Title: The Shaw Prize Lecture in Mathematics 2025: From Shape to Space.
Date: Oct 22, 2025
Speaker: Prof. Kenji FUKAYA
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Category: mathematics – Page 2
Music of the Spheres and the Lessons of Pythagoras
I. Using simple mathematics, Pythagoras was able to describe the basis of almost all musical scales, including the pentatonic, the Western, the chromatic and the Arabic scales. This shows the power and excitement of science. For the first time, Pythagoras could answer the question, WHY? Why are these notes and scales special? The answer is that they are formed in a simple, systematic, and mathematical manner. Most importantly, Pythagoras showed that the notes are not random or arbitrary and that they could be understood on a deeper level.
II. Pythagorass discoveries bring up a deeper psychology question: scales were first developed by ear: we and the Neanderthals choose these particular notes before there was any understanding of mathematics or physics. The notes were chosen simply because they were pleasing to the ear. But, as it turns out, the scales also follow basic mathematical constructs. So the question is, what does this say about our likes and emotions? Is there a mathematical/physical basis to them, as well?
III. The power of spectroscopy. What Pythagoras did was look a physical system (the musical scale), found characteristic frequencies (pitches/notes) and found simple mathematical relationships between the frequencies (ratios of 3/2, for example). This process actually became a fundamental part of physics, and modern physics, in particular.
Statistical mechanics method helps machines better understand complex systems
A study by University of Hawaiʻi researchers is advancing how we learn the laws that govern complex systems—from predator-prey relationships to traffic patterns in cities to how populations grow and shift—using artificial intelligence (AI) and physics.
The research, published in Physical Review Research, introduces a new method based on statistical mechanics to improve the discovery of equations directly from noisy real-world data. Statistical mechanics is a branch of physics that explains how collective behavior emerges from individual particles, such as how the random motion of gas molecules leads to predictable changes in pressure and temperature.
In this new work, statistical mechanics is used to understand how different mathematical models “compete” when trying to explain a system. This matters because many scientific fields rely on understanding how systems change over time, whether tracking disease spread, analyzing climate change or predicting the stock market. But real-world data is often messy, and traditional AI models can be unreliable when the data gets noisy or incomplete.
Mathematical model reveals why cracks sharpen during rapid rubber fracture
A research group from the University of Osaka, Zen University, and the University of Tokyo has mathematically uncovered the mechanism that causes crack tips to sharpen during the rapid fracture of rubber.
The bursting of rubber balloons or tire blowouts is caused by rapid fracture, a phenomenon in which a small crack propagates instantaneously. During this process, the crack tip sharpens, accelerating the fracture. However, the reason behind this sharpening had long remained unexplained. Traditionally, it was believed to result from the material’s complex nonlinear effects.
The research group—comprising Hokuto Nagatakiya, a doctoral student; Shunsuke Kobayashi, assistant professor; and Ryuichi Tarumi, professor at the University of Osaka; along with Naoyuki Sakumichi, associate professor at Zen University and project associate professor at the University of Tokyo—has mathematically solved the problem of crack propagation. They derived equations that describe both the shape of the crack and the overall deformation of the material.
A mathematical ‘Rosetta Stone’ translates and predicts the larger effects of molecular systems
Penn Engineers have developed a mathematical “Rosetta Stone” that translates atomic and molecular movements into predictions of larger-scale effects, like proteins unfolding, crystals forming and ice melting, without the need for costly, time-consuming simulations or experiments. That could make it easier to design smarter medicines, semiconductors and more.
In a recent paper in Journal of the Mechanics and Physics of Solids, the Penn researchers used their framework, stochastic thermodynamics with internal variables (STIV), to solve a 40-year problem in phase-field modeling, a widely used tool for studying the shifting frontier between two states of matter, like the boundary between water and ice or where the folded and unfolded parts of a protein join.
