Four Young Mathematicians Awarded the 2026 Fields Medals

Scientific American reports that the 2026 Fields Medals went to four mathematicians for work ranging from the theory of knots to the motion of fluid: The Fields Medals went to Hong Wang of New York University and France's Institute of Advanced Scientific Studies (IHES), Yu Deng of the University of Chicago, John Pardon of Stony Brook University and Jacob Tsimerman of the University of Toronto. In the awards' 90-year history, Wang is only the third woman to win one, after mathematicians Maryam Mirzakhani and Maryna Viazovska in 2014 and 2022, respectively. Wang and Deng represent the prizes' only Chinese-born recipients besides mathematician Shing-Tung Yau, who won a Fields Medal in 1982. Hong Wang co-proved the three-dimensional Kakeya conjecture, establishing a fundamental limit on how little space is needed to rotate a line through every possible direction. Mathematician Nets Katz called it the field's "holy grail" problem and said the achievement made her "a central figure" in the area.

Yu Deng and his collaborators reconciled the microscopic and macroscopic mathematics of fluid motion, proving that equations describing chaotic molecular interactions and large-scale fluid behavior are fundamentally connected. N.Y.U. mathematician Scott Armstrong called it "a truly spectacular, singular result."

John Pardon made an early breakthrough in knot theory by proving that certain sequences of knots can have arbitrarily large "distortion," a measure of how difficult they are to traverse. Princeton mathematician David Gabai said the problem had "attracted much interest among mathematicians during the previous 25 years."

Jacob Tsimerman and two collaborators proved the Andre-Oort conjecture, giving mathematicians a stronger way to understand special points on complex geometric objects known as Shimura varieties. Collaborator Jonathan Pila described him as "a brilliant mathematician" known for his "brilliance and resourcefulness." Tsimerman has also advanced Hodge theory and hopes pure mathematics can help researchers better understand AI.

A bit more about Kakeya

[JoshuaZ]

A bit more about the history of the Kakeya problem, which Hong Wang worked on. It comes from originally a more concrete question, namely how small an area do you need to rotate a 1 unit long needle 360 degrees if it is allowed to also move around as you rotate it? The naive thing is to use a circle, but you can also do a bit better by using a triangle, and then rotating it a little bit at the corner, moving the needle to the next corner, and then continuing the rotation. But you can do a bit better than this by cutting out small regions from the sides of the triangle. You instead use a deltoid https://en.wikipedia.org/wiki/Deltoid_curve [wikipedia.org]. Kakeya asked if this was the best possible construction But Beiscovitch made a very clever construction to show that there is no minimal area. That is, for any epsilson>0, you can make a region of area less than epsilon where you can rotate the need. https://en.wikipedia.org/wiki/Kakeya_set#Besicovitch_needle_sets [wikipedia.org] . Thinking about these ideas in abstract settings and higher dimensions lead to what was called the Kakeya conjecture, even though it post-dated Kakeya's own work. Hong Wang Joshua Zahl (not me, different mathematician with the same first name and last initial) proved that conjecture.

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[sabbede]

What I don't understand about it is the Minkowski dimension part. Not being a mathematician, seeing "Minkowski dimension 3" makes me think, "well, duh, if you're rotating something in three dimensions, you need three dimensions." If that's obvious to me, an idiot, I must not know what it actually means or it wouldn't be an open problem. Looking up what a Minkowski dimension is, I'm puzzled.

You seem to know what you're talking about, so if you know and it isn't any trouble, can you explain that and how

Re:A bit more about Kakeya

[JoshuaZ]

I think I can explain this. I'm not sure how well, I'll do but I can try.

For "nice" objects, we all have an intuition about how many dimensions the object is, which is roughly how many variables it takes to specify a given location on the object. For example, on a line embedded in your standard Cartesian two dimensions, say y=3x+2, you can just tell someone the x coordinate and they know exactly where they are. This works in general, so typical 3 dimensional space needs 3 coordinates. Etc. Similar, a circle (even though it lives in 2 dimensions) just needs 1 variable to tell someone where you are on it because you can take an angle from the center.

However, lots of objects turn out to have a notion of dimension that doesn't quite match up this way. This is roughly what people mean when they call something a "fractal" (although some people use fractal more to mean a thing deifned by a recursive procedure. Defining it precisely is tough.) Now, to talk about objects that are fractals, like say the Koch snowflake https://en.wikipedia.org/wiki/Koch_snowflake [wikipedia.org] we want a notion of dimension that makes sense for them. Ideally, for a given object we want a notion of "dimension" which gives the correct value of dimension when we apply it to an object that is a traditional object with a known dimension, like a line, or sphere or ball. The Minkowski dimension is one such approach. It is easiest to explain in 2 dimensions, but the same basic idea looks in any number of dimensions. Roughly speaking, you take the object you want, and you pick some tiny number epsilon, and then you make a grid on the plane of little boxes each which are epsilon by epsilon, and then you ask how many tiny boxes does it take to cover the object? Then, you take that number and divide it by log (epsilon), and then you let epsilon go to zero. This essentially measures how pointy thick the object is. If the object is thick this will just be two. If it is just a few isolated points (even infinitely many but not a lot) this will be zero. It isn't obvious that this is the right thing to count but turns out to work pretty well. You then generalize this to any number of dimensions for your space your object lives in by using boxes of the corresponding dimension number. So, for a fractal which lives on the real line, you use intervals of length epsilon, in 3 dimensions you use cubes of side length epsilon and so on.

