2025 marked a historic year in mathematics. Researchers solved a major case of Hilbert’s ambitious sixth problem, proved a sweeping new theorem about hyperbolic surfaces, and settled the longstanding three-dimensional Kakeya conjecture.
*https://www.youtube.com/watch?v=hRpcWpAeWng
**https://300.ya.ru/v_QRQiTqOE
таймкоды
00:00:06 Шестая задача Гильберта
- Дэвид Гильберт предложил 23 задачи для будущего математики, включая задачу о математическом доказательстве законов физики.
- Задача заключалась в доказательстве связи между различными уравнениями, описывающими поведение газов.
- Три исследователя разработали доказательство, связывающее уравнения в больших временных масштабах.
00:00:53 Масштабы моделирования газов
- Микроскопический масштаб: частицы газа моделируются как бильярдные шары, подчиняющиеся законам Ньютона.
- Мезоскопический масштаб: уравнение Больцмана описывает поведение групп частиц.
- Макроскопический масштаб: уравнения Навье-Стокса описывают поведение газа как единого вещества.
- Доказательство связи между микроскопическим и мезоскопическим масштабами было сложной задачей.
00:01:48 Сложности столкновений частиц
- Столкновения частиц в газе, подчиняющемся законам Ньютона, вызывают множество проблем.
- История столкновений представлена в виде диаграмм, которые сложно анализировать.
- Оскар Лэнфорд показал связь между бесконечной суммой моделей столкновений и уравнением Больцмана для коротких временных интервалов.
00:03:06 Преодоление трудностей
- Главным препятствием были «воспоминания» — сценарии, в которых частицы сталкиваются более одного раза.
- Математики Юй Данг Цзы и Сяо Ма разработали алгоритм для разбиения сложных диаграмм на более мелкие фрагменты.
- Они показали, что модели столкновений с множеством «воспоминаний» маловероятны даже в длительных временных масштабах.
00:04:56 Результаты и перспективы
- Команда опубликовала результаты, показывающие сходимость микроскопических уравнений Ньютона к мезоскопическому уравнению Больцмана.
- Решение шестой задачи Гильберта — важная веха в математической физике.
- В будущем математики могут изучать газы в более реалистичных условиях, например, с использованием квантовой механики.
00:06:28 Гиперболические поверхности
- Гиперболические поверхности имеют отрицательную кривизну и являются строительными блоками многих математических дисциплин.
- Марьям Мирзахани разработала инструменты для изучения их геометрии.
- Налия Нараман и Лаура Монк развили её работу, доказав знаковый результат о спектральном разрыве гиперболических поверхностей.
00:08:30 Спектральный разрыв гиперболических поверхностей
- Спектральный разрыв измеряет взаимосвязь гиперболической поверхности.
- Большой спектральный разрыв означает, что поверхность хорошо взаимосвязана.
- Анантхарман и Монк пытались доказать, что у большинства поверхностей спектральный разрыв составляет почти одну четверть.
00:09:15 Проблемы и вдохновение
- Некоторые поверхности имеют запутанные геодезические линии, искажающие расчёты спектрального разрыва.
- Исследователи обратились к работе Джоэла Фридмана, который доказал знаковый результат о спектральном разрыве в случайных графах.
- Формула обращения Мебиуса, использованная Фридманом, стала ключевым инструментом для решения проблемы.
00:11:22 Спектральный разрыв гиперболических поверхностей
- Монк и Анонтерман адаптировали формулу Фридмана для вычисления среднего спектрального разрыва гиперболических поверхностей.
- Они показали, что почти все гиперболические поверхности имеют спектральный разрыв в одну четверть, что означает их высокую математическую связность.
- Новая работа может помочь в понимании теории чисел, динамики и квантового хаоса.
00:12:32 Гипотеза Каака
- В 1917 году японский математик Сай Каа задал вопрос о наименьшей области, охватываемой бесконечно тонкой иглой при вращении во всех направлениях.
