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. Read about more math breakthroughs from this year at Quanta Magazine: www.quantamagazine.org/the-year-in-math-20241216/
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.❤
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
It's a fascinating problem in terms of how simple it is and how it developed calculus of variations and continuous optimization techniques, but it's not as important as the geometric Langlands conjecture or sphere packing or even the combinatorics of Erdos sets (maybe) in terms of how many mathematicians were researching that area and expecting solutions.
Has the proof been officially recognized ? I mean I have little doubt about its legitimacy, but perhaps Quanta wants it to gain credibility among the mathematician community, just as a precaution 🤷♂️ It will most likely be in next year’s edition tho
All of these videos are quite subjective. For instance, they could've also included the new zero density estimates for the Riemann zeta function that happened earlier this year. They also could've included the telescope conjecture (and several others) in the video last year. I think the three put in this video are fine candidates for the biggest breakthroughs of the year.
@@skylardeslypere9909 I think, that you are correct. It is still in the process of being validated. It is a 100+ page proof so it will take a bit of time.
As A high school student i hear often bout Great past mathemticians (In classes n YT videos) but i rarely hear bout any great modern Mathematicians or breakthrough which made me unconsiously assume tht Mathematics and Science had ripened and there are very little progress tht can be made and it's not worth dedicating time over it. But I do love the subject and aspire to be one great mathematician myself oneday Which is y I desperately needed videos like this Thanks❤
Mathematicians are fascinated by Platonic reality. This allows them spend far more time than others thinking about it. In the process they develop a stronger than usual ability to focus and spot patterns. Most talents are developed, not innate. One should note that this obsession with inner reality can result in, or be caused by, a dissatisfaction with practical day to day reality, especially concerning social interactions, which are generally complex beyond reason.
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!
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.
Math is getting so large that memorizing all the results is becoming difficult for the human brain. We need a digitalization formalism for all mathematics and input discovered theorems and results in them. That way you could just check online what has already been proven in your line of work
@@solverapproved A piece of software that's the complete map of mathematics, updated in real time by AI scraping of journals and all published mathematical literature. A man can dream.
The ideas in this video strangely complement each other very well! The Langlands program can be used to study the distribution of primes within certain arithmetic progressions, and here we have another work on the other angle of density of arithmetic progressions! These are bound (get it?) to connect very soon.
Small observation on the Fourier Series Decomposition - the frequency of each harmonic is a transformed input parameter, the two results for each harmonic are its amplitude and phase.
Great video. The last line was funny. I don’t think anyone believes there is a shortage of mathmatical problems. In fact, much harder, is imagining what the ideal end of inquiry will look like when math is done? [edited]
Dam today's a good Day. I have been thinking about this from a different perspective and really enjoy the I sights hoping to be able to use this for a great Tool 👍😎
I have solved the Collatz Conjecture and the Goldbach Conjecture.. They are simple yet nobody is willing to accept them. As an Indian,. my favourite mathematician is Srinivasa Ramanujan... I have found 2 new ways to arrive at 1729 - - > Ramanujan Number 1.8465 Parse :84 65 84-65 = 19 Reverse 🔀 91 19*91 = 1729 2.6748 Parse :67 48 67-48 = 19 Reverse 🔀 91 19*91 = 1729 VIOLA 😮
Good question. My thoughts are that in order to really know how difficult a problem is you would have to understand it completely but if you understand it completely then you already probably know how to solve the problem. So the issue would be knowing what it is about a problem that you don't know and that has "known unknowns" and "unknown unknowns" the latter of which, by definition, you don't know. So how difficult a problem is comes down to the point at which you ask the question and you could define that as the level of difficulty before you know anything (an absolute difficulty) or the level of difficulty assuming all required inputs to it are required (a relative level of difficulty). So then, finally, each problem, assuming all required input "knowns" are known, comes down to how big is the space of potential solutions. This value might be computable but in reality we would not know at any point whether there were any more "unknown unknowns" which are needed as required inputs to a problem. So I think we can calculate how difficult a problem is but only after whe have solved it :D
@@christophersinger9149 I think it is completely subjective. Not only depending on one's knowledge but also intuition. If you just cant think of it, its impossible. If it somehow just strucks, it does and its easy then.
