The 2026 Fields Medalists:
An Institutional Research Briefing

What Changed in Mathematics?

On July 23, 2026, at the International Congress of Mathematicians (ICM) in Philadelphia, the International Mathematical Union awarded four Fields Medals — often called the "Nobel Prize of mathematics" — to four mathematicians under 40 (Simons Foundation). Each medalist solved a problem that had resisted the world's best mathematicians for decades, and in some cases, for over a century.

Think of mathematics as an immense mountain range. There are peaks everyone can see — famous unsolved problems looming over the landscape. There are hidden ridgelines connecting peaks that look unrelated from below. And there are entire valleys of technique that most climbers don't even know exist, yet which turn out to hold the key routes to the summit.

The four 2026 Fields Medalists didn't just summit peaks. They built new climbing routes that changed the map itself:

• Yu Deng proved that the chaotic dance of billions of particles genuinely gives rise to the smooth, predictable equations of fluid mechanics — completing a dream David Hilbert articulated in 1900 (UChicago News).
• John Pardon proved that two completely different ways of counting curves on six-dimensional shapes — shapes that may model our universe in string theory — are actually counting the same thing, settling a 20-year-old conjecture (Quanta Magazine).
• Jacob Tsimerman imported a tool from mathematical logic — a concept called "o-minimality" that makes precise what "tame geometry" means — and used it to prove conjectures in number theory that had been open since the 1970s (Plus Magazine).
• Hong Wang proved that a needle spinning in every direction in three-dimensional space must sweep out a set of full dimension — resolving a problem posed in 1917 and opening the door to a tower of conjectures in harmonic analysis (Plus Magazine, IHES).

This briefing covers each medalist in depth: the field they work in, the problem they solved, why it matters, and the 10 most influential works — theirs and their predecessors' — that define the research lineage.

Animation (author's aid): four new routes draw themselves into the range; four flags land on the peaks that fell — they changed the map itself.

Yu Deng — Partial Differential Equations

Affiliation: University of Chicago (formerly USC Dornsife; Princeton PhD 2015)
Age at award: 37

Official Citation

"For his work in partial differential equations, including the rigorous derivation of the Boltzmann equation from hard-sphere dynamics for rarefied gases, the derivation of wave kinetic equations from nonlinear dispersive systems, and probabilistic approaches to nonlinear Schrödinger dynamics."
— (Simons Foundation)

The Field: Partial Differential Equations (PDEs)

Imagine you're swimming in the ocean. The water around you isn't static — it moves, swirls, and crashes in patterns that seem chaotic up close but somehow organized at larger scales. The equations that describe how fluids, gases, and waves evolve are called partial differential equations (PDEs). A PDE describes how something changes across both space and time simultaneously — not just "how fast is the ball falling?" (that's an ordinary differential equation) but "how does temperature spread across a metal plate as time passes?" (UChicago News).

PDEs are the language physics speaks. The heat equation, the wave equation, the Navier-Stokes equations of fluid dynamics, the Schrödinger equation of quantum mechanics — all PDEs. But PDEs are notoriously difficult to solve rigorously. You can write down the equation, but proving that solutions exist, that they behave the way physics predicts, and that you can actually say something precise about them — that's where mathematics gets hard.

Deng's work lives at the intersection of PDEs and probability. He brings the flexible, probabilistic viewpoint into the rigid, structured world of wave equations — like bringing the fluid adaptability of a martial artist into a rigid kata (form). As Deng himself says: "There is a lot of randomness in this world, and we need a mathematical way to capture it" (UChicago News).

