenAI. This tool, which is now available for public experimental use, takes as input a request or other statement from the user, then responds. It employs a dialogue format, which makes it possible for ChatGPT to answer followup questions, and even admit its mistakes.

It is clear, even from a cursory examination, that ChatGPT represents a rather dramatic advance in artificial intelligence. Some of the results are rather

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Machine learning program finds new matrix multiplication algorithms

Credit: Wikimedia

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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

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Breakthrough Prizes honor AlphaFold and quantum computing pioneers

High-level 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 early-career awards have smaller stipends, typically US$50,000). The Breakthrough Prizes were founded by Sergey Brin (co-founder of Google, now Alphabet), Pricilla Chan, Mark Zuckerberg (co-founder 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

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2022 Fields Medalists: Diverse backgrounds, breakthrough mathematics

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

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Advances in artificial intelligence raise major questions

Artificial intelligence in technology; credit: iStock-metamorworks

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

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Where are the extraterrestrials? Fermi’s paradox, diversity and the origin of life

Delicate Arch at night; credit:


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

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PiDay 2022 crossword puzzle

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 custom-constructed crossword puzzle to commemorate the occasion.

This year’s puzzle commemorates a well-known pi-related 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

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Latest experimental data compounds the Hubble constant discrepancy

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The standard model and the Lambda-CDM 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 Lambda-CDM or Λ-CDM), which governs the evolution of the entire universe from the

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