2025's Biggest Breakthroughs in Mathematics

The video showcases three monumental mathematical breakthroughs achieved in 2025, each resolving a long-standing open problem. It begins with the solution to Hilbert's sixth problem, proving that Newton's laws for infinitely many particles converge to the Boltzmann equation by showing recollisions are negligible over long timescales. It then highlights a landmark result in hyperbolic geometry, building on Mirzakhani's work, where researchers used Möbius inversion to prove that almost all hyperbolic surfaces achieve the maximum spectral gap of one-quarter, a key question for quantum chaos and dynamics. Finally, it covers the proof of the 3D Kakeya conjecture, demonstrating that every Kakeya set has fractal dimension 3—a "once-in-a-century" advance with foundational implications for harmonic analysis, signals, and waves. Together, these breakthroughs open new frontiers across physics and mathematics, from realistic gas models to number theory and the structure of random surfaces.

• ▶ 0:07 Hilbert's sixth problem asked to mathematically prove the laws of physics, specifically linking different gas equations, and remained unsolved for 125 years until a 2025 proof.
• ▶ 1:29 The hardest link was microscopic-to-mesoscopic: proving Newton's laws for infinitely many particles converge to the Boltzmann equation, which required taming explosion of collision-history diagrams.
• ▶ 4:22 The breakthrough came from ignoring "bad" collision histories and proving recollisions are negligible, allowing the team to finally connect Newton's laws to Boltzmann's equation over long timescales.
• ▶ 5:33 Long-time gas behavior was a bottleneck, not the endpoint; new techniques now open up harder questions about realistic gases, non-hard-sphere collisions, and quantum effects, which remain very open.
• ▶ 6:26 Hyperbolic surfaces curve like a saddle everywhere, make parallel lines diverge, and twist so wildly they cannot exist in ordinary 3D space — only abstractly.
• ▶ 6:54 Maryam Mirzakhani pioneered the study of these surfaces; after her death, Nalini Anantharaman and Laura Monk advanced her work, producing a landmark result for mathematics and physics.
• ▶ 8:06 Mirzakhani’s 2004 thesis derived a formula for counting closed geodesics up to a given length on hyperbolic surfaces, enabling probabilistic questions about random surfaces.
• ▶ 8:33 A key goal was showing that most random hyperbolic surfaces have a spectral gap near the maximum possible value of 1/4, meaning they are as connected as possible.
• ▶ 9:30 Rare, tangled geodesics distorted calculations so severely that the proof stalled at 3/16, making it impossible to push through to 1/4 without a way to exclude problematic surfaces.
• ▶ 10:30 Friedman's landmark 2002 proof showed most random graphs have the largest possible spectral gap, making them expanders—a property foundational to many algorithms.
• ▶ 11:06 The key tool was Möbius inversion, which the researchers adapted to filter tangled geodesics and compute the average spectral gap across all hyperbolic surfaces.
• ▶ 11:43 Almost all hyperbolic surfaces achieve a spectral gap of one-quarter, and the work promises broad applications in number theory, dynamics, and quantum chaos.
• ▶ 12:32 The section introduces Kakeya's 1917 needle-rotation puzzle, which grew into the Kakeya conjecture—one of the most influential problems in modern mathematics.
• ▶ 13:13 After decades of resistance, a "once-in-a-century proof" in early 2025 resolved higher-dimensional cases, called the biggest harmonic analysis development in at least 20 years.
• ▶ 15:42 Fefferman's work connected the Kakeya conjecture to the Fourier transform, making geometry essential to analysis and placing the conjecture at the foundation of major results about signals and waves.
• ▶ 16:24 Wang and Zahl first proved the 3D Kakeya conjecture for "sticky sets," where same-direction tubes stay close, giving strong evidence for their approach.
• ▶ 16:58 Larry Guth's insight that any counterexample must be "grainy" let them bound how much tubes can overlap, yielding an initial dimension lower bound of 2.5.
• ▶ 17:35 Using "induction on scales" with graininess controlling losses, they iteratively improved the bound until completing the proof in early 2025 that every 3D Kakeya set has fractal dimension 3.

• ▶ 0:07 Hilbert's Sixth Problem: From Challenge to Proof (0:07 - 5:33) - Hilbert's sixth problem asks whether gas behavior can be derived from Newtonian mechanics; after a 125-year struggle, a diagram-taming proof completes it.
• ▶ 5:33 Gas Challenges and Hyperbolic Surfaces (5:33 - 8:00) - Follow-up gas questions lead to hyperbolic surfaces, whose saddle-like geometry is difficult to visualize.
• ▶ 8:00 Mirzakhani's Formula, Spectral Gaps, and Tangled Geodesics (8:00 - 10:30) - Mirzakhani's geodesic formula enables spectral-gap tests, but tangled geodesics stall the proof.
• ▶ 10:30 Random Graphs, Expanders, and the Spectral Gap (10:30 - 12:32) - Friedman's random-graph work inspires the final expander-based approach to spectral gaps on hyperbolic surfaces.
• ▶ 12:32 The Kakeya Conjecture and Harmonic Analysis (12:32 - 16:04) - The rotating-needle problem becomes a family of hard harmonic-analysis questions about sets containing lines in every direction.
• ▶ 16:04 Breakthrough on the 3D Kakeya Conjecture (16:04 - 19:37) - Wang and Zahl prove the 3D conjecture by analyzing sticky sets, using Guth's clue and induction on scales.