The Award-Winning Contribution

Deng's Fields Medal recognizes three interlinked breakthroughs:

1. Deriving the Boltzmann equation from hard-sphere dynamics (Hilbert's Sixth Problem)

In 1900, David Hilbert posed 23 problems that would shape mathematics for the next century. His sixth problem asked: can we derive the equations of physics — particularly fluid mechanics — from first principles, starting from the motion of individual particles? (Plus Magazine)

Here's the picture: imagine a gas made of billions of tiny hard spheres bouncing off each other like billiard balls. At the microscopic level, each particle follows Newton's laws — deterministic, predictable for any single particle. But at the macroscopic level, the gas behaves according to the Boltzmann equation (1872), which describes temperature, pressure, and velocity in terms of statistical averages. Then at the largest scale, the gas flows as a continuous fluid described by the Navier-Stokes equations (UChicago News).

The chain is: Newton → Boltzmann → Navier-Stokes. Hilbert wanted mathematicians to prove each link rigorously.

The second link (Boltzmann → Navier-Stokes) was already understood. The first link (Newton → Boltzmann) was the hard one. In 1975, Oscar Lanford proved it — but only for an extremely short time, so short that "only one in a million particles is expected to have had one collision" (Plus Magazine). Over longer times, particles collide repeatedly, building up complicated shared histories — like sparring partners who've fought so many rounds that each one anticipates the other's moves. The independence assumption that Boltzmann relied on breaks down.

Deng, together with Zaher Hani (University of Michigan) and Xiao Ma (Princeton), cracked this problem. Their key insight — which came to Deng in a Korean fried chicken stall in Providence, Rhode Island (UChicago News) — was to break long time intervals into short ones, then use an elaborate "molecule cutting" algorithm to control the combinatorial explosion of collision patterns. They represented collision histories as tree-like diagrams (reminiscent of Feynman diagrams in physics), then showed that even though the trees grow infinitely complex over long times, their contributions cancel and combine in controlled ways (Quanta Magazine).

The result: in 2024–2025, Deng, Hani, and Ma published a 200-page proof deriving the Boltzmann equation from hard-sphere dynamics for arbitrarily long times (as long as the relevant Boltzmann solution exists, under the Boltzmann-Grad assumptions), and then completed the rigorous pipeline from Newton to Navier-Stokes in this model setting. Hilbert's sixth problem — at least for the case of a rarefied hard-sphere gas in the Boltzmann-Grad limit — was resolved (Plus Magazine, UChicago News).

2. Deriving the wave kinetic equation from nonlinear dispersive systems

Before tackling gases, Deng and Hani worked on an analogous problem for waves. Physicists had long believed that if you start with the nonlinear Schrödinger equation (which describes how waves interact in quantum systems), and you zoom out to look at the average behavior of many waves, you should get the wave kinetic equation — a higher-level description of wave turbulence, like how energy cascades through ocean waves (Quanta Magazine).

Deng and Hani proved this rigorously. They represented the wave solution as a sum of tree-like diagrams, then estimated each tree's contribution with almost surgical precision. As Jalal Shatah of Courant Institute put it: "These estimates were beyond anyone's ability" (Quanta Magazine). This result, published in Inventiones mathematicae in 2023, was the wave analog of Lanford's theorem — and it established the diagrammatic methodology that Deng later adapted for the Boltzmann derivation.

3. Probabilistic approaches to nonlinear Schrödinger dynamics

Deng also made fundamental contributions to the random data problem for PDEs. When you start a PDE with random initial data — waves that oscillate wildly or have discontinuities — can you still prove the equation has a solution? Deng, with Andrea Nahmod (UMass Amherst) and Haitian Yue, proved that the Gibbs measure (a probability rule modeling thermal equilibrium) is invariant for the 2D nonlinear Schrödinger equation. Their key innovation was the theory of random tensors — a framework for analyzing the wildly oscillating pieces of the solution that had been too complicated for previous methods (Quanta Magazine).

Think of it like this: in martial arts, when you face an opponent who moves unpredictably, you can either try to predict each individual move (the traditional, deterministic approach) or you can learn the statistical patterns — which moves are likely, which combinations are rare — and build your strategy on the probabilities. Deng brought this probabilistic thinking into the deterministic world of wave equations.