I don't think so, theres time complexity analysis but that tell you how hard is to compute a solution to the problem not the difficulty of the problem itself
the answer is obvious, if metals lattices have max packing fraction is 0.74, we just need to reversely add it from different perspectives which can exceed 74% where we get states where it can be double or same depending on the dimensions
It’s been solved in 3 dimensions for a while. The high dimensional problem is useful for a bunch of data analysis problems. High dimensional math shows up alot in data analysis. Neural networks rely on a bunch of properties in high dimensions (like spheres being pointier)
Dude, SO much math has practical educations that're found out later Like "sedenions", which are "16-dimensional noncommutative and nonassociative algebra over the real numbers", are used in traffic and weather forecasting 16-dimensional, with real-world applications. Where's the anti-academic sentiment coming from?
this is pure mathematics, they don't care about applications. it's so abstract. You'll need to do graduate work then another x amount of years to understand the basic outline of a proof. No mere mortal will full be able to understand it. They are not obligated to do anything that they don't enjoy. If you want practical applications you might want to look at other areas of Maths or other sciences.
The correct largest subset of [20] avoiding a 5-term ar. progression is 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19. It does have 16 members. For more information check Sloane's OEIS.
In the second discovery, the szemerdi problem, the video says for a 5 term progression not to appear in a pool of 20 numbers, the highest density possible is 80 percent i.e. 16 number. But with in 16 numbers in the set, there is one 5 terms progression of 3. Or may they were talking about 5 terms' progression of one particular number i.e. 4 in the case
I hardly understand these problems, however it's cool to see now in Maths it's too complicated that most of the time you need cooperation to solve problems.
It's crazy that I can use Fourie's transform for physics/engineering without ever understanding what it represented until now. Just goes to show how powerful these tools we have are! Similar to how you don't need to be a mechanic to drive a car, I don't need to be a mathematician to be able to work with frequency. Although, I often feel like a monkey with a screwdriver 😅
The first problem is by far not a stupid question. My mom once asked if there was any software that could pack a bunch of 2D figures into a fabric plane, since she needed to cut down some pieces of a fabric and wanted to waste the least amount of it. Now think about how much material and waste could be spared with a software programmed with such solution.
5:25 Randomness has to do with how the numbers are generated. It has nothing to do with what they are, or with what patterns (if any) there are in them.
It always amazes me how the brightest minds spend their time and energy at the most useless tasks... I wonder if there is a connection between this fact and the state we find the world is at...
Why are the breakthroughs always in combinatorics, number theory, discrete math etc? I wish you would cover fields such as geometric analysis, probability etc
I always enjoy these videos. Well done! Thank you Quanta. A correction: >"Any wave can be broken down into an infinite sum of sine waves" This is fundamentally wrong. Some waves contain more information than others. Almost all waves, even at infinite resolution, can be expressed in just a few bits. The more complex the wave, the more bits it requires. There is a limit to the information physical waves have since physical systems are fundamentally discrete (Planck's constant, quantum mechanics). There's only infinite information in some abstract, theoretical waves. For example, a wave that somehow encoded pi π would contain infinite information, although from the top of my head I couldn't detail a method to do this. Real waves, and all mathematically described waves that I can think of, have limited resolution. Just because a wave is continuous does not mean it has infinite composite waves. A pure sine wave is an example. The fact that Fourier decomposition is broken down into component waves also shows this. Each component wave does not have infinite information, otherwise it would be useless as a component. The whole utility of Fourier decomposition is that it breaks down complex waves into simpler, well-defined components with finite information content. A wave with finite information content cannot be made up of an infinite number of independent components, as that would imply infinite information content. Also, don't forget the Nyquist sampling theorem, which also says that for a given bandwidth, *all information* may be described using limited sampling/limited information. There is a strict limit to the information potentially in a given band. As someone who works in information theory, I notice that mathematicians ignore information theory, and its consequences, all the time. It makes me suspicious that there's low understanding and application of information theory in wider mathematics. It also explains why I experience resistance with some mathematicians when discussing the fundamentality of information theory, like the Church-Turing Thesis. Mathematicians don't typically think in terms of information even though it is fundamental to their work. I think there's a general attitude that since they work in abstract concepts, information theory is somehow an implementation from abstract to application and thus can be ignored. I suspect there's a lot of low hanging fruit in mathematics that could progress the science if information theory was taken seriously and applied.
I swear every time I see videos with "the latest math research" type content, it always includes some "packing" problem. Why is this such a popular thing? I'm assuming it's a problem in industry or something. E: Makes me wonder who are the "brokers" between mathematicians and those who identify a problem in industry that can be solved with math.
The topics you see in math journalism aren’t very representative of what mathematicians care about. Sphere packing is one of those things that’s really easy to explain to a lay audience, so it gets covered a lot, same with tilings of the plane. I would say that there are probably 10 topics or more that you could put in a video like this based on their significance in the subfield, it’s just that 90% of math research is incomprehensible to those without specialized training.
