Showcasing practical examples, projects, and case studies demonstrating how AI models and tools are being applied across various fields.
For decades, mathematicians had good reasons to believe that a certain kind of algebra could never contain zero divisors. The argument seemed plausible: remove finite-order elements from the underlying group, and the familiar route to producing those troublesome products disappears. But plausibility isn’t a proof. On September 23, 2026, OpenAI released A Torsion-Free Group Algebra … Read more
An infinite collection of equations, a coin-flip prediction, and a proof that has resisted mathematicians for decades. Goldfeld’s Conjecture sounds deceptively tidy: take one elliptic curve, build its quadratic twists, and analytic ranks zero and one should each appear half the time. The average should approach one-half. OpenAI now claims arguments establishing both conclusions for … Read more
A mathematical question can survive 126 years not because nobody knows how to count, but because nobody knows whether the counting can ever get out of control. That is the tension behind Hilbert’s 16th Problem, and it sits at the center of a striking new OpenAI manuscript. Released in OpenAI’s October 2026 mathematics collection and … Read more
Can a mathematical structure forget how many independent ingredients built it? For free group factors, that question has resisted decades of research. OpenAI’s proposed answer is yes, and the interesting part is how it makes the extra ingredient recoverable from the others. The free group factor isomorphism problem asks whether operator algebras associated with free … Read more
A mathematical object can be easy to describe and remarkably difficult to understand. Thompson’s group F fits that description almost too well. Its elements stretch and compress an interval according to a short list of rules. Yet a basic question about averaging over those transformations resisted resolution for decades. OpenAI’s manuscript, dated September 23, 2026, … Read more
OpenAI claims a positive answer to the unitary synthesis problem: a quantum computer can approximate any quantum unitary efficiently when supplied with a suitably chosen classical Boolean oracle. The qualification belongs in the opening sentence. The paper does not show how to construct that oracle efficiently. That still makes this a striking claim. The problem … Read more
The fraction 355/113 gets astonishingly close to π using just three digits in its denominator. It also points to a question mathematicians have struggled with for decades: how often can fractions achieve that kind of accuracy as their denominators grow? OpenAI’s manuscript The Irrationality Exponent of π Is 2, dated September 24, 2026, claims an … Read more