- Эта гипотеза стала основой для многих идей в гармоническом анализе.
- В начале 2025 года два исследователя опубликовали доказательство гипотезы Каака, что считается крупным достижением в математике.
00:13:30 Открытие Бесаковича и вклад Дэвиса
- Абрам Бесакович показал, что на плоскости можно вращать иглу, оставаясь внутри нулевой области.
- Рой Дэвис доказал, что каждый набор Каа имеет дробное измерение, равное двум.
- Это открытие вдохновило на гипотезу Каака о дробной размерности в n-мерных пространствах.
00:14:24 Сложности доказательства в трёх измерениях
- В трёх измерениях семейство направлений богаче, что усложняет доказательство гипотезы Каака.
- Для доказательства в трёх измерениях используются тонкие трубки, направленные в разные стороны.
- Необходимо рассмотреть все возможные конфигурации трубок и показать, что каждая из них занимает большое пространство.
00:15:25 Связь с преобразованием Фурье
- Чарльз Пфефферман связал гипотезу Каака с преобразованием Фурье.
- Понимание геометрии стало ключом к решению вопросов в аналитике.
- Гипотеза Каака стала основой для иерархии гипотез о поведении преобразования Фурье в высоких измерениях.
00:16:23 Доказательство Ванга и Золя
- В 2022 году Хонг Ван и Джошуа Золь доказали гипотезу Каака для липких наборов.
- Ларри Гух показал, что контрпример к гипотезе должен быть зернистым.
- Зернистость дала Вану и Чжао необходимую структуру для доказательства.
00:17:14 Индукция по шкалам и завершение доказательства
- Ван и Чжао использовали индукцию по шкалам для контроля потерь при доказательстве.
- На каждом шаге они улучшали нижнюю границу размерности наборов Каа.
- В начале 2025 года они завершили доказательство: каждое множество Каа в трёхмерном пространстве имеет дробную размерность три.
00:18:53 Перспективы дальнейших исследований
- Доказательство Ванга и Золя даёт уверенность в правильности многих предположений.
- Ожидается значительный прогресс в решении оставшихся гипотез.
- Математики надеются использовать полученный результат для доказательства более амбициозных гипотез.
В этом видео
Hilbert’s Sixth Problem Solved
0:06
At the dawn of the 20th century, the mathematician David Hilbert presented a
0:11
23-problem roadmap for the future of mathematics. His sixth problem was among the most ambitious:
0:17
to mathematically prove the laws of physics. It was a tall order, but Hilbert offered a
0:23
starting point. Physicists model the behavior of gases using different sets of equations.
0:29
The challenge was to prove that these equations describe the same reality,
0:33
just seen through different lenses. For 125 years, mathematicians struggled with
0:39
the problem. This year, three researchers devised a proof that finally connects these equations at
0:45
long timescales. It’s one of the most significant advances in mathematical physics in decades.
0:53
At the microscopic scale, physicists model gas particles as tiny billiard balls that
0:58
follow Newton’s laws of motion. Zoom out to the mesoscopic scale,
1:03
where the Boltzmann equation describes how groups of particles are likely to behave over time.
1:08
Zoom out again to the macroscopic level, where you study the behavior of
1:12
the entire gas as a single substance using the Navier-Stokes equations.
1:18
Mathematicians had already proved that a mesoscopic description of a gas gives rise
1:23
to a macroscopic one in various settings, like for a dilute gas sealed inside a box.
1:29
But proving the microscopic-to-mesoscopic connection was vastly harder. To do this,
1:35
researchers would have to show that if they started with Newton’s laws
1:38
and increased the number of particles in their gas to infinity, then their
1:42
description of the gas’s likely behavior would converge to the Boltzmann equation.
1:49
The hardest part is dealing with particle collisions. In a gas where every particle
1:53
follows Newton’s laws, the vast number of possible collisions introduces huge headaches.
