Quanta Magazine called 2025 a historic year in mathematics and picked out three major breakthroughs. One was work on the spectral gap of hyperbolic surfaces: Nalini Anantharaman of the Collège de France and Laura Monk of the University of Bristol settled a conjecture Peter Buser had posed in 1984. The argument is laid out across a two-part preprint series on arXiv.

The proof builds on the work of Maryam Mirzakhani, the Iranian mathematician who died in 2017 at the age of 40.

What Are Hyperbolic Surfaces? The Saddle-Shaped World

The easiest handle on a hyperbolic surface is a horse's saddle. Unlike ordinary flat planes or spherical surfaces, hyperbolic surfaces curve upward in one direction and downward in another, possessing what mathematicians call negative curvature.

These hyperbolic surfaces exist in a special space that cannot be handled by the Euclidean geometry we learn in school. They have properties that defy intuition, such as different behaviors of parallel lines and triangles whose interior angles sum to less than 180 degrees. However, this seemingly strange geometry plays crucial roles in various fields, from the structure of the universe to quantum mechanics and even internet design.

What Is the Spectral Gap? Measuring a Surface's "Connectivity"

At the heart of this research is the "spectral gap," a number that quantifies how easily information spreads across a surface and how well-connected it is.

Picture a dumbbell: two large balls joined by a thin bar. This shape has poor connectivity because moving from one end to the other requires passing through the narrow bar. Such shapes have small spectral gaps.

In contrast, surfaces where the entire surface is evenly connected and you can efficiently move from any point to any other point have large spectral gaps. Mathematicians knew the theoretical ceiling was 1/4 as the genus grows, and whether randomly chosen hyperbolic surfaces get close to it had been argued over for decades. That such surfaces exist at all was shown in 2021 by Will Hide and Michael Magee, using a probabilistic construction.

Seven Years of Research Culminates in Proof

The research by Anantharaman and Monk took approximately seven years to complete, beginning in 2018 when Monk started as Anantharaman's graduate student. They built upon the "Weil-Petersson measure" technique developed by Mirzakhani and introduced a new mathematical tool called "Friedman-Ramanujan functions."

What they proved is this: as the genus (the number of holes, a measure of a surface's complexity) grows, the probability that a surface has spectral gap at least 1/4 minus epsilon tends to 1, for any epsilon greater than zero. The distinction matters. They did not show that surfaces hit 1/4 exactly; they showed you can get arbitrarily close. In practical terms, pick a large-genus surface at random and you will almost certainly land on one with near-optimal connectivity.

The key was a filtering step that sets aside surfaces with tangled geodesics (shortest paths on a surface) as exceptions that do not move the overall result. It transplants the reasoning Joel Friedman used on graphs to settle Alon's conjecture into a geometric setting.

Real-World Applications

What potential impact could this pure mathematics discovery have on our daily lives?

1. Understanding Quantum Chaos

In quantum mechanics, there is a phenomenon called "quantum chaos" where particles move irregularly. The theory of spectral gaps plays an essential role in understanding quantum chaos. This could contribute to the development of quantum computers and advances in quantum cryptography.

2. Efficient Communication Network Design

Complex networks, including the internet, are known to have deep connections with hyperbolic space structures. Research on spectral gaps could be applied to more efficient network design and the development of routing algorithms that allow information to travel quickly. Research has shown that representing the internet's structure in hyperbolic space requires far fewer dimensions than representing it in traditional Euclidean space.

3. Quantum Error Correction

Quantum computers are extremely sensitive to noise, making error correction a major challenge. Research on quantum error correction codes utilizing hyperbolic geometry is advancing, and this achievement could contribute to this field as well.

4. Machine Learning and AI

In recent years, machine learning in hyperbolic space has gained attention. Because it can efficiently represent hierarchical data structures (such as semantic relationships between words in natural language processing), it may contribute to the advancement of AI technology.

The Legacy of Women Mathematicians

Behind this research lies a remarkable story of succession among women mathematicians. Maryam Mirzakhani was the first woman to receive the Fields Medal (often called the Nobel Prize of mathematics) in 2014. After her death, her research was carried forward by many mathematicians, leading to this result.

The Breakthrough Prize includes the Maryam Mirzakhani New Frontiers Prize, established in her honor to support early-career women in mathematics.

Update: On July 23, 2026, at the International Congress of Mathematicians in Philadelphia, Hong Wang was awarded a Fields Medal for work including the three-dimensional Kakeya problem. She is the third woman to receive it in the prize's 90-year history, after Mirzakhani in 2014 and Maryna Viazovska in 2022.

What Abstract Mathematics Opens Up

The fact that seemingly abstract mathematical discoveries far removed from our daily lives are actually fundamental to the technologies that support our society reaffirms the importance of basic research.

Japanese researchers have kept turning up in this territory too: Kyoto University's Takuro Mochizuki received the Breakthrough Prize in Mathematics in 2022.

How has your country reacted to such mathematical discoveries? What is the state of social support and interest in pure mathematics research in your region? We'd love to hear your thoughts!

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