The E8-lattice (optimal unit ball packing in 8D) for example was recently used in a paper called QuIP# about a new LLM quantization algorithm, which are used for compressing a large model, so that it fits on smaller GPUs
The process is straightforward: thoroughly squash the oranges, then place them into any container of your choosing. Avoid overcomplicating a task that is inherently simple.
The close-packing sphere problem, as reported, is WRONG without mentioning the two distinct ways of close-packing. Only one is pyramidal and grows around a center and repeats every third layer; the other has an identical density, repeats every second layer, and grows into a column along an axis. Connecting the centers of the spheres of close-packing spheres yields a close-packing tetrahedra-octahedral lattice. When the octahedra only share edges, they create pyramidal close-packing, leaving tetrahedral voids, also known as A-B-C in crystallography, because the vertices repeat every third layer. When octahedra share faces stacking into columns, they leave voids of pairs of tetrahedra sharing faces, known as A-B in crystallography. I learned this from a sculptor in an art class. I was always astonished at how sloppy mathematicians are compared to artists on this issue. A professor of mathematics friend who taught the sphere of equal diameters close-packing proof to university students had yet to realize there were two ways of stacking the spheres in 3D with equal density. It turns out the proof is not affected by which of the two ways the spheres are stacked (she said).
if you were to take a box , and place spheres in a pyramid fashion , that would still leave some space at the top of the box, wouldn't it?(provided we are letting them touch the walls, ofcourse)
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.
Definitely not your "only" portal. Dig deeper.
@@rickyd3550You could share some with us if you know about other portals 😊
@@rickyd3550 ?
suggestions?
اتفق معاك يا دكتور مصطفي و ان شاء الله math club يكون بداية التفكير في الرياضيات بشكل ابداعي
paul erdos is the most consistent side character in the math universe
He's the Guinan of the math world
@@javen9693My type of guy to connect Erdos to Guinan.
Up there with Euler
I didn't expect a pirate like Walter Raleigh to have anything to do with Johannes Kepler.
He's like a traveling merchant, appearing when least expected but always there.
imagine Tao saying you made a very impressive achievement, that is peak.
Everyone should be respected, but no one idolized. - Albert Einstein
Don't put your fingers in the toaster, you'll be toast. - The toast of science
Don't put your wiener into a microwave, it will hurt
- Albert Microsoft
@@kamalapatiprajapati440 Believe everything you read on the internet - Albert Einstein
Didn't realize that was Terence Tao at first..
Please never stop posting this kind of videos for each scientific discipline at the end of the year. They're so good!
I look forward to these videos every year!
I am more excited about these than my spotify wrapped
I'd watch Quanta's breakthroughs for hours!
Same!
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. Read about more math breakthroughs from this year at Quanta Magazine: www.quantamagazine.org/the-year-in-math-20241216/
No time for the sofa problem 😂😂😂
Atleast pin your comment
Hey, are you sure, that on 2:55 it is Marina's photo? I'm quite sure that's not her...
Let's hope that we will be able to calculate the curvature of the Earth and show that we don't live on a spinning ball.
*Cool, now prove they are the biggest and show your working please*
Do your research yourself
@@black_crest he was joking
There's papers about this, quanta just tells you it exists. You're at the wrong place for formal proofs.
Wow maths look so simple
This exercise is left to the viewer
More on the Langland proof breakthrough!!!!
A longer video is coming in 2025
@@QuantaScienceChannel Nice!
fourier was phusicist no?
@@hundrethnameofalli many things, but mostly engineer and mathematician
@@QuantaScienceChannel 😊
2023's breakthrough video feels like just yesterday :(
It really does
FACTS
It really does, just like my acad...😞
Was feeling the same when I started watching!
Maybe you are standing too close to a massive black hole?! :)
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.❤
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
It's a fascinating problem in terms of how simple it is and how it developed calculus of variations and continuous optimization techniques, but it's not as important as the geometric Langlands conjecture or sphere packing or even the combinatorics of Erdos sets (maybe) in terms of how many mathematicians were researching that area and expecting solutions.
Has the proof been officially recognized ? I mean I have little doubt about its legitimacy, but perhaps Quanta wants it to gain credibility among the mathematician community, just as a precaution 🤷♂️
It will most likely be in next year’s edition tho
Has it been checked yet? Last I checked (last week or so), it was still being peer reviewed.
