A Field Day in Mathematics
This blog post is about the 2026 Fields medal, one of the most prestigious awards in mathematics given to persons below the age of 40.
The prize (sometimes called the Nobel prize for mathematics) is named after the Canadian mathematician John Charles Fields, and is given to 2,3 or 4 mathematicians once every 4 years. This year the ceremony took place at the International Congress of Mathematicians in Philadelphia in July.
This year's awardees in last name reverse-alphabetical order (because I want to mention the female awardee first) and their work:
Hong Wang: She was awarded for her work in two areas. The first was Harmonic Analysis, a more advanced version of Fourier analysis, which breaks complex signals or waves down into simple sine and cosine components. Wang worked in Fourier restriction theory.
The second (related) area was Geometric Measure Theory, where Wang solved the Kakeya conjecture. The Kakeya problem is a geometric puzzle. Its 2-dimensional version asks: What is the smallest possible area in space required to completely turn a needle of length 1 around 360 degrees?
It sounds like introductory geometry (one might think that the answer is a circle), but it turns out to be incredibly deep (by clever overlapping, one can turn it around using an area much smaller than a circle!). Wang (along with Joshua Zahl) solved the 3-dimensional version of this conjecture, a problem that puzzled top mathematicians for half a century.
Wang is only the third woman to ever win a Fields Medal, after Maryam Mirzakhani in 2014 and Maryna Viazovska in 2022.
Jacob Tsimerman: He was awarded for making several contributions at the intersection of number theory and algebraic geometry, including the solution (with other collaborators) of the André–Oort conjecture.
The work basically answered the question: when do abstract geometric shapes contain infinitely many points with clean, rational number coordinates? What does that mean? I found it easier to approach the statement backwards.
First consider a famous and important result in number theory, Fermat's Last Theorem: There are no positive whole numbers a,b,c that satisfy the equation a^n + b^n = c^n for integer values of n greater than 2. In number theory, such special points (whole number or rational solutions of polynomial equations) are considered important.
Now algebraic geometry can graph these polynomial equations into geometric shapes. By analyzing these shapes, Tsimerman was able to narrow down the solutions of many polynomial equations.
Tsimerman has famously taken leave from academia to join OpenAI.
John Pardon: He was awarded for Topology and Symplectic Geometry. Topology can be thought of as "rubber-sheet geometry"—studying what properties of a geometric object remain unchanged when you stretch, twist, or crumple it without cutting or gluing it.
Pardon's work was in differential geometry (the geometry of smooth shapes and spaces) and advanced knot theory. He resolved a major bottleneck by defining something called virtual fundamental cycles for highly abstract geometric spaces known as Fukaya categories. Essentially, for geometric spaces too jagged or chaotic to be studied using standard calculus or algebraic tools, Pardon created a mathematical technique to cleanly smooth them out and analyze them.
Yu Deng: Awarded for his work on Partial Differential Equations (PDEs) and Mathematical Physics.
PDEs (see my recent post) track how quantities of physical interest vary over time and with space, which suits them well for describing fluid dynamics, thermodynamics, and quantum mechanics among other areas.
Deng bridged the gap between microscopic physics (how individual atoms or molecules bounce around like billiard balls) and macroscopic laws (how gas or fluids behave as a whole). He provided rigorous mathematical proofs for equations that describe wave turbulence and how chaotic gas particles stabilize over long timelines.
Afterword
Next big one coming up - the Nobel prizes - to be announced in the first week of October. We'll do a warm up for this event in the coming weeks.