Why It Is Transformative

Deng's work bridges three scales of physics — microscopic (particle dynamics), mesoscopic (kinetic theory), and macroscopic (fluid mechanics) — and does so with mathematical rigor that mathematicians had been seeking for over a century. The IMU citation describes it as "a leap forward in a centuries-long quest by mathematicians and physicists to derive the basic laws of physics from first principles" (Plus Magazine).

The methodology Deng developed — the tree-diagram decomposition, the "molecule cutting" algorithm, the random tensor theory — is not just a one-trick solution. It's a new toolkit that is already being applied across PDE theory, statistical physics, and stochastic analysis. The Séminaire Bourbaki, the most prestigious expository venue in mathematics, dedicated a seminar to this work within two years of its appearance — a powerful indicator of its impact (arXiv:2602.04407).

Animation (author's aid): the key idea and mechanism — jittering particles (micro · Newton) are welded by "molecule cutting + cancelling tree diagrams" to the statistics histogram (meso · Boltzmann), then to the smooth wave (macro · fluid); the three scales light up in sequence and the flowing arrows mark the rigorously proven direction of derivation.

10 Most Influential Related Works

Core Papers by Yu Deng

Foundational Predecessor Works

Current Impact and Follow-Up

Seminar Discussion Questions

John Pardon — Symplectic Geometry

Affiliation: Stony Brook University, Simons Center for Geometry and Physics (PhD Stanford 2015, Princeton AB 2011)
Age at award: 37

Official Citation

"For his achievements in symplectic geometry, including new approaches to virtual fundamental cycles, Fukaya categories of Liouville manifolds and counting holomorphic curves, and for his contributions to other areas of geometry and topology, including group actions on 3-manifolds and knot theory."
— (Simons Foundation)

The Field: Symplectic Geometry

Imagine you're climbing a rock face. Your position on the wall and your momentum (how fast and in what direction you're moving) together describe your complete state at any moment. The space of all possible (position, momentum) pairs is called phase space — and symplectic geometry is the geometry of phase space (Quanta Magazine).

In phase space, position and momentum are locked in a dance: change one and the other must respond. This relationship — a symplectic structure — is more flexible than rigid Euclidean geometry (where shapes have fixed angles and distances) but more constrained than topology (where you can stretch and compress shapes freely). Think of it as the difference between a rigid climbing route with fixed holds (Euclidean geometry), a route where holds can shift but the overall structure stays (symplectic geometry), and a completely free-form scramble where anything goes (topology).

Symplectic geometry originated in classical mechanics (Hamilton's formulation) but has become one of the deepest and most active areas of modern mathematics, with connections to string theory, mirror symmetry, and quantum physics.

The Award-Winning Contribution

Pardon's Fields Medal recognizes a series of breakthroughs across symplectic geometry, knot theory, and geometric topology:

1. Virtual fundamental cycles (dissertation work)

Many problems in symplectic geometry come down to counting — specifically, counting certain curves (called pseudo-holomorphic curves) on symplectic manifolds. It's like counting the number of distinct routes up a climbing wall that satisfy certain constraints. But there's a problem: sometimes the curves don't intersect the manifold "transversely" (cleanly crossing through), but instead merely "kiss" it tangentially — and when that happens, your count goes wrong (Quanta Magazine).

Mathematicians had developed various workarounds called virtual fundamental cycles — techniques for perturbing or reorganizing the counting to handle these problematic tangencies. But each approach had limitations. Pardon, in his 2015 Stanford dissertation under Yakov Eliashberg, developed a completely new, algebraic approach using the notion of an "implicit atlas" — a sheaf-theoretic framework providing local finite-dimensional reductions. As Leonid Polterovich described it: "It was clean. It was powerful" (Quanta Magazine).

The impact was immediate. Mohammed Abouzaid said the thesis "made it possible for many people to stop worrying about things. It was immediately impactful" (Quanta Magazine). Eliashberg noted that "a kind of foundation for certain parts of symplectic field theory are built on his machinery." In a field that had been in a "slow-burn crisis" over foundational rigor, Pardon's work resolved disputes and provided a solid platform.

