DeepSeek: A breakthrough in AI for math (and everything else)

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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

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Behind the scary headlines: Sustained scientific and social progress

Updated 1 February 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

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AI weather forecaster beats the best operational system

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

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The “Hubble tension”: A growing crisis in cosmology

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

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2024 Nobel Prize in Chemistry awarded to computational protein folding pioneers

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

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Terence Tao’s vision of AI assistants in research mathematics

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

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New developments in the origin of life on Earth

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

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Data science and physics give Olympic swimmers the edge

Credit: Mykal McEldowney, The Indianapolis Star/ USA TODAY Network

What do American Olympic swimmers Kate Douglass, Gretchen Walsh and Paige Madden, French gold-medalist Leon Marchand, and Australian Olympian Kyle Chambers have in common? In their training, these five leading swimmers and numerous others have employed (and, in the case of Douglass, personally helped develop) advanced data science and physics techniques to fine-tune their athletic performance.

Many of these top swimmers, who have traditionally relied on their “feel” in the water, now employ accelerometers strapped to their bodies, measuring movements in 3-D up to 512 times per second, thus capturing

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Computation, AI and the future of mathematics

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Optimal stacking of oranges.

Computer tools in mathematics: From foot dragging to full embrace

The present author recalls, while in graduate school at Stanford nearly 50 years ago, hearing two senior mathematicians (including the present author’s advisor) briefly discuss some other researcher’s usage of computation in a mathematical investigation. Their comments and body language clearly indicated a profound disdain for such efforts. Indeed, the prevailing mindset at the time was “real mathematicians don’t compute.”

Some researchers at the time attempted to draw more attention to the potential of computer tools in mathematics. In the

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Pi Day 2024 crossword puzzle

Yes, Pi Day (March 14), that day of mathematical trivia and pie-eating, is almost here again.

As is the custom on this site, we present below a crossword puzzle constructed on a theme of mathematics, computing and the digits of pi. Enjoy!

This puzzle has been constructed in conformance with New York Times crossword puzzle standards. In terms of overall difficulty, my mathematician daughter and my genealogist spouse agreed that it would likely rate as a Tuesday (the New York Times grades its puzzles, with Monday puzzles being the easiest and Saturday puzzles being the most challenging).

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