All of these videos are quite subjective. For instance, they could've also included the new zero density estimates for the Riemann zeta function that happened earlier this year. They also could've included the telescope conjecture (and several others) in the video last year. I think the three put in this video are fine candidates for the biggest breakthroughs of the year.
@@skylardeslypere9909 I think, that you are correct. It is still in the process of being validated. It is a 100+ page proof so it will take a bit of time.
Le vidéo que j'attends toute l'année! Merci Quanta Magazine, j'en prendrais pendant des heures!
As A high school student i hear often bout Great past mathemticians (In classes n YT videos) but i rarely hear bout any great modern Mathematicians or breakthrough which made me unconsiously assume tht Mathematics and Science had ripened and there are very little progress tht can be made and it's not worth dedicating time over it. But I do love the subject and aspire to be one great mathematician myself oneday
Which is y I desperately needed videos like this
Thanks❤
As someone who struggled with fractions, I am amazed and the mathematical talent of mathematicians.
Mathematicians are fascinated by Platonic reality. This allows them spend far more time than others thinking about it. In the process they develop a stronger than usual ability to focus and spot patterns. Most talents are developed, not innate.
One should note that this obsession with inner reality can result in, or be caused by, a dissatisfaction with practical day to day reality, especially concerning social interactions, which are generally complex beyond reason.
Congrats Marcelo wonderful work!
Great to see the contributors are invited - it really illuminates the documentary. Would love to see more such videos.
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**
They should pin this.
Doing god's work here
@@falnica This is likely AI generated :(
@@j-maffe one of the more innocuous use cases of LLM's, if so.
Higher dimensions: *random beats ordered*: Did someone solve 8-D and 24-D which are ordered, I think?
Marcelinho and Julian being in two consecutive years on the quanta recap is so iconic
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!
More to come in 2025
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.
Math is getting so large that memorizing all the results is becoming difficult for the human brain. We need a digitalization formalism for all mathematics and input discovered theorems and results in them.
That way you could just check online what has already been proven in your line of work
@@solverapprovedThat's coming with HoTT
@@solverapproved A piece of software that's the complete map of mathematics, updated in real time by AI scraping of journals and all published mathematical literature. A man can dream.
And now AI is here to make that worse.
We really need something like an "artificial brain" real quick
Best time of the year
ha ha ha
Once again great video
Thank you for simplifying complicated things.
Though still I didn't understand completely.
❤
The ideas in this video strangely complement each other very well! The Langlands program can be used to study the distribution of primes within certain arithmetic progressions, and here we have another work on the other angle of density of arithmetic progressions! These are bound (get it?) to connect very soon.
I understand maybe 5% of this, but I still stayed and actually enjoyed watching it. Amazing how smart these young mathematicians are.
Very, very interesting BUT I want more! Please consider expanding on this subject in many more videos. Thank You!
This is definitely a wonderful recap 🌞
I've been looking forward to this! Even better than I expected :))
You’ve gotten my thumbs up regardless of your findings. Many gratitudes and welcomes to you.
I love the animated visual analogy for non-numerical neural networks within the brain…brilliant!
The video I wait for all year long! I'd take a full hour, this is so amazing! Thank you Quanta Magazine!
It is an excellent way to end the year, seeing great advances in different fields of knowledge.👍🏻
15:00 -- Mathematics is infinite; once you solve these, some new paradigms appear!
This was awesome! I’m looking forward for the Physics video!
Wow! This much knowledge in such a short period of time . Just amazing ❤
Man I wait the whole fucking year every year for this video and I’m never disappointed
I absolutely love Quanta Magazine and these videos. Although for the most part I am like an ape in the Louvre, staring at the pretty stuff.
As someone who never learned or studied math I find these videos extremely interesting!
You are a polymath
Great video, thank you!
14:30 It's not often my favorite moment appears right near the end, but take my like for the algorithm and may existence treat you well.
Amazing achievements. Langlands needs a video all on its own
This is amazing for maths. Great work.
Awesome content, as always.👏
Progress on the Langlands project is truly exciting
I love you guys for these
Excited for these breakthprughs, i think ill make a hobby of collecting yourbreakthrough videos
My favorite videos of the year :)
Small observation on the Fourier Series Decomposition - the frequency of each harmonic is a transformed input parameter, the two results for each harmonic are its amplitude and phase.
Great video. The last line was funny. I don’t think anyone believes there is a shortage of mathmatical problems.
In fact, much harder, is imagining what the ideal end of inquiry will look like when math is done? [edited]
in 14 years
great article
Doctors: What can we do about this disease?
Mathematicians: How can you pack a bunch of oranges in a infinite dimensional cube?