2. The MNOP conjecture (2023)

This is Pardon's most celebrated single result. On a Calabi-Yau 3-fold — a shape of complex dimension 3 (real dimension 6) that appears in certain string-theory models as the geometry of hidden compactified dimensions — there are two completely different ways to count curves:

• Gromov-Witten invariants (developed by Gromov and Witten): count curves organized by their genus (how many "holes" the curve has)
• Donaldson-Thomas invariants (developed by Donaldson and Thomas): count curves organized by a different property

These two methods give completely different sequences of numbers. But in 2003, four mathematicians — Maulik, Nekrasov, Okounkov, and Pandharipande — conjectured that these two sequences are actually coefficients of the same function, expanded in two different ways (Plus Magazine, Quanta Magazine).

Think of it like discovering that two completely different scoring systems in martial arts — one based on strikes landed, another based on control time — are secretly measuring the same underlying performance, just from different angles. The MNOP conjecture said: these two countings are the same thing in disguise.

For 20 years, no one could prove it. Then in summer 2023, Pardon posted a proof online, establishing the MNOP conjecture for all complex threefolds with nef anti-canonical bundle (including all Calabi-Yau threefolds) with primary insertions. Jim Bryan, an expert in the field, said it came "out of nowhere" and that he "would have dismissed it entirely if not for Pardon's reputation." The proof used tools from a completely different area of math and created a new mathematical structure now called a "Pardon algebra" (Quanta Magazine). Bryan called it "the biggest result in enumerative algebraic geometry for the last 20 years" and said it "rewires the way he thinks about the entire subject."

3. Fukaya categories of Liouville manifolds (with Ganatra and Shende)

The Fukaya category is an algebraic structure that encodes the symplectic geometry of a manifold — it's like a dictionary that translates geometric information (about Lagrangian submanifolds and their intersections) into algebra (about objects, morphisms, and compositions). Pardon, with Sheel Ganatra and Vivek Shende, constructed wrapped Fukaya categories of Liouville sectors with a crucial property called covariant functoriality — meaning the construction behaves well under maps between spaces, which is essential for actually computing things (Quanta Magazine).

4. Knot theory and group actions on 3-manifolds

Even before his symplectic geometry work, Pardon made headlines as a Princeton undergraduate. He solved a 1983 problem of Gromov about knot distortion — the ratio between the distance along a knot and the straight-line distance between two points on it. Pardon proved that torus knots can have arbitrarily large distortion, meaning they can be arbitrarily tangled, no matter how you try to rearrange them. This was published in the Annals of Mathematics — a rare achievement for an undergraduate (Quanta Magazine). He also proved the Hilbert-Smith conjecture for three-manifolds, showing that every locally compact group acting faithfully on a 3-manifold must be a Lie group (arXiv:1112.2324).

Why It Is Transformative

Pardon's work is characterized by seeing connections between different areas of mathematics that others miss. He didn't just solve individual problems — he built frameworks. His virtual fundamental cycles work gave the entire field of symplectic geometry a rigorous foundation. His MNOP proof didn't just settle a conjecture; it introduced entirely new mathematical structures (Pardon algebras) that are already being used by other researchers to push further. And his work on Fukaya categories provided the computational infrastructure that makes symplectic geometry actually usable in practice.

As the Fields Medal citation states: "Pardon is a mathematician of extraordinary depth and originality, who has repeatedly had a significant impact on the fields of topology and symplectic geometry. He has both contributed to fundamental structure and solved problems that had stymied the community for decades" (Plus Magazine).

Animation (author's aid): the key idea and mechanism — blue circles (GW · by genus) and green squares (DT · by another property) count separately; two beams converge into a single pulsing "partition function Z" card: two sequences, one function, two expansions.

