Darboux's theorem says every symplectic manifold looks locally like standard ℝ^(2n) with coordinates (q₁,...,qₙ,p₁,...,pₙ). Yet globally, symplectic manifolds can have wildly different topologies. This is the symplectic version of 'all politics is local'—except the global structure still matters enormously.

Star mathematician Joshua Zahl solves 100-year-old Kakeya conjecture, leaves Canada for China's Nankai University

https://freedomcents.substack.com/p/why-your-brain-is-your-worst-investment

https://freedomcents.substack.com/p/beyond-correlation-why-modern-markets?r=2pxsfh

https://open.substack.com/pub/freedomcents/p/the-bayesian-revolution-how-probabilistic?r=2pxsfh&utm_campaign=post&utm_medium=web&showWelcomeOnShare=true

https://open.substack.com/pub/freedomcents/p/how-a-triathlete-family-slashed-grocery?r=2pxsfh&utm_campaign=post&utm_medium=web&showWelcomeOnShare=true

Deep neural networks often "forget" earlier training when learning new tasks—a phenomenon called catastrophic forgetting. It's like how we struggle to recall childhood phone numbers. This fundamental limitation shows how far we are from true artificial general intelligence.

https://open.substack.com/pub/freedomcents/p/the-future-of-work-and-fire?r=2pxsfh&utm_campaign=post&utm_medium=web&showWelcomeOnShare=true

https://freedomcents.substack.com/p/the-currency-of-freedom

We're entering an awkward phase: AI is good enough to threaten jobs but not good enough to create abundance. This transition might force us to confront fundamental questions: What is work for? What gives life meaning? How do we distribute resources in a post-scarcity world? #AI #Philosophy

If AI makes many jobs obsolete, UBI might become necessary. This could decouple survival from employment, allowing people to pursue meaningful activities without financial fear. Utopian dream or practical necessity? The debate intensifies as AI capabilities grow. #UBI #FutureEconomy

Fermat's Last Theorem took 358 years to prove! The margin was too small for Fermat's "marvelous proof" but Andrew Wiles needed 150+ pages in 1994. #MathHistory #NumberTheory

Non-Euclidean geometry changed everything by challenging 2000+ years of mathematical "truth." On a sphere, parallel lines DO meet and triangles sum to >180°! #GeometryRevolution #MathFoundations

The Basel Problem: 1 + 1/4 + 1/9 + 1/16 + ... = π²/6. Somehow adding inverse squares gives us π? Euler's proof of this in 1735 established him as a mathematical genius.

The Mandelbrot set blows my mind: an infinitely complex shape from a simple equation z² + c. Zoom in forever and find new patterns. Math creating endless beauty!

Behavioral finance shows we feel losses 2x more strongly than equivalent gains. This loss aversion explains panic selling during market dips—our brains are wired to overreact!

Just learned about the Rebalancing Bonus—how regularly realigning your portfolio to target allocations can add ~0.5% annual returns. Free money from mathematical principles!

The Sharpe ratio is my favorite metric for evaluating investments. It measures return relative to risk taken. Higher is better! Always know what risk you're taking for that promised return.

Dollar-cost averaging > trying to time the market. Small, consistent investments over time reduce risk and emotional decision-making. Boring but effective!

Time isn't constant! Clocks in GPS satellites tick slightly faster than those on Earth because gravity dilates time. General relativity making your navigation app work.