- **Hilbert’s Sixth Problem (microscopic → mesoscopic dynamics):**
In 2025 Yu Deng, Zaher Hani and Xiao Ma proved that, in the limit of infinitely many particles whose size shrinks to zero, Newton’s laws for hard‑sphere gases converge to the Boltzmann equation not only for very short times (as shown by Lanford in 1975) but for arbitrarily long times. Their breakthrough was an algorithm that splits enormously complicated collision‑history diagrams into simple pieces and a focus on “well‑behaved’’ histories, showing that problematic recollisions become negligible over long timescales.
- **Hyperbolic surfaces and the spectral gap:**
Building on Maryam Mirzakhani’s formula for the growth of closed geodesics, Nalini Anantharaman and Laura Monk adapted Joel Friedman’s Möbius‑inversion technique to eliminate rare, highly tangled geodesics that distort averages. They proved that for a typical hyperbolic surface the spectral gap – a number between 0 and ¼ measuring how well‑connected the surface is – is almost the maximal value ¼. Hence almost all hyperbolic surfaces are as connected as mathematically possible, opening avenues for applications in number theory, dynamics and quantum chaos.
- **The 3‑dimensional Kakeya conjecture:**
Wang and Zahl (early 2025) established that every Kakeya set in ℝ³ has full Hausdorff dimension 3, confirming the conjecture in three dimensions. Their proof combined Larry Guth’s “graininess’’ insight – which limits how tightly tubes can overlap – with an induction‑on‑scales argument that gradually raises the lower bound on dimension. This resolves a long‑standing bottleneck in harmonic analysis and gives confidence that related higher‑dimensional conjectures (e.g., on Fourier‑transform restriction) will soon see progress.
1. David Hilbert presented a 23‑problem roadmap for mathematics at the dawn of the 20th century.
2. Hilbert’s sixth problem sought a mathematical proof of the laws of physics.
3. Physicists model gases with different equations at microscopic, mesoscopic, and macroscopic scales.
4. The challenge was to prove these equations describe the same reality.
5. Mathematicians struggled with Hilbert’s sixth problem for 125 years.
6. In 2025, Yu Deng, Zaher Hani, and Xiao Ma devised a proof connecting the equations at long timescales.
7. At the microscopic scale, gas particles are modeled as tiny billiard balls obeying Newton’s laws of motion.
8. At the mesoscopic scale, the Boltzmann equation describes how groups of particles behave over time.
9. At the macroscopic scale, the Navier‑Stokes equations describe the entire gas as a single substance.
10. Mathematicians had already proved that a mesoscopic description yields a macroscopic one in settings such as a dilute gas in a box.
11. Proving the microscopic‑to‑mesoscopic connection required showing that, as particle number →∞, the description converges to the Boltzmann equation.
12. The main difficulty is handling particle collisions; the number of possible collision histories becomes enormous.
13. Each collision history can be represented as a diagram (graph) with nodes marking collisions and lines showing particle motion between collisions.
14. In 1975, Oscar Lanford showed that the infinite sum of collision patterns yields Boltzmann’s equation, but only for extremely short time intervals.
15. For longer times, the number of possible collision diagrams explodes, causing the method to collapse.
16. The principal obstacle for long times is recollisions, where the same particles collide more than once.
17. To prove convergence for long times, one must show that collision patterns involving many recollisions are vanishingly rare.
18. Lanford proved this for very short periods; extending it to longer times is much harder.
19. Deng, Hani, and Ma developed an algorithm to break enormous collision diagrams into smaller, manageable pieces.
20. Using this algorithm, they showed they needed only to consider well‑behaved collision histories.
21. They demonstrated that collision patterns with many recollisions are extremely unlikely even over longer time scales.
22. In March 2025, the team posted their result showing that Newton’s microscopic equations converge to Boltzmann’s mesoscopic equation over long times.
23. They proved that, in the limit of particle number →∞ and particle diameter →0, the density satisfies Boltzmann’s equation.
24. Their work completes the solution to Hilbert’s Sixth problem, a milestone more than a century in the making.
25. Reaching long times was a bottleneck that needed to be overcome to access further questions in the field.
26. Future work could apply the new techniques to gases in more realistic settings, such as non‑dilute gases, non‑hard‑sphere interactions, or quantum mechanics.
27. Maryam Mirzakhani developed tools to explore the geometry of hyperbolic surfaces.
28. In her 2004 Ph.D. thesis, Mirzakhani derived a formula estimating the number of closed geodesics on a hyperbolic surface up to a given length.
29. Mathematicians study the spectral gap of a hyperbolic surface, a number between 0 and 1/4 measuring how connected the surface is.
30. A large spectral gap indicates the surface is well connected; a small gap indicates poor connectivity.
31. Anantharaman and Monk aimed to compute the spectral gap of a typical hyperbolic surface.
32. They sought to prove that most hyperbolic surfaces have a spectral gap of almost one‑quarter, meaning they are as connected as possible.
33. Some surfaces possess very tangled geodesics that distort spectral gap calculations, stalling previous attempts at 3/16.
34. To overcome this, the researchers adapted Friedman’s Möbius inversion formula to filter out tangled geodesics.
35. By doing so, they showed that almost all hyperbolic surfaces have a spectral gap of one‑quarter.
36. The new work may be used to answer questions in number theory and dynamics, including quantum chaos.
37. In 1917, Sōichi Kakeya asked: rotating an infinitely thin needle through all directions, what is the smallest region it can sweep out?
38. This question became the Kakeya conjecture, a central problem in harmonic analysis.
39. In the plane, Abram Besicovitch showed one can rotate a needle inside a set of zero area.
40. Roy Davies proved in the 1970s that every Kakeya set has full fractal dimension 2.
41. The Kakeya conjecture asserts that in every dimension, Kakeya sets are full‑dimensional objects.
42. Mathematicians struggled to prove the conjecture in higher dimensions.
43. Progress on the three‑dimensional case was slow for decades.
44. In 2022, Hong Wang and Joshua Zahl proved the 3D Kakeya conjecture for sticky sets (tubes pointing in the same direction stay close together).
45. Larry Guth showed that any 3D counterexample to the Kakeya conjecture would have to be grainy, with many tubes overlapping in tiny regions.
46. Wang and Zahl used this graininess to control losses in an induction‑on‑scales argument.
47. At each step, they improved the lower bound on the dimension of any Kakeya set.
48. In early 2025, they completed the proof: every Kakeya set in 3D has fractal dimension 3.
49. The result gives confidence that higher‑dimensional Kakeya conjectures may be approached similarly.
50. Mathematicians hope to use the proof to tackle more ambitious conjectures in the hierarchy.