Mathematicians Crack a Fractal Conjecture on Chaos
10 days ago
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- #chaos-theory
- Randomness and chaos influence everything in the universe, from galaxies to subatomic particles.
- French mathematician Vincent Vargas studied random fluctuations leading to global effects, culminating in the Garban-Vargas conjecture in 2023.
- Gaussian multiplicative chaos (GMC) is a mathematical tool used to detect patterns in randomness, applicable in fields like quantum chaos and Brownian motion.
- Jean-Pierre Kahane first developed GMC in 1985, but its significance was later revived by Vargas and others.
- GMC models multiscale randomness, such as turbulent fluids with eddies breaking into smaller eddies, acting as a fractal measure.
- GMC reveals that small-scale events can govern entire systems, with fractal structures shaping chaos universally across scales.
- Beyond a critical threshold of randomness, GMC collapses, marking a phase transition similar to ice melting into liquid.
- Garban and Vargas used harmonic analysis to study GMC, matching dimensions like patterns and clumpiness in snowflakes.
- Their conjecture linked correlation and harmonic dimensions in GMC systems, but they couldn't prove it initially.
- In 2024, mathematicians Zhaofeng Lin, Yanqi Qiu, and Mingjie Tan proved the conjecture using higher-dimensional martingales.
- Their proof conserves energy across scales, validating the Garban-Vargas formula and opening doors for more complex fractal models.
- Challenges remain, especially at critical phase-transition points where current methods fail, requiring new ideas for further progress.