10 Most Influential Related Works

Core Papers by John Pardon

Foundational Predecessor Works

Current Impact and Follow-Up

Seminar Discussion Questions

Jacob Tsimerman — Number Theory and Arithmetic Geometry

Affiliation: University of Toronto (PhD Princeton 2011, advised by Peter Sarnak)
Age at award: 38

Official Citation

"For his role in the vast extension of the scope of o-minimal techniques within arithmetic and complex algebraic geometry, including the proof of Griffiths' conjecture on the algebraicity of images of the period maps."
— (Simons Foundation)

The Field: Number Theory and Arithmetic Geometry

Number theory is the study of integers and their properties — prime numbers, divisibility, solutions to equations. Arithmetic geometry brings geometry into the picture: instead of working with equations directly, you study the shapes (varieties) defined by those equations. It's like how a rock climber doesn't just think about the individual holds on a route, but about the overall geometry of the wall — the overhangs, the cracks, the dihedral — because the geometry reveals where the path must go.

A central theme in arithmetic geometry is the study of special points — points on geometric shapes that carry especially rich arithmetic information. Think of these as the "pressure points" of the geometric body: small, specific locations where deep structure is concentrated. The question is: when you find many such special points, what does that tell you about the shape they live on?

Shimura varieties are a class of geometric spaces that parameterize families of abelian varieties (higher-dimensional generalizations of elliptic curves). They are extraordinarily rich objects that encode deep arithmetic information. Hodge theory, developed in the mid-20th century, translates geometric problems into analytic ones — like translating a climbing problem from "where are the holds?" to "what forces are acting?" — and this translation is the setting for Tsimerman's work.

The Award-Winning Contribution

Tsimerman's Fields Medal recognizes his transformative use of o-minimality — a concept from mathematical logic — to solve deep problems in number theory and algebraic geometry:

1. The André-Oort conjecture

Formulated in the 1990s by Yves André and Frans Oort, the conjecture concerns special points on Shimura varieties. The prediction: if an irreducible subvariety of a Shimura variety contains a dense set of special points (called CM points), then the subvariety itself must be special (ScientiaMag). In other words: if you find enough "pressure points" clustered on a piece of geometry, that piece must itself be a special, highly structured object.

This is analogous to a principle in martial arts: if you find that a fighter consistently exposes a particular target, it's because their stance and movement pattern are built around protecting something else — the vulnerability reveals the structure.

The conjecture was attacked using the Pila-Zannier strategy, introduced in 2008: count the special points from above (using o-minimal geometry) and from below (using Galois orbit estimates), then compare. Tsimerman proved the conjecture for the moduli space of abelian varieties (A_g) in 2018, and then — with Jonathan Pila and Arul Shankar — proved the full André-Oort conjecture for all Shimura varieties in 2021 (arXiv:2109.08788, ScientiaMag).

2. Griffiths' conjecture on period maps

In 1970, Phillip Griffiths conjectured that images of period maps — maps that translate geometric data about algebraic varieties into analytic (Hodge-theoretic) data — should always be algebraic varieties (shapes defined by polynomial equations). This would mean that even when you translate geometry into analysis, the underlying algebraic structure persists (Plus Magazine).

Tsimerman, with Benjamin Bakker and Yohan Brunebarbe, proved this in 2018–2023. Their key innovation was "o-minimal GAGA" — a GAGA-type theorem (named after Serre's Geometrie Algébrique et Géométrie Analytique) for coherent sheaves on complex algebraic spaces that are definable in an o-minimal structure. This fundamentally connects o-minimal geometry to algebraic geometry and Hodge theory (arXiv:1811.12230).

3. Ax-Schanuel for Shimura varieties

With Ngaiming Mok and Jonathan Pila, Tsimerman proved the Ax-Schanuel theorem for all (pure) Shimura varieties — a functional transcendence statement controlling the algebraic part of the preimage of subvarieties under the uniformization map. This is a necessary ingredient for the Pila-Zannier strategy, analogous to knowing exactly which parts of an opponent's defense can be "gotten through" before you plan your attack (arXiv:1711.02189).

What is o-minimality?