1:59
You’re talking about a very large number of particles – a number of particles that in the
2:02
limit goes to infinity. So those collision histories are very, very, very complicated.
2:08
Each collision history can be turned into a diagram. Mathematically,
2:13
this is a graph with nodes that mark the moments when particles collide,
2:16
and lines that show how the particles move between those collisions.
2:20
Those diagrams record the collision histories, like saying that particle one collided with
2:27
particle two, and before that it collided with particle three. We want to give estimates on the
2:32
probability of such collision histories, which are probabilities of those very huge graphs.
2:39
In 1975, Oscar Lanford showed that the infinite sum of those collision patterns does give rise
2:44
to Boltzmann’s equation – but only for extremely short time intervals. Beyond that tiny window,
2:51
the number of possible collision diagrams explodes, and the whole method collapses.
2:57
There are big challenges that one encounters if one wants to go to long times, which was
3:03
something that has never been done before. The main obstacle was recollisions:
3:08
scenarios where the same particles run into each other more than once.
3:13
To show that the infinite sum behaves like the Boltzmann equation, mathematicians have
3:18
to show that collision patterns involving lots of recollisions are vanishingly rare.
3:24
Lanford showed this for very short time periods. But as time goes on and particles
3:29
have more chances to recollide, this gets much harder to prove.
3:33
This is the dangerous mathematical issue which has to be tackled in order to prove
3:39
the validity of the convergence for a long time. And so the mathematicians Yu Deng, Zaher Hani
3:46
and Xiao Ma set out to tame the explosion of diagrams that made the problem intractable.
3:52
What is really causing this divergence? We were zooming for hours, going to bed
3:57
thinking about it, waking up, thinking about it. It required such intense work.
4:03
The team developed an innovative mathematical
4:06
algorithm to break enormous collision diagrams into smaller pieces that are easier to study.
4:11
The algorithm is really about how to break a very big diagram – a very complicated collision
4:17
history – into some small pieces that are so simple that we can compute by hand.
4:22
Then came the breakthrough: the mathematicians realized that they didn’t need to follow every
4:27
possible collision history. Instead, they learned to focus only on the well-behaved ones.
4:33
This was the flip of the switch that made me believe that we can get to longer times.
4:39
Now, with a manageable number of diagrams, the team could show that collision patterns
4:43
with lots of recollisions are extremely unlikely, even over longer time scales.
4:49
So it turned out that [these] recollisions were actually not important. They can be neglected.
4:57
In March 2025, the team posted their result, showing that Newton’s microscopic
5:02
equations converge to Boltzmann’s mesoscopic equation over long times.
5:07
We were able to show that indeed as you take the limit of the number of particles going to infinity
5:14
and the diameter of the particle is going to zero, this density does satisfy Boltzmann’s equation.
5:21
Their work finally completes the solution to Hilbert’s Sixth problem – a milestone more than
5:26
a century in the making. Mathematicians are excited about what comes next.
5:32
Going to long times – this was well understood as not the end point of the research in this field,
5:39
but rather that it was somewhat of a bottleneck that we needed to overcome in order to access
5:47
even more interesting questions that lie ahead. Going forward, mathematicians could try to
5:52
use the new techniques to study gases in more realistic settings.
5:56
There are many settings that we had to impose on ourselves in order to achieve this. So what
5:57
happens for a gas that is not as dilute? What happens to a gas of particles that
6:03
are not exactly colliding with each other like hard spheres? What if I use the laws
6:09
of quantum mechanics instead of Newton’s laws as my fundamental law of physics?
6:14
So all those are questions that are still very open. It is very exciting
6:18
that we are in this period where we can start talking seriously about the next challenges.
6:25
Imagine a surface that curves like a saddle at every point,
Hyperbolic Geometry
6:32
causing parallel lines to diverge. Defying geometric intuition,
6:36
it twists and folds so wildly that it can’t live in ordinary space. This is a hyperbolic surface.
