2024's Biggest Breakthroughs in Math

We investigate three of 2024’s biggest breakthroughs in mathematics, including a better way to pack spheres in high dimensions, a new way to avoid forming patterns of numbers, and an 800-page proof of the so-called geometric Langlands conjecture.

00:04 High-Dimensional Sphere Packing Problem
The sphere-packing problem is simple to state: How do you arrange identical spheres to fill as much volume as possible without overlapping? In three dimensions, you can arrange spheres in a pyramid-shaped pile, the way oranges get stacked in a grocery store. But what about in higher dimensions?
In April, Quanta reported on the first significant advance on this version of the sphere-packing problem in 75 years. The result improved on the efficiency of previous packings, while making use of a novel approach: Rather than packing spheres in a nice, organized way, the mathematicians used graph theory to pack spheres in a very disorderly fashion.

05:10 Inevitable Patterns in Big Number Sets
Three graduate students proved a better estimate of the largest size that sets can reach before they must inevitably contain arithmetic progressions, which are evenly spaced patterns of numbers. The work, which deals with how order inevitably emerges from disorder in mathematics, marks the first progress on “Szemeredi’s problem” in decades.

09:33 Monumental Geometric Langlands Proof
In May, a team of nine mathematicians announced a major breakthrough. They had proved what is called the geometric Langlands conjecture — a central component of a broader research program to build a “grand unified theory” of mathematics. The proof, which totaled more than 800 pages and marked the culmination of 30 years of work, was, in the words of one mathematician, a “crowning achievement.”

[mostafatouny]

As an Egyptian student, residing in Egypt, These open-access resources are my sole portal to know what people around the world are doing. Thank you.

[pricesainterneta]

paul erdos is the most consistent side character in the math universe

[ivanleon]

imagine Tao saying you made a very impressive achievement, that is peak.

[Wheezy_calyx]

I look forward to these videos every year!

[moumous]

More on the Langland proof breakthrough!!!!

[Mane]

Please never stop posting this kind of videos for each scientific discipline at the end of the year. They're so good!

[tanmayshukla]

2023's breakthrough video feels like just yesterday :(

[totsamykotory]

That's funny that Quanta just had no time to prepare for the moving sofa problem solution that came in late november
Actually the biggest breakthrough of the year in my opinion

[carmelwolf]

as soon as you described the geometric Langlands conjecture and said they proved something i got so excited. i remember thinking this was impossible, or at least i wouldn't see any progress on this problem during my life. so please please please tell us more! this is so big!

[xvegitto]

This channel always delivers with great videos, always look forward to the end of year compilations, thanks to all the team for the great presentations, I always love the graphics as they help in the understating of often very complex subjects.❤

[drgamerj]

*Cool, now prove they are the biggest and show your working please*

[imperialdragon]

Sphere Packing:
- 2D optimal: **honeycomb**
- 3D optimal: **pyramid (74.05%)**
- Higher dimensions: **random beats ordered**
- Proof: geometry→graphs→**Rödl nibble**

Arithmetic Progressions:
- Core: **max set size pre-pattern emergence**
- Advance: **3 students improved bounds**
- Impact: **technique generalizes**

Geometric Langlands:
- Core: **math unification via Fourier**
- Proof: **eigensheaves + fundamental group reps**
- Key: **Poincaré sheaf = complete container**

[snappycattimesten]

As someone who struggled with fractions, I am amazed and the mathematical talent of mathematicians.

[emilev]

Le vidéo que j'attends toute l'année! Merci Quanta Magazine, j'en prendrais pendant des heures!

[emilev]

The video I wait for all year long! I'd take a full hour, this is so amazing! Thank you Quanta Magazine!

[ShermerHighSchool]

Very, very interesting BUT I want more! Please consider expanding on this subject in many more videos. Thank You!

[gustavobertozzimotta]

Man I wait the whole fucking year every year for this video and I’m never disappointed

[brunoalejandroandrades]

Marcelinho and Julian being in two consecutive years on the quanta recap is so iconic

[EneldoSancocho]

We are in hard times for math progress.
I myself discovered a few marvelous results, only to find they were already discovered hundreds of years before I was even born.
I'm afraid that math is getting harder faster than we are getting smarter.

[Perplaxus]

Amazing achievements. Langlands needs a video all on its own