“Phase-field modeling is about predicting what happens at the thin frontier between phases of matter, whether it’s proteins folding, crystals forming or ice melting,” says Prashant Purohit, Professor in Mechanical Engineering and Applied Mechanics (MEAM) and one of the paper’s co-authors. “STIV gives us the mathematical machinery to describe how that frontier evolves directly from first principles, without needing to fit data from experiments.”
Algorithm precisely quantifies flow of information in complex networks
Networks are systems comprised of two or more connected devices, biological organisms or other components, which typically share information with each other. Understanding how information moves between these connected components, also known as nodes, could help to advance research focusing on numerous topics, ranging from artificial intelligence (AI) to neuroscience.
To measure the directional flow of information in systems, scientists typically rely on a mathematical construct known as transfer entropy, which essentially quantifies the rate at which information is transmitted from one node to another. Yet most strategies for calculating transfer entropy developed so far rely on approximations, which significantly limits their accuracy and reliability.
Researchers at AMOLF, a fundamental physics institute in the Netherlands, recently developed a computational algorithm that can precisely quantify transfer entropy in a wide range of complex networks. Their algorithm, introduced in a paper published in Physical Review Letters, opens new exciting possibilities for the study of information transfer in both biological and engineered networks.
These Tiny Robots Can Swarm, Adapt, and Heal Themselves
Scientists designed microrobots that use sound to swarm, adapt, and heal themselves — working together like a living organism. The discovery could transform medicine, environmental cleanup, and robotics.
Nature’s Blueprint for Robot Swarms
Animals such as bats, whales, and insects have long relied on sound to communicate and find their way. Drawing inspiration from this, an international group of scientists has developed a model for tiny robots that use sound waves to move and work together in large, coordinated swarms that behave almost intelligently. According to team leader Igor Aronson, Huck Chair Professor of Biomedical Engineering, Chemistry, and Mathematics at Penn State, these robotic collectives could eventually take on challenging missions like exploring disaster areas, cleaning polluted environments, or performing medical procedures inside the human body.
Generation of harmful slow electrons in water is a race between intermolecular energy decay and proton transfer
When high-energy radiation interacts with water in living organisms, it generates particles and slow-moving electrons that can subsequently damage critical molecules like DNA. Now, Professor Petr Slavíček and his bachelor’s student Jakub Dubský from UCT Prague (University of Chemistry and Technology, Prague) have described in detail one of the key mechanisms for the creation of these slow electrons in water, a process known as Intermolecular Coulombic Decay (ICD). Their powerful mathematical model successfully explains all the data from complex laser experiments conducted at ETH Zurich (Hans-Jakob Woerner team).
The work, which deepens the fundamental understanding of radiation chemistry, has been published in the journal Nature Communications.
A detailed knowledge of the processes in aqueous solutions, combined with advances in research technologies using high-energy radiation, is transforming the field of radiation chemistry. In the future, these insights could lead to significant changes in various fields, including medicine, particularly in developing more sensitive and controllable applications for devices based on ionizing radiation.
How do you trust a robot you’ve never met?
Many of the environments where human-facing universal robots can provide benefits — homes, hospitals, schools — are sensitive and personal. A tutoring robot helping your kids with math should have a track record of safe and productive sessions. An elder-care assistant needs a verifiable history of respectful, competent service. A delivery robot approaching your front door should be as predictable and trustworthy as your favorite mail carrier. Without trust, adoption will never take place, or quickly stall.
Trust is built gradually and also reflects common understanding. We design our systems to be explainable: multiple AI modules talk to each other in plain language, and we log their thinking so humans can audit decisions. If a robot makes a mistake — drops the tomato instead of placing it on the counter — you should be able to ask why and get an answer you can understand.
Over time, as more robots connect and share skills, trust will depend on the network too. We learn from peers, and machines will learn from us and from other machines. That’s powerful but just like parents are concerned about what their kids learn on the web, we need good ways to audit and align skill exchange for robots… Governance for human–machine societies isn’t optional; it’s fundamental infrastructure.