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Category: mathematics

Math, Inc.

The Math Inc. team is excited to introduce Gauss, a first-of-its-kind autoformalization agent for assisting human expert mathematicians at formal verification. Using Gauss, we have completed a challenge set by Fields Medallist Terence Tao and Alex Kontorovich in January 2024 to formalize the strong Prime Number Theorem (PNT) in Lean (GitHub).

The translation of human mathematics into verifiable machine code has long been a grand challenge. However, the cost of doing so is prohibitive, requiring scarce human expertise. In particular, after 18 months, Tao and Kontorovich recently announced intermediate progress in July 2025 toward their goal, obstructed by core difficulties in the field of complex analysis.

In light of such difficulties, we are pleased to announce that with Gauss, we have completed the project after three weeks of effort. Gauss can work autonomously for hours, dramatically compressing the labor previously reserved for top formalization experts. Along the way, Gauss formalized the key missing results in complex analysis, which opens up future initiatives previously considered unapproachable.

The Universal Law Behind Market Price Swings

Analysis of a large dataset from the Tokyo Stock Exchange validates a universal power law relating the price of a traded stock to the traded volume.

One often hears that economics is fundamentally different from physics because human behavior is unpredictable and the economic world is constantly changing, making genuine “laws” impossible to establish. In this view, markets are never in a stable state where immutable laws could take hold. I beg to differ. The motion of particles is also unpredictable, and many physical systems operate far from equilibrium. Yet, as Phil Anderson argued in a seminal paper [1], universal laws can still emerge at the macroscale from the aggregation of widely diverse microscopic behaviors. Examples include not only crowds in stadiums or cars on highways but also economic agents in markets.

Now Yuki Sato and Kiyoshi Kanazawa of Kyoto University in Japan have provided compelling evidence that one such universal law governs financial markets. Using an unprecedentedly detailed dataset from the Tokyo Stock Exchange, they found that a single mathematical law describes how the price of every traded stock responds to trading volume [2] (Fig. 1). The result is a striking validation of physics-inspired approaches to social sciences, and it might have far-reaching implications for how we understand market dynamics.

String Theory Inspires a Brilliant, Baffling New Math Proof

When the team posted their proof in August, many mathematicians were excited. It was the biggest advance in the classification project in decades, and hinted at a new way to tackle the classification of polynomial equations well beyond four-folds.

But other mathematicians weren’t so sure. Six years had passed since the lecture in Moscow. Had Kontsevich finally made good on his promise, or were there still details to fill in?

And how could they assuage their doubts, when the proof’s techniques were so completely foreign — the stuff of string theory, not polynomial classification? “They say, ‘This is black magic, what is this machinery?’” Kontsevich said.

Magic moments with John Bell

This was a monumental breakthrough in the philosophy and foundations of quantum mechanics. Bell derived a mathematical inequality that showed if there were any local “hidden variables” (underlying, deterministic factors) explaining the “spooky” correlations in quantum entanglement, those correlations would have to obey certain limits. Experiments inspired by his theorem (starting with Alain Aspect in the early 1980s) have repeatedly shown that these limits are violated, confirming that quantum entanglement is real, non-local, and that nature fundamentally disagrees with Einstein’s idea of “local realism.”

John Bell, with whom I had a fruitful collaboration and warm friendship, is best known for his seminal work on the foundations of quantum physics, but he also made outstanding contributions to particle physics and accelerator physics.

100+ Years Old Debate About Quantum Reality Settled With Experiment. Really?

Go to https://ground.news/sabine to get 40% off the Vantage plan and see through sensationalized reporting. Stay fully informed on events around the world with Ground News.

In quantum physics, a wave function is a mathematical way to describe everything in the universe. But since quantum physics emerged, physicists have argued about whether or not the wave function is real. A group of physicists recently conducted a test of a theorem that describes the mechanics of the wave function, and they’ve told the press that they’ve settled the question: Yes, the wave-function is real. Let’s take a look.

Paper: https://arxiv.org/abs/2510.

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Maryna Viazovska

Viazovska was born in Kyiv, the oldest of three sisters. Her father was a chemist who worked at the Antonov aircraft factory and her mother was an engineer. [ 6 ] She attended a specialized secondary school for high-achieving students in science and technology, Kyiv Natural Science Lyceum No. 145. An influential teacher there, Andrii Knyazyuk, had previously worked as a professional research mathematician before becoming a secondary school teacher. [ 7 ] Viazovska competed in domestic mathematics Olympiads when she was at high school, placing 13th in a national competition where 12 students were selected to a training camp before a six-member team for the International Mathematical Olympiad was chosen. [ 6 ] As a student at Taras Shevchenko National University of Kyiv, she competed at the International Mathematics Competition for University Students in 2002, 2003, 2004, and 2005, and was one of the first-place winners in 2002 and 2005. [ 8 ] She co-authored her first research paper in 2005. [ 6 ]

Viazovska earned a master’s from the University of Kaiserslautern in 2007, PhD from the Institute of Mathematics of the National Academy of Sciences of Ukraine in 2010, [ 2 ] and a doctorate (Dr. rer. nat.) from the University of Bonn in 2013. Her doctoral dissertation, Modular Functions and Special Cycles, concerns analytic number theory and was supervised by Don Zagier and Werner Müller. [ 9 ]

She was a postdoctoral researcher at the Berlin Mathematical School and the Humboldt University of Berlin [ 10 ] and a Minerva Distinguished Visitor [ 11 ] at Princeton University. Since January 2018 she has held the Chair of Number Theory as a full professor at the École Polytechnique Fédérale de Lausanne (EPFL) in Switzerland after a short stint as tenure-track assistant professor. [ 4 ] .

How Ramanujan’s formulae for pi connect to modern high energy physics

Most of us first hear about the irrational number π (pi)—rounded off as 3.14, with an infinite number of decimal digits—in school, where we learn about its use in the context of a circle. More recently, scientists have developed supercomputers that can estimate up to trillions of its digits.

Now, physicists at the Center for High Energy Physics (CHEP), Indian Institute of Science (IISc) have found that pure mathematical formulas used to calculate the value of pi 100 years ago has connections to fundamental physics of today—showing up in theoretical models of percolation, turbulence, and certain aspects of black holes.

The research is published in the journal Physical Review Letters.

New control system teaches soft robots the art of staying safe

Imagine having a continuum soft robotic arm bend around a bunch of grapes or broccoli, adjusting its grip in real time as it lifts the object. Unlike traditional rigid robots that generally aim to avoid contact with the environment as much as possible and stay far away from humans for safety reasons, this arm senses subtle forces, stretching and flexing in ways that mimic more of the compliance of a human hand. Its every motion is calculated to avoid excessive force while achieving the task efficiently.

In the MIT Computer Science and Artificial Intelligence Laboratory (CSAIL) and Laboratory for Information and Decisions Systems (LIDS) labs, these seemingly simple movements are the culmination of complex mathematics, careful engineering, and a vision for robots that can safely interact with humans and delicate objects.

Soft robots, with their deformable bodies, promise a future where machines move more seamlessly alongside people, assist in caregiving, or handle delicate items in industrial settings. Yet that very flexibility makes them difficult to control. Small bends or twists can produce unpredictable forces, raising the risk of damage or injury. This motivates the need for safe control strategies for soft robots.

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