Hong Wang and the 3D Kakeya Conjecture: A Fields Medal Moment in Geometry

Podcast overview

The Quanta audio edition spotlights Hong Wang, a 2026 Fields Medal winner, and her landmark solution to the three-dimensional Kakeya problem. The episode weaves together Wang’s personal journey from Guilin to MIT and NYU with the deep mathematics behind Kakeya type problems, harmonic analysis, and geometric measure theory. It also introduces the broader world of connections between pure math and real-world ideas.

• Hong Wang’s century-spanning Kakeya proof reframes how we think about dimension and space in three dimensions.
• The episode places Wang within a web of ideas spanning harmonic analysis, incidence geometry, and partial differential equations.
• Her collaborative approach with Joshua Zahl and the trajectory from a blog-suggested strategy to a public, peer-reviewed breakthrough are highlighted.
• The host and guests discuss the personal discipline and resilience needed to pursue such a monumental result and its implications for the math community.

Overview

The podcast from Quanta Magazine presents an in-depth portrait of Hong Wang, a Fields Medal winner in 2026, and her proof of the 3D Kakeya conjecture, a landmark result at the crossroads of harmonic analysis, geometric measure theory, and partial differential equations. The episode frames the story as a blend of monumental mathematics and personal perseverance, tracing Wang’s path from early life in Guilin, China, through her education in Beijing and Paris, to MIT and beyond. It situates her work within a family of Kakeya-type problems that explore how line segments pointing in every direction can occupy space, and how such questions connect to fundamental objects like the Fourier restriction phenomenon in harmonic analysis. The conversation also touches on the broader narrative of mathematics as a living discipline, where abstract proofs can reveal real-world connections and drive a cascade of prizes and recognition.

Background and the journey to excellence

The episode sketches Wang’s early fascination with math as a constant, with stories from Guilin that emphasize math as a stable, enduring pursuit. It highlights her instinct to push beyond comfort zones, including switching majors from earth sciences to mathematics in college, studying in France, and ultimately pursuing graduate work at MIT where she encountered Larry Guth, a pivotal mentor in her development. The narrative situates her within the Kakeya problem’s history, explaining how Soichi Kakeya’s original question about rotating a needle in all directions expanded into a broad family of mathematical questions that lie at the interface of several domains of mathematics.

The Kakeya problem and the mathematical landscape

The core mathematical landscape, as described in the podcast, revolves around Kakeya sets and their dimensionality. Basickovich’s work showed that Kakeya sets in higher dimensions can have zero volume, yet still occupy space in a precise sense. The podcast explains the three-dimensional Kakeya set conjecture, the role of tube-like coverings in measuring dimension, and how fractal-like behavior can challenge naive dimensional intuition. The discussion then connects Kakeya-type questions to Fourier analysis via the Fourier restriction conjecture, which contemplates how waves with the same wavelengths but different directions behave when combined. These connections illuminate why Kakeya-type problems are central to harmonic analysis and geometric measure theory, and why solving the 3D version represents a major milestone in the field.

From Falconer distance to 3D Kakeya: a collaborative strategy

The podcast details Wang’s collaboration with Joshua Zahl, a narrative arc that begins with reading a blog post by Terence Tao and Katz on a potential strategy to tackle the Kakeya problem. Wang and Zahl pursued a strategy built on the idea that a hypothetical counterexample would possess a rigid structure causing a contradiction in the interplay between addition and multiplication. They focused on sticky Kakeya sets, where line segments with similar orientations cluster together, making it feasible for thicker tubes to contain them. Their method aimed to show that sticky structures cannot form counterexamples in 3D, and then extended the analysis to Kakeya sets with a dimensionality less than 3, using a recursive, scale-based argument. This inductive framework, combined with geometric-measure-theoretic tools, led to the eventual conclusion that Kakeya sets in 3D space must be three-dimensional, resolving a long line of inquiries in the field.

Timeline, validation, and impact

The episode traces the process from 2014 through 2024, outlining the rigorous chain of checks and reviews that culminated in the public announcement in 2025 and the cascade of prizes in 2025 and 2026, including the Fields Medal. It emphasizes the community’s mixed feelings about the victory: the Holy Grail of Kakeya is achieved in significant ways, but many related problems remain open in 3D, 4D, and higher. The interviewees reflect on how the techniques developed during this journey open paths to addressing the 3D restriction and local smoothing conjectures, and potentially many other deep questions in harmonic analysis.

Personal dimensions and future directions

Beyond the math, the podcast highlights Wang’s personal discipline and the emotional landscape of doing high-stakes research. It discusses her meticulous routine, her connections to friends and dogs in New York City, and the way she navigates the demands of travel and public attention while maintaining focus on deep problems. The conversation ends with an eye toward the future: the math community will build on these techniques to attack higher-dimensional Kakeya-type conjectures and related harmonic-analysis questions, and Wang herself remains motivated by the idea that the next breakthroughs will come from applying these ideas to new realms of mathematics and related disciplines.