The Greatest Mathematical Achievements of 2025
The Proof of the Geometric Langlands Conjecture
This monumental achievement is described as a central part of the Langlands program, an ambitious research effort often referred to as a potential "grand unified theory of mathematics" for its power to connect disparate mathematical fields.
Scope of the Work
The proof was a "gargantuan" effort, resulting from the collaboration of nine mathematicians across five research papers spanning almost 1,000 pages.
The Award
The work's significance was confirmed by the awarding of the 2025 Breakthrough Prize in Mathematics—often called the "Oscars of Science"—and its $3 million award to mathematician Dennis Gaitsgory for his foundational role in the proof.
The conjecture establishes a profound, one-to-one correspondence between two completely different types of mathematical objects. Experts believe this will have deep implications for number theory, algebraic geometry, and mathematical physics, opening new avenues for research across these fundamental mathematical disciplines.
Other Major Mathematical Breakthroughs of 2025
The proof of the geometric Langlands conjecture was part of a year rich with significant discoveries. Here are other key achievements from 2025:
Breakthroughs in Longstanding Problems
Kakeya Conjecture
Researchers settled the long-standing three-dimensional Kakeya conjecture, which deals with the minimum volume of shapes that contain a line segment pointing in every direction.
Hilbert's 6th Problem
A major step was taken toward solving David Hilbert's sixth problem (from 1900) by successfully unifying three physical theories to explain fluid motion.
Moving Sofa Problem
A solution was found for the "moving sofa problem," which seeks the largest shape that can turn a right-angled corner in a narrow hallway.
Advances in Core Mathematical Fields
Geometry & Topology
The discovery of a new shape called the "noperthedron" disproved an old geometrical conjecture. In knot theory, the discovery of a knot simpler than the sum of its parts overturned a long-held assumption about knot complexity.
Number Theory
Multiple breakthroughs involved prime numbers, including new methods for finding large primes and for estimating their distribution within any given range.
The year's top achievements were also highlighted by other prestigious mathematics awards, which often focus on groundbreaking research.
New Horizons in Mathematics Prize: Awarded to Ewain Gwynne for his contributions to conformal probability and the understanding of random fractal surfaces, with applications in theoretical physics.
Maryam Mirzakhani New Frontiers Prize: Awarded to Si Ying Lee for her work on Shimura varieties, contributing to the broader Langlands program.
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