An o-minimal structure is a universe of definable sets whose one-dimensional definable subsets are finite unions of points and intervals — no infinitely oscillating boundaries, no fractal pathologies. Higher-dimensional definable sets inherit strong tameness properties from this one-dimensional condition. As Tsimerman explains: "It makes precise the notion of what a tame geometry is" (Plus Magazine).

The genius of Tsimerman's contribution was recognizing that this tool from mathematical logic — developed to study the foundations of geometry — could be wielded as a powerful weapon in number theory. It's like a swimmer who discovers that a technique from rock climbing — reading the texture and structure of a surface — gives them a better understanding of water flow patterns around pool walls. The cross-domain transfer is what makes the work transformative.

Why It Is Transformative

Tsimerman didn't just solve individual problems. He built an entire infrastructure — definability of period maps, Ax-Schanuel theorems, o-minimal GAGA — that has become the standard toolkit for arithmetic geometry. His results are deeply related to the Hodge conjecture, one of the seven million-dollar Millennium Prize Problems (Simons Foundation). The o-minimal framework is now being applied to mixed period maps, the Shafarevich conjecture, birational geometry, and the study of Calabi-Yau varieties.

The University of Toronto notes that his work "opened entirely new directions of research" and influenced "functional transcendence theory, the Ax-Schanuel theorem for Hodge structures, and the study of so-called unlikely intersections" (U of T News).

Animation (author's aid): the key idea and mechanism — the infinitely oscillating "wild" curve on the left passes through the o-minimal lens and becomes tame geometry: finitely many points and intervals. The twin blue arrows squeeze from above (o-minimal point counting) and below (Galois orbit bounds) — the heart of the Pila-Zannier strategy.

10 Most Influential Related Works

Core Papers by Jacob Tsimerman

Foundational Predecessor Works

Current Impact and Follow-Up

Seminar Discussion Questions

Hong Wang — Harmonic Analysis and Geometric Measure Theory

Affiliation: NYU Courant Institute and IHES (PhD MIT 2019, advised by Larry Guth)
Age at award: 35 (third woman to receive a Fields Medal, after Maryam Mirzakhani in 2014 and Maryna Viazovska in 2022)

Official Citation

"For her work in harmonic analysis and geometric measure theory, including applications of multiscale and decoupling techniques to the local smoothing conjecture for the planar wave equation, and major advances in Fourier restriction, Falconer distance sets, Furstenberg sets in the plane, and the Kakeya problem in three dimensions."
— (Simons Foundation)

The Field: Harmonic Analysis and Geometric Measure Theory

Harmonic analysis is the art of breaking things into waves. Joseph Fourier discovered in the 19th century that any sound — no matter how complex — can be decomposed into a sum of simple sine waves. This is why an MP3 can compress music: it stores the wave components rather than the raw signal. Harmonic analysis generalizes this idea to far broader mathematical settings (Plus Magazine).

Think of it like a martial artist breaking down a complex combination into individual techniques: jab, cross, hook, kick. Each individual move is simple; the complexity comes from how they're combined. Harmonic analysis studies both the decomposition (breaking into waves) and the synthesis (putting waves back together), and asks: what can we say about the original object from its wave components?

Geometric measure theory extends the tools of calculus to shapes that aren't smooth — like fractals, soap bubble clusters, or the wildly irregular sets that arise in harmonic analysis. The key concept is Hausdorff dimension, which measures how the "size" of an object scales. A line has Hausdorff dimension 1; a square has dimension 2. But fractals can have non-integer dimensions — the Koch snowflake, for instance, has dimension about 1.26 (Plus Magazine).

The Award-Winning Contribution

Wang's Fields Medal recognizes a constellation of breakthroughs, crowned by one landmark result:

1. The Kakeya conjecture in three dimensions (with Joshua Zahl)

In 1917, the Japanese mathematician Sōichi Kakeya asked: what is the minimum area needed to rotate a needle so it points in every possible direction? You might think a circle works, but Kakeya found a smaller shape. Two years later, Abram Besicovitch showed something shocking: you can make the area arbitrarily small — even zero — by sliding and slightly twisting the needle in a fractal-like branching pattern (Plus Magazine).