6:44
You can’t realize these surfaces in three dimensions, but you can
6:48
realize them abstractly, and they are the building blocks of a lot of mathematics.
6:53
The mathematician Maryam Mirzakhani developed tools to explore the strange
6:58
geometry of hyperbolic surfaces. But when she died in 2017,
7:02
at just 40 years old, her quest to illuminate these unimaginable shapes was left unfinished.
7:08
She was a super-superstar. You immediately recognize her genius just talking to her.
7:12
Now, mathematicians Nalini Anantharaman and Laura Monk have advanced Mirzakhani’s pioneering work,
7:18
proving a landmark result with deep implications for mathematics and physics.
7:24
So hyperbolic surfaces are quite enigmatic things. We can’t really draw them very well.
7:29
The plane is flat and the sphere is positively curved. And, to the contrary,
7:34
hyperbolic surfaces – they’re negatively curved.
7:38
And so as opposed to the sphere, there’s more directions to explore. So if you have
7:42
a little character here and it’s walking on this plane, there’s actually a lot of
7:46
different directions it can go to. The space is actually growing very, very fast around it.
7:51
To understand hyperbolic surfaces, mathematicians study closed
7:55
geodesics – paths that follow the shortest route and loop back to their starting point.
8:00
The search for such closed geodesics or what are called periodic orbits has always been central in
8:06
the study of mechanics and the study of geometry. In her 2004 Ph.D. thesis, Mirzakhani derived a
8:13
formula estimating how many closed geodesics a hyperbolic surface has
8:17
up to a given length, offering a hint of its overall geometric structure.
8:22
She started to ask questions that are very natural, like if you take
8:26
a random surface, does it have some property? Mathematicians are interested in how connected
8:32
a hyperbolic surface is – a property measured by a number between 0 and 1/4 called the spectral gap.
8:40
A large spectral gap means the surface is well connected, with many paths linking its regions.
8:46
A small gap means it’s poorly connected and harder to move across.
8:50
So, what we wanted to do is to compute the spectral gap of a typical hyperbolic surface.
8:56
So take a very big bag and put all of them in the bag, take a surface, and wonder, what
9:02
does it look like? There’s a magic number, one quarter, which is the best possible spectral gap,
9:08
and what we wanted to show is that most surfaces have a spectral gap of almost one quarter. So
9:12
it means they are as connected as possible. Anantharaman and Monk hoped to prove this by
9:18
building on Mirzakhani’s formula to get a more accurate estimate of geodesics.
9:23
What struck me when I started reading her works is how beautiful it was.
9:27
But they soon encountered a problem:
9:29
Some surfaces have very tangled geodesics that wind around the same region for a long time.
9:35
When these rare geodesics crop up, they appear in such large numbers that they distort calculations
9:41
of the spectral gap. Because of this, attempts to prove the maximum spectral gap stalled at 3/16ths.
9:50
Tangles – they make life very difficult for you to push through from 3/16 to a quarter.
9:56
They completely blow up the average and make it completely impossible to do our technique.
10:02
So we need to remove them. I want to take my big bag and I want to remove
10:07
those surfaces which I don’t like from the bag. But all the tools we have – we’re not really
10:11
allowed to take an average on some of the bag. We were manipulating very complicated formulas
10:20
of hyperbolic trigonometry, and I remember that early 2022 we thought that we should give up.
10:29
For inspiration, the researchers turned to mathematician Joel Friedman,
10:33
who in 2002 proved a landmark result about the spectral gap in
10:37
random graphs – networks with points and edges that are connected according to probability.
10:43
His proof was very long. I don’t know how many
10:46
people even studied his proof because it was so complicated.
10:50
Friedman showed that most random graphs have the largest possible spectral gap. This means they
10:56
are expanders — networks that stay highly connected even with relatively few edges.
11:02
Everywhere you turn, algorithms are based on expanders.
