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Category: mathematics – Page 2

Researchers Simulate Alzheimer’s Progression Across the Entire Brain

This hypothesis has led to a new “whole-brain” mathematical and computational model developed at the MOX Laboratory of the Department of Mathematics of Politecnico di Milano. The model is designed to describe in an integrated way the interaction between the spread of amyloid beta and the functioning of the cerebral vascular network. The aim is to provide a tool capable of simulating, on the scale of the whole organ, how small biological or vascular alterations can evolve over time and contribute to neurodegeneration. The model and the accompanying scientific study have been published in the prestigious scientific journal Computer Methods in Applied Mechanics and Engineering.

The model integrates two scales of analysis. On the one hand, it describes the dynamics of the production, transformation, diffusion and elimination of the healthy and pathological forms of amyloid beta. On the other, it describes blood flow through a “macroscopic” description of arteries, capillaries and veins, treating brain tissue as a porous medium perfused by blood vessels, through a macroscopic compartmental model. The two components are then connected to represent the possible mechanism of mutual reinforcement between protein accumulation and vascular dysfunction.

The simulations show a particularly relevant result: the brain can evolve into different states depending on the initial conditions. Small localised amounts of amyloid beta can be eliminated, allowing the system to return to a healthy state. Conversely, larger amounts can trigger a self-sustained spread of the pathology at brain scale.

New twist on the Einstein problem reveals unexpected physics

A shape that captured worldwide attention for solving a decades-old mathematical puzzle has returned to the spotlight. While the shape’s properties allowed it to solve previous puzzles, little is known about its other associated properties, creating opportunities for further discovery. These unexplored properties may also help solve new physics mysteries, such as how to twist light into striking chiral patterns.

In an article published in Nature Communications, researchers from the Institute of Industrial Science, The University of Tokyo, and collaborating institutions created optical structures based on the “Smith hat,” an unusual shape that solves the so-called Einstein problem in mathematics. By shining laser light onto these structures, the team discovered diffraction behaviors never before observed in conventional quasicrystals.

The long-standing Einstein problem asks whether a single tile shape, or “monotile,” can be used to cover an entire surface in a nonrepeating pattern. While periodic tilings such as honeycombs or checkerboards repeat regularly, an aperiodic monotile can cover all of space without ever producing a repeating arrangement. In 2023, the first such monotile, the Smith hat, was discovered, sparking widespread interest in the scientific community.

Physicists link the Riemann Hypothesis to phase transitions in quantum systems

A new study in Nature Communications has established a link between the Riemann Hypothesis and dynamical phase transitions in engineered quantum systems, demonstrating the effect on a quantum processor.

First posed in 1859, the Riemann Hypothesis is one of the longest-standing unsolved problems in mathematics. It underpins parts of cryptography, as well as more than a thousand theorems proved on the assumption that it is true.

Physicists have previously proposed physical counterparts to this mathematical statement. The aim was to map the Riemann Hypothesis to something concrete, such as the energy levels of a quantum system. The new study ties the hypothesis to how a quantum system evolves over time.

Claude AI finds cryptography weaknesses human experts missed

AI company Anthropic has announced that its unreleased model, Claude Mythos Preview, has discovered previously unknown mathematical weaknesses in cryptographic algorithms that human researchers had missed for years.

Cryptography uses sophisticated algorithms to protect web traffic, email, software updates and sensitive information like banking details. It does this by rendering the data completely unreadable.

The Anthropic team describes its work in two papers.

Trembling Photons in Non-Abelian Electric Fields

A ring of optical fiber can be made to host phenomena that originated in the realm of high-energy physics.

In non-Abelian gauge theories, particles interact with each other; operation order matters; and a so-called gauge symmetry ensures the invariance of physical laws under local transformations via fields mediated by photons, gluons, and the W and Z bosons. Such theories describe the strong and weak forces; provide the mathematical backbone of the standard model; and underpin efforts to understand the early Universe, quantum gravity, and exotic topological phases of matter. For decades, subjecting these theories to direct experimental scrutiny often meant resorting to enormous particle accelerators. Recently, an experimental approach known as photonic synthetic dimensions has offered a tabletop alternative. Applications of the approach have already realized a non-Abelian magnetic field [1]. Now Shu Yang of the University of Hong Kong and his colleagues have synthesized the corresponding non-Abelian electric field [2].

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.”

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