This raised a deeper question: if the area can be zero, what about the dimension? The Kakeya conjecture predicts that even though a Kakeya set can have zero volume, its Hausdorff dimension must still be as large as possible — equal to the dimension of the space it lives in. In 2D, this was proved by Roy Davies in 1971 in just a few pages. In 3D and higher, it remained open for over 50 years.

In February 2025, Wang and Joshua Zahl (Nankai University) published a 127-page proof that every Kakeya set in R³ has Hausdorff dimension exactly 3 (arXiv:2502.17655, IHES). The proof works by showing that the only way a collection of thin tubes in 3D space can heavily overlap (which is what happens in a Kakeya set) is by clustering into convex sets — and then bounding the dimension of those clusters.

The proof built on several earlier breakthroughs. First, Wang and Zahl proved the "sticky Kakeya" case — sets with multi-scale self-similarity — in 2022 (arXiv:2210.09581). Then, in 2025, they showed that the general case reduces to the sticky case. The full argument combines multiscale geometric analysis, incidence combinatorics, and a deep understanding of how tubes in 3D space can overlap.

2. The Furstenberg set conjecture in the plane (with Kevin Ren)

A Furstenberg set is a set that, in many directions, contains fractal pieces of lines in that direction — a Kakeya-like condition, but one that concerns dimensional intersections with lines rather than rotating a unit needle. Wang and Kevin Ren fully resolved the Furstenberg set conjecture in R² in 2023, proving sharp dimension bounds (arXiv:2308.08819). The multi-scale "sticky reduction" ideas developed here directly influenced the 3D Kakeya strategy.

3. The local smoothing conjecture for the planar wave equation (with Larry Guth and Ruixiang Zhang)

The local smoothing conjecture asks: when a wave spreads out, does it "smooth out" locally — that is, does it become more regular in any given region than it was at the start? Wang, Guth, and Zhang proved this for the wave equation in 2+1 dimensions via a sharp L⁴ square-function estimate for the cone in R³ (arXiv:1909.10693). This was the first complete resolution of a local smoothing conjecture in any dimension.

4. Falconer distance set problem (with Guth, Iosevich, and Ou)

The Falconer conjecture asks: if you have a set of points in the plane with Hausdorff dimension greater than 1 (the general threshold is d/2 for R^d), must the set of distances between pairs of points have positive measure? Wang and collaborators proved that if the dimension exceeds 5/4, then the pinned distance set has positive measure — a major advance on the conjecture (arXiv:1808.09346).

5. Fourier restriction (with Shukun Wu)

The Fourier restriction conjecture asks: can you restrict the Fourier transform to a curved surface (like a sphere) and still get a bounded operator? This is fundamental to understanding how waves interact with curved boundaries. Wang and Wu introduced a new framework combining decoupling theory with two-ends Furstenberg inequalities, yielding new restriction estimates (arXiv:2411.08871).

Why It Is Transformative

Wang describes her field as having "a tower of conjectures" (Plus Magazine). The Kakeya conjecture sits at the bottom: it implies the Fourier restriction conjecture, which implies the local smoothing conjecture, which implies further results about PDEs. If you disprove the Kakeya conjecture, the whole tower collapses. By proving the 3D Kakeya conjecture, Wang didn't just solve one problem — she provided "foundational strength" to an entire tower of results in harmonic analysis, PDEs, and geometric measure theory.

Moreover, the methods Wang developed — the multiscale analysis, the sticky reduction, the interaction between incidence combinatorics and geometric measure theory — are already being applied to higher dimensions and related problems. As Wang herself put it: "In Fourier analysis we have a tower of conjectures. The Kakeya conjecture lies at the bottom of three other conjectures, one implying the other. If you disprove the Kakeya conjecture, then you disprove the whole tower" (Plus Magazine).