Now, the Minkowski dimension is not the only notion of dimension we have that works this way. There's a related idea called the Hausdorff dimension which is defined in a more complicated fashion but turns out to often be easier to calculate. For example, it is not too hard to show that the Hausdorff dimension of the Koch snowflake is (ln 4)/(ln 3) which is about 1.26. (This reflects the intuition one might have that the Koch snowflake is much closer to being a thing made of lines than it is to being a thing with normal area). Now, it turns out that the Minkowski is always at least as large as the Hausdorff dimension, and they are equal in many situations. But there are some annoyingly simple situations where they are not. For example, if you take the interval of points between 0 and 1 and just take the rational points, then it has has Hausdorff dimension zero and Minkowski dimension one, which is about as far as you could want. Sometimes we'll calculate Haussdorff dimension for an object but really care about the Minkowski dimension, where finding that takes more work.

In the case of Hong Wang's work. She and Zahl proved that if you had what is called a Kakeya set in 3 dimensions (where you want a set that can fit a needle in every direction but you don't care about rotating it around, you just want a needle to fit at any orientation you choose), is that the Haussdorf dimension of the set, and the Minkowski dimension have to be equal and in fact equal to 3. But note that this isn't obvious in part because one doesn't have the rotation requirement here.

Does that help/make sense?

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[sabbede]

Yes, or at least it's starting to. Certainly, why the Haussdorf and Minkowski dimension matters, as it must mean that the needle's movement describes a very smooth shape, and I suppose a very solid one with the set of possible needle orientations completely filling that space, right? There must be a slew of interesting implications I couldn't even guess at, given my relatively paltry mathematical knowledge.

I blame school scheduling for that. From Jr High on through college, all my math classes were fi

It is math and not bean counting

[will4]

We should be beyond this repeated reporting of person X won an award - mention person demographics news reporting.

The nonprofit, UN, NGO, agitator groups who benefit from this reporting all need to have numbers to claim discrimination, oppression, or victimhood to keep the grievance industry and grievance jobs program going.

Imagine making a world class math discovery and then spending the next 50 years hearing "second woman Fields Award winner" always being first and foremost in reporting, introductions and

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[sabbede]

You mean something whose usefulness we might not be able to predict?

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[dsgrntlxmply]

"What is the use of a newborn baby?" -- Benjamin Franklin

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[Woeful Countenance]

Can't imagine wasting my time on something that useless

As useless as ... posting on Slashdot?

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[aRTeeNLCH]

I've not finished wrapping my head around this, but is my intuitive assumption correct that for a 360 degree rotation, one should pass through the point of 180 degrees? Or does one stay in the same 2d plane?

In visualising this, the triangle is actually how you invert a parked car. Well, doing it twice might be just for exercise...

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[JoshuaZ]

In the original 2-d problem, you do need to stay in the same 2-d plane the whole time.

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[aRTeeNLCH]

Yeah, thank you for answering. I was still a bit sleepy in morning public transport,...

I later realised it's like seeing a twirling baton moving forward in stroboscopic light, then figuring out what area it paints.

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[JoshuaZ]

Yeah! That's a very good analogy. I may steel it for future use when explaining this problem.

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[aRTeeNLCH]

No need to steel, just share, and I'm glad to have made sense. Happens all the time to myself, but to others, well ... ;-)

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[Buchenskjoll]

A very good explanation. I'm looking forward to the practical applications of this when carrying sofas up staircases.

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[cellocgw]

I'm looking forward to the practical applications of this when carrying sofas up staircases.

See e.g. https://en.wikipedia.org/wiki/... [wikipedia.org]

packaging

[noshellswill]

No mathematician myself ... I  can imagine  useful applications of this work for "packaging"  tasks, where  surface area translates into cost.

Deng, Hani, Ma: thanks!

[excelsior_gr]

I will not pretend I understand the paper, but this is a big deal! I find it annoyingly funny that the very mathematical definition of the continuity hypothesis (on which fluid dynamics is based), that the density equals the limit of DeltaMass/DeltaVolume for DeltaVolume going to zero, will break up if you actually apply it in reality: for volumes down at the atomic level the mass is not equally distributed anymore unless you use probabilities. This is very helpful in explaining some fluid dynamic principle

Congratulations! But why is reporting obsessed ...

[quenda]

... with identity politics?
2 Chinese-born, a woman, but poor Jacob Tsimerman doesn't rate a mention. Being Jewish doesn't make headlines when you are number 18.
Is all this really so important that it leads the reporting before any mention of what they did?

Young Mathemeticians?

[ISoldat53]

Isn't this award limited to young mathematicians? Nothing to take away from their achievements but do older people not make contributions to the field?

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[quenda]

but do older people not make contributions to the field?

Great mathematicians can certainly make major contributions late in life. Euler notably.
But like gymnastics, you have to start young.