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Category: mathematics

How a 99yearold mathematician unraveled a centuryold braid mystery

In the 1930s Werner Burau, a German mathematician, introduced a twisted geometrical mystery that would stand for nearly a century.

Previously, mathematicians had shown that knots could be reformulated into something more relatable: braids. A “braid” starts with a collection of strands. To make the braid, one dangles the strands vertically and weaves them downward however they like. Any type of knot, no matter how complicated, can be translated into a braid.

As part of his investigation, Burau neatly translated braid structures into algebraic objects, making them much easier to manipulate mathematically. The objects, called matrices, are grids of numbers that function much like a spreadsheet. But mathematicians of the day worried that his elegant translation was losing information. Did some of these matrices represent more than one braid? If so, they were dubbed “unfaithful.” The problem was determining which, if any, of his braid representations were unfaithful.

Terence Tao on AI summary

A curated summary of Terence Tao’s current thinking on AI, with practical guidance and links to source material. Positions are distilled from ~70 Mastodon posts, some sixteen interviews and talks, six long-form essays and lectures, ~55 of his own blog comments, and a direct interview; not everything he has said appears here, by design — omission is editorial. Voice is third person; scope is confined to what is obviously about AI.

Latest: his ICM 2026 public lecture “Mathematics in the age of AI” (July 24, 2026) gathers much of this into one argument — the slides are linked here now, and its content will be folded into this summary once the recording is available.

How this page was made. This summary was compiled and drafted by an AI assistant (Claude) from Terence Tao’s public writing, talks, and interviews, then reviewed and corrected by him; the companion interview was conducted by that assistant, with his answers reproduced verbatim (lightly edited). It is a living document, revised as his views develop. Like the rest of tao-web, it is maintained with AI assistance.

Vernor Vinge: We Can Surpass the Wildest Dreams of Optimism

Fifteen years ago, I spent an hour with the man who gave us the word.

Vernor Vinge coined “technological singularity.” In his 1993 paper for NASA, he put a clock on it: within thirty years we would have the technological means to create superhuman intelligence. Shortly after, he wrote, the human era would end.

That deadline came and went. Vinge died in March 2024, past his own due date, which means he never got to grade his own paper. The rest of us are holding the pencil now.

The title of our conversation was his line, not mine:

We can surpass the wildest dreams of optimism.

From most people, that would be marketing. From a hard #ScienceFiction writer who spent thirty years teaching math and computer science, it was a hypothesis with conditions attached, and he was just as specific about what happens when the conditions are not met.

Continue reading “Vernor Vinge: We Can Surpass the Wildest Dreams of Optimism” | >

Physicists turn to the universe’s ‘piano notes’ to detect hidden particles

It’s been said that a finely tuned ear knows the size and shape of a piano by merely listening to the instrument’s notes. An international team of physicists has now devised an analogous approach to detect the universe’s hidden particles at high energies—opening a potential pathway for discovering new laws of physics.

The work, which will appear in the journal Physical Review Letters, outlines how effective field theory (EFT) coefficients, which quantify how new laws of physics would influence known particle interactions at low energies, can be transformed into information about the nature of these hidden particles. CERN’s Large Hadron Collider, the scientists note, already searches for values of EFT coefficients through its measurement of particle collisions, thereby providing ready-to-use data for this approach.

“Like deducing the shape and mechanism of a piano from the sound of its notes, this breakthrough provides the means to use collider measurements to deduce the details of hidden particles at high energies,” explains Grant Remmen, the James Arthur Postdoctoral Fellow at New York University and one of the paper’s authors. “This solves a classic open problem in particle physics in an elegant and useful way, providing powerful and sharp mathematical tools that bridge high-energy theory and particle physics experiments.”

A new ‘golden age’ of mathematics may be dawning, thanks to AI and human ingenuity

In May 2026, OpenAI released a new math result that sent shock waves throughout the world of mathematical research. A major unsolved problem called the “unit distance conjecture” had just been resolved by generative AI.

Since then, there has been a steady drumbeat of new results that either partially or completely leverage artificial intelligence to solve research-level mathematics problems. However, most new math results published in any given month are still generated by humans.

So where is this going? How good, and how quickly, will AI capabilities grow? Will most mathematical research be predominantly artificial intelligence? Or, as some mathematicians suggest, will AI combine with human ingenuity and other computer tools to create a golden age of mathematics?

Quantum Neural Networks Face the Hardware Test

Artificial neural networks have become powerful tools for finding patterns in complex data, from classifying images to predicting protein structures and assisting mathematical discovery. Yet their success has so far relied almost entirely on classical hardware. Recent developments in quantum-computing technologies make it timely to ask whether trainable models can also make use of quantum effects such as superposition and the intrinsic uncertainty associated with quantum measurements. What’s more, running neural networks on real quantum processors could potentially turn these networks into probes, revealing how different hardware architectures shape networks’ behaviors.

Philosophy Of Physics (@PhilosophyOfPhy) on X

The continuity equation was not the work of a single physicist. Its development grew from early hydraulic studies and the work of Daniel and Johann Bernoulli. In the eighteenth century, Jean le Rond d’Alembert produced the first partial-differential expression of mass conservation in fluid motion, and Leonhard Euler soon placed it in the general mathematical framework that became the foundation of modern fluid mechanics. It should therefore not be attributed solely to Giovanni Battista Venturi, whose later work concerned flow through constricted tubes. Its general form is ∂ρ/∂t + ∇·(ρv) = 0 where ρ is fluid density and v is the velocity field. The equation says that mass cannot simply appear or disappear: any change in the amount of fluid inside a region must be explained by fluid entering or leaving it. For steady flow through a pipe, this becomes ρ₁A₁v₁ = ρ₂A₂v₂ If the fluid is effectively incompressible, its density remains constant, giving the familiar form: A₁v₁ = A₂v₂ The meaning is simple. The same volume of fluid must pass through every section of the pipe each second. When the pipe becomes narrower, the fluid must move faster; when it becomes wider, the fluid slows down. This equation is fundamental to the study of pipes, nozzles, rivers, aircraft flow, circulation systems and computational fluid dynamics. More broadly, continuity equations appear throughout physics wherever something locally conserved, such as mass or electric charge, moves through space. The equation is not merely about fluids; it is the mathematical language of the principle that what flows into a region must either flow out or remain inside.

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