11:05
Buried in the middle of Friedman’s paper was a key mathematical tool, the Möbius inversion formula,
11:12
which he used to extract information about the average spectral gap of expanders.
11:17
We really understood that what we were trying to do was similar to what he did.
11:23
By adapting Friedman’s formula, Monk and Anantharaman were able to filter out tangled
11:28
geodesics to compute the average spectral gap across all hyperbolic surfaces. They showed
11:34
that almost all hyperbolic surfaces have a spectral gap of one-quarter,
11:38
meaning they are as connected as mathematically possible.
11:43
Mathematicians hope to use the new work to answer questions in number theory and dynamics,
11:48
including about how quantum systems create chaos.
11:52
The thinking is that hyperbolic surfaces are very good toy models. You have your
11:59
model – it’s too complicated. Well, let’s think about something that’s
12:02
simpler and see what we can do there. Quantum chaos is a very, very complicated topic. But
12:07
hyperbolic surfaces have a lot of things going on for them. There’s a lot of algebraic tools.
12:10
That’s an area that I think will be impacted tremendously in the next few years.
12:15
Myriam Mirzakhani – I think she would have been very excited. Seeing how big
12:21
the field has become over the last few years would have made her very happy, doubtless.
3D Kakeya Conjecture
12:32
In 1917, Japanese mathematician Sōichi Kakeya asked a deceptively simple question:
12:39
If you rotate an infinitely thin needle through all possible directions, what’s
12:43
the smallest region it can sweep out? This playful geometric puzzle grew
12:48
into the Kakeya conjecture, one of the most influential problems in modern mathematics.
12:55
Today, the conjecture sits at the base of a towering set of ideas in harmonic analysis,
13:00
the study of the mathematics of signals and waves.
13:03
Everything’s resting on this Kakeya conjecture. If that’s wrong,
13:07
then the whole tower of implications kind of collapses, right? Everything fails.
13:12
For decades, higher-dimensional cases of the Kakeya conjecture eluded the world’s greatest
13:17
mathematical minds. But in early 2025, two researchers published a once-in-a-century proof.
13:24
In harmonic analysis, this is probably the biggest development in at least 20 years.
13:30
In the early 20th century, Abram Besicovitch stunned
13:33
mathematicians by showing that in the plane, you can rotate a
13:37
needle through every direction while remaining inside a set of zero area.
13:42
Roughly speaking, what Besicovitch found was that if you take a million-point U-turn – if
13:45
you do a ridiculous number of back and forths – you can make the area used as
13:49
small as possible. So that was unintuitive. These complex objects, called Kakeya sets,
13:55
aren’t “large” in any ordinary sense. To understand their true size, mathematicians
14:00
had to look beyond area and study how they fill space in fractal dimensions.
14:06
In the 1970s, the mathematician Roy Davies proved
14:09
that every Kakeya set, no matter how thin, still has a full fractal dimension of 2.
14:16
That revelation inspired the Kakeya conjecture,
14:19
a bold claim that in every dimension, these bizarre sets somehow fill the whole space.
14:25
These Kakeya sets can have small areas,
14:27
but they have to be full-dimensional objects. They can never be kind of
14:30
compressed into a lower-dimensional space. While this statement may seem simple,
14:36
mathematicians struggled to prove it in higher dimensions.
14:39
The family of different directions in three dimensions is much richer
14:42
and much more complicated. And as a consequence of this, there’s a lot of
14:46
phenomena that occur in three dimensions that you don’t really see at all in two dimensions.
14:51
In his 2D proof, Davies thickened needles into little rectangles and
14:55
studied their intersections. In 3D, you get extremely thin tubes instead.
15:01
And in the Kakeya conjecture, what you’re trying to prove – what it really boils down
15:05
to – is showing that if you have a collection of tubes pointing in different directions, then they
15:10
can’t intersect very much. You have to consider all possible ways you can arrange these needles
15:15
in different regions. And there’s an infinite number, basically, of all these configurations.