The problem remains open for dimensions 4 and higher — the frontier beckons.

Animation (author's aid): the key idea and mechanism — the needle rotates through every direction in the disc (area can be zero); the dimension meter is pushed to a full 3.00; the tower of conjectures lights up from its base (Kakeya): if the base falls the tower falls; the base proved, the tower stands.

10 Most Influential Related Works

Core Papers by Hong Wang

Foundational Predecessor Works

Seminar Discussion Questions

What the 2026 Fields Medals Tell Us About Mathematics

Looking across the four medalists, several patterns emerge:

1. Bridges between fields. Every medalist built a bridge between areas that seemed unrelated. Deng connected probability and PDEs. Pardon connected algebra and symplectic geometry. Tsimerman connected mathematical logic and number theory. Wang connected geometric measure theory and harmonic analysis. The most transformative mathematics happens at the boundaries between fields — like a martial artist who borrows techniques from judo, boxing, and wrestling to create something new.

2. Deep problems solved, not just advanced. Each medalist solved a problem that had been open for decades — Hilbert's sixth (1900), the MNOP conjecture (2003), the André-Oort conjecture (1990s), the Kakeya conjecture (1917). These weren't incremental improvements; they were completions of stories that had defined their fields.

3. New tools created. Beyond solving specific problems, each medalist created new mathematical machinery — random tensors (Deng), Pardon algebras and implicit atlases (Pardon), o-minimal GAGA (Tsimerman), sticky reduction and multiscale tube analysis (Wang) — that is already being used by other mathematicians to push further.

4. Physics and mathematics intertwined. Deng's work grounds the equations of physics in first principles. Pardon's Calabi-Yau manifolds appear in string-theory models of hidden dimensions. Wang's Kakeya conjecture underlies our understanding of wave propagation. The unreasonable effectiveness of mathematics in describing the physical world remains as mysterious and fruitful as ever.

5. The global nature of mathematics. The 2026 medalists hail from China (Deng, Wang), the United States (Pardon), and Canada (Tsimerman, born in Russia). Two are Chinese nationals — the first time mathematicians holding Chinese citizenship have won the Fields Medal (Shing-Tung Yau, who won in 1982, is of Chinese descent but was a US citizen at the time). Wang is only the third woman to receive the medal in its 90-year history (CNN, Simons Foundation). Mathematics is, increasingly, a truly global endeavor.

Looking Forward

The 2026 Fields Medalists have not only solved long-standing problems but opened new frontiers. Deng's methods may extend to more complex fluids and plasmas. Pardon's framework could unlock further enumerative geometry conjectures. Tsimerman's o-minimal toolkit is pushing into birational geometry. And Wang's 3D Kakeya proof has raised the urgent question: can these methods go to dimension 4 and beyond?

For the young mathematician — the one who sees waves in the pool, routes on the wall, and strategy on the mat — the lesson is clear: the deepest mathematics lives where different worlds meet. The 2026 Fields Medalists didn't just climb higher; they found new mountains.

Animation (author's aid): four bridges arc into place — probability↔PDE (Deng), algebra↔symplectic geometry (Pardon), logic↔number theory (Tsimerman), geometric measure theory↔harmonic analysis (Wang).

Quick Reference — The Four Medalists

This briefing was compiled from official IMU citations, university press releases, Quanta Magazine profiles, Plus Magazine explainers, and arXiv preprints. arXiv identifiers and journal references have been cross-checked against multiple sources.

Sources: Simons Foundation · Quanta Magazine — Deng · Quanta Magazine — Pardon · Plus Magazine — Deng · Plus Magazine — Pardon · Plus Magazine — Tsimerman · Plus Magazine — Wang · IHES — Wang · UChicago News — Deng · U of T News — Tsimerman · ScientiaMag — Tsimerman · Princeton University · Stanford Mathematics · Scientific American · CNN · NYT — Tsimerman · NYT — Wang · NYT — Deng · arXiv