2025 marked a historic year in mathematics. Researchers solved a major case of Hilbert’s ambitious sixth problem, proved a sweeping new theorem about hyperbolic surfaces, and settled the longstanding three-dimensional Kakeya conjecture.
00:00 - Hilbert's Sixth Problem Solved
In a landmark achievement, mathematicians have put the laws of fluid dynamics on firmer mathematical footing. By proving how individual molecules create the complex motion of fluids, they've helped advance David Hilbert’s 125-year-old mathematical program.
Read more at Quanta Magazine: https://www.quantamagazine.org/epic-effort-to-ground-physics-in-math-opens-up-the-secrets-of-time-20250611/
Associated paper:
https://arxiv.org/abs/2503.01800
06:26 - Hyperbolic Geometry
A powerful new proof extends the late Maryam Mirzakhani’s work on hyperbolic surfaces, showing that most of them have an important geometric property. The result greatly advances our understanding of these exotic shapes. Watch our full Kakeya Conjecture explainer: https://www.youtube.com/watch?v=5J3tYU_-IZI
Read more at Quanta Magazine: https://www.quantamagazine.org/years-after-the-early-death-of-a-math-genius-her-ideas-gain-new-life-20250303/
Associated paper: https://arxiv.org/abs/2502.12268
12:27 - 3D Kakeya Conjecture
Two mathematicians have resolved a deceptively simple problem that had stymied researchers for decades, opening up new avenues in analysis and geometric measure theory.
Read more at Quanta Magazine: https://www.quantamagazine.org/once-in-a-century-proof-settles-maths-kakeya-conjecture-20250314/
Associated paper: https://arxiv.org/abs/2502.17655