Credit: Lawrence Berkeley National Lab
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The Hubble tension
One experimental anomaly that has bedeviled physicists and cosmologists for years is the discrepancy in values of the Hubble constant based on different experimental approaches. This discrepancy is now known as the Hubble tension.
The Hubble constant $H_0$ is a measure of the rate of expansion of the universe, and is directly connected to estimates of the age $A$ of the universe via the relation $A = 1 / H_0$. Units must be converted here, since the age of the universe is
Continue reading New galaxy map challenges cosmological models
So soon? Yes, it is that time of year again: Pi Day (March 14, in North American month/date notation) is nearly upon us.
Each year the Math Scholar site celebrates Pi Day with a custom-written crossword puzzle on a mathematical theme. This year’s edition (see below) highlights a memorable comment made by Isaac Newton about his own computation of pi (see puzzle entries 19 Across, 32 Across, 56 Across, 77 Across and 96 Across). However, no particular mathematical knowledge is required to solve the puzzle.
This puzzle is written to the standards of New York Times crossword puzzles (see here),
Continue reading Pi Day 2025 crossword puzzle
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Credit: Vatican Museum
NOTE: A typeset PDF version of this article, with references, is available HERE.
Introduction
The first rigorous mathematical calculation of $\pi$ was due to Archimedes (ca. 250 BCE), who used a scheme of inscribed and circumscribed polygons to obtain the bounds $3 \frac{10}{71} < \pi < 3 \frac{1}{7}$. The fifth century Indian mathematician Aryabhata produced four digits; the Chinese mathematician Tsu Chung-Chih produced seven. Following the discovery of calculus by Newton and Leibniz in the 1600s, several new formulas for $\pi$ were found.
Continue reading Simple proofs: Pi is transcendental
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DeepSeek performance versus OpenAI
DeepSeek’s big splash
By now, many readers have likely heard about DeepSeek, a new AI software system developed by a team in China. The latest version (R1) was introduced on 20 Jan 2025, while many in the U.S. were preoccupied by Donald Trump’s inauguration. DeepSeek is variously termed a generative AI tool or a large language model (LLM), in that it uses machine learning techniques to process very large amounts of input text, then in the process becomes uncannily adept in generating responses to new queries. It represents yet another
Continue reading DeepSeek: A breakthrough in AI for math (and everything else)
Updated 14 April 2025
Life expectancy, 1770-2023; source: Our World in Data
First the bad news
Today there is no shortage of discouraging news headlines, among them (as of 21 Jan 2025):
A war in Ukraine that has dragged on for three years. A war in Gaza that raises questions about the long-term stability of the Middle East. Simmering conflicts in Sudan, Myanmar, Niger, Afghanistan, Colombia and Somalia. A constitutional crisis in South Korea. Wildfires in the Los Angeles area that are among the costliest natural disasters in history. A never-ending stream of floods, hurricanes and other destructive
Continue reading Behind the scary headlines: Sustained scientific and social progress
Schematic of GenCast’s procedure to generate a weather forecast. Credit: Ilan Price, Alvaro Sanchez-Gonzalez, Ferran Alet, Tom R. Andersson, Andrew El-Kadi, Dominic Masters, Timo Ewalds, Jacklynn Stott, Shakir Mohamed, Peter Battaglia, Remi Lam and Matthew Willson [Nature]
Recent advances in machine learning (ML) and artificial intelligence (AI) have been nothing short of astonishing. Here are just a few notable milestones from the past decade:
In 2011, IBM’s “Watson” computer system defeated two premier champions of the American quiz show Jeopardy!. In 2017, AlphaGo Zero, developed by DeepMind, a subsidiary of Alphabet (Google’s parent company), defeated an earlier program, also developed
Continue reading AI weather forecaster beats the best operational system
Credit: NASA Goddard Space Flight Center
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The standard model of mathematical physics
The standard model, namely the framework of laws at the foundation of modern mathematical physics, has reigned supreme since the 1970s, having been confirmed to great precision in a vast array of experimental tests. Among other things, the standard model predicted the existence of the Higgs boson, which was experimentally discovered in 2012, nearly 50 years after it was first predicted.
Yet physicists have recognized for many years that the standard model cannot be the final answer. For example, quantum theory and
Continue reading The “Hubble tension”: A growing crisis in cosmology
Model of human nuclear pore complex, built using AlphaFold2; credit: Agnieszka Obarska-Kosinska, Nature
AlphaFold 2: A breakthrough in computational protein folding
The 2024 Nobel Prize in Chemistry has been awarded to David Baker of the University of Washington and to Demis Hassabis and John Jumper of Google DeepMind. Hassabis and Jumper, now at the Google DeepMind Research Center in the U.K. (a subsidiary of Alphabet, Google’s parent company), developed an artificial intelligence-based software package that predicts with remarkable accuracy the structure of proteins. Baker, now at the University of Washington in Seattle, was recognized for his work on designing
Continue reading 2024 Nobel Prize in Chemistry awarded to computational protein folding pioneers
Credit: Valentin Tkach for Quanta Magazine
Computer-assisted tools for research mathematics
In a previous Math Scholar article, we highlighted some recent developments in sophisticated computer tools being applied to the enterprise of research mathematics. These tools include:
Typesetting tools (usually LaTeX), combined with tools such as MathJax for embedding typeset mathematics into web pages. Collaboration tools such as blogs, FaceTime, Skype, Zoom, Slack and Microsoft Teams to facilitate communication and collaboration between researchers. Symbolic mathematical software such as Mathematica, Maple and Sage to perform increasingly powerful symbolic manipulations and derivations, and to generate graphics illustrating results. Custom-written code, often
Continue reading Terence Tao’s vision of AI assistants in research mathematics
Updated 24 September 2024 (c) 2024
Credit: Walker, Packer, Cody, “Re-conceptualizing the origins of life,” https://royalsocietypublishing.org/doi/10.1098/rsta.2016.0337
Introduction
In the past few decades, modern science has uncovered a universe that is far vaster and more awe-inspiring than ever imagined before, together with a set of elegant natural laws that deeply resonate with the idea of a cosmic lawgiver. In spite of these exhilarating developments, some writers, principally of the creationist and intelligent design communities, prefer a highly combative approach to science, particularly to geology and evolution.
In addition to citing deeply flawed probability-based arguments (see Probability), these evolution skeptics
Continue reading New developments in the origin of life on Earth
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