15:19
And you want to show that every single one of these has to occupy a large amount of space.
15:25
Around the same time Davies published his 2D result, Charles Fefferman published a surprising
15:30
proof that connected the Kakeya conjecture to the Fourier transform, a powerful tool from the field
15:35
of harmonic analysis that allows mathematicians to study complicated signals or functions.
15:41
It was a fundamental new connection. It became increasingly clear that the
15:45
path forward to solving questions in analysis was to understand geometry.
15:49
The Kakeya conjecture was no longer geometric curiosity. Suddenly, it was foundational to
15:55
a hierarchy of major conjectures about how the Fourier transform behaves in higher dimensions.
16:00
So it became increasingly important to understand the Kakeya conjecture because
16:04
it’s part of a whole family of really difficult conjectures in many fields.
16:09
For decades, mathematicians worked on the Kakeya conjecture in 3 dimensions, but progress was slow.
16:15
People were stuck for a long time. I spent many years working on this
16:18
problem. And we had some partial results, but we were missing big chunks of the program.
16:23
Then, in 2022, Hong Wang and Joshua Zahl joined the effort. Their first target was
16:29
a special family of Kakeya sets called sticky sets, where tubes pointing in the
16:34
same direction stay close together in space. This structure made the sets easier to study.
16:40
In 2022, Wang and Zahl proved the 3D Kakeya conjecture for sticky sets.
16:45
That was really strong evidence that they were closing in on the goal.
16:49
The harder challenge was the non-sticky case,
16:52
where sets exhibit an irregular geometry, with tubes scattered in all directions.
16:57
A clue came from mathematician Larry Guth. He showed that any counterexample to the Kakeya
17:03
conjecture in 3D would have to be grainy, or full of tiny regions where many tubes overlap.
17:09
We spent quite a bit of time trying to understand how these grains interact with each other.
17:14
Graininess gave Wang and Zahl the structure they needed. They showed that no point in space
17:19
can lie in too many grains, limiting how efficiently tubes can overlap,
17:24
and preventing the set from compressing into a smaller-dimensional object.
17:28
They could use this argument to show that no 3D Kakeya set can have a dimension below 2.5
17:34
for example. Then, they expanded their proof with an argument called induction on scales.
17:40
There has been this dream for a while to prove this conjecture
17:43
about what’s called induction on scales. It’s a way to get from A to B, when A and B are
17:48
very far apart, by just little steps. Previous attempts to prove the Kakeya
17:53
conjecture with the induction on scales had resulted in inefficiencies and losses.
17:58
A classic example is the game Chinese Whispers, where you get a lot of people
18:01
in a row and someone whispers a sentence to the next person who whispers a sentence to the next
18:06
person. If they had perfect reproduction, then by induction, the person at the end would have
18:11
exactly the same sentence as what they started with. But if you only have even a tiny amount
18:15
of loss in each transition step, then the end result can be worthless – or at least hilarious.
18:24
Wang and Zahl figured out that graininess was the key to controlling these losses.
18:29
And so you can slightly increase, with each time you run this argument,
18:33
your best estimate on the dimension of sets until eventually you get all the way up to three.
18:39
At each step, Wang and Zahl improved the lower bound on the dimension of any Kakeya set. Then,
18:45
in early 2025, they completed the proof: Every Kakeya set in 3D has a fractal dimension 3.
18:53
You really try hard to not let yourself get carried away by the excitement.
18:58
Mathematicians are hopeful that they can use the result to help build up
19:01
proofs of the more ambitious conjectures above it.
19:05
So it’s now given us a lot more confidence that all these beautiful conjectures we
19:13
have – that they’re going to hopefully be true. People are just going to start climbing this
19:17
hierarchy. And I think one by one, these other conjectures are going
19:20
to either be solved completely or you will see a lot of dramatic progress.