Paradoxically, those two things are very close. Turns out diseases can be solved by designing a protein that packs correctly into its target
@@questmarq7901they weren’t ready for your response. They think doctors are more important but the math is.
A lot of abstract questions in math end up being used later in important fields such as fourier analysis
I have discovered something in Mathematics, number theory to be specific and I will be publishing my discovery in a paper soon
Could you post the link here please?
Dam today's a good Day. I have been thinking about this from a different perspective and really enjoy the I sights hoping to be able to use this for a great Tool 👍😎
"Imagine a pool of prime numbers" 😂 Fkn mathematicians! Love it!
I have solved the Collatz Conjecture and the Goldbach Conjecture.. They are simple yet nobody is willing to accept them.
As an Indian,. my favourite mathematician is Srinivasa Ramanujan...
I have found 2 new ways to arrive at 1729 - - > Ramanujan Number
1.8465
Parse :84 65
84-65 = 19
Reverse 🔀 91
19*91 = 1729
2.6748
Parse :67 48
67-48 = 19
Reverse 🔀 91
19*91 = 1729 VIOLA 😮
It seems like choosing prizes winners is quite not easy this year!
They're just purely beautiful works!
This is awesome! I have a small pasture and need to fit a LOT of spherical cows into it....
I'm in awe of what Fourier achieved in 1820. He's up there with Gauss, Euler and Newton.
Campos and Julian are awesome!
Thanks.
Thanks God I took Mathematics in my intermediate Education
I'm sure the spheres packing was a difficult problem. But
Can we have a formal definition or measure of how hard a problem can be?
Good question. My thoughts are that in order to really know how difficult a problem is you would have to understand it completely but if you understand it completely then you already probably know how to solve the problem. So the issue would be knowing what it is about a problem that you don't know and that has "known unknowns" and "unknown unknowns" the latter of which, by definition, you don't know. So how difficult a problem is comes down to the point at which you ask the question and you could define that as the level of difficulty before you know anything (an absolute difficulty) or the level of difficulty assuming all required inputs to it are required (a relative level of difficulty). So then, finally, each problem, assuming all required input "knowns" are known, comes down to how big is the space of potential solutions. This value might be computable but in reality we would not know at any point whether there were any more "unknown unknowns" which are needed as required inputs to a problem. So I think we can calculate how difficult a problem is but only after whe have solved it :D
@@christophersinger9149 I think it is completely subjective. Not only depending on one's knowledge but also intuition. If you just cant think of it, its impossible. If it somehow just strucks, it does and its easy then.
I don't think so, theres time complexity analysis but that tell you how hard is to compute a solution to the problem not the difficulty of the problem itself
Im aware of a formal concept measuring the minimal length of a proof
Very good Video🎉❤
This year was a banger wow
technology is helping advance fields. If it wasn't for the internet the three mathematicians wouldn't have found each other so easily.
The decomposition of any signal into a sum of sines and cosines is a Fourier Series and not the Fourier Transform if I recall correctly.
the answer is obvious, if metals lattices have max packing fraction is 0.74, we just need to reversely add it from different perspectives which can exceed 74% where we get states where it can be double or same depending on the dimensions
Great video,
Inspirational!
to my modest opinion the best of them all are the complex breakthroughs of canon-mathematicon!!
Oh oranges example, finally mathematicans began to solve some usefull problems.. 2 seconds later "what about bilion dimensions?"
It’s been solved in 3 dimensions for a while. The high dimensional problem is useful for a bunch of data analysis problems.
High dimensional math shows up alot in data analysis. Neural networks rely on a bunch of properties in high dimensions (like spheres being pointier)
Dude, SO much math has practical educations that're found out later
Like "sedenions", which are "16-dimensional noncommutative and nonassociative algebra over the real numbers", are used in traffic and weather forecasting
16-dimensional, with real-world applications.
Where's the anti-academic sentiment coming from?
Yeah, applying such an anti-intellectual sentiment towards mathematics of all things is absurd.
this is pure mathematics, they don't care about applications. it's so abstract. You'll need to do graduate work then another x amount of years to understand the basic outline of a proof. No mere mortal will full be able to understand it. They are not obligated to do anything that they don't enjoy. If you want practical applications you might want to look at other areas of Maths or other sciences.
saying that only some math is useful because you’re too stupid to not know a use for everything else is total bs
Whats the reason to study more than 3 dimension?
Hentai
At 7:34: Seems to me that 8,11,14,17,20 is a 5-term a.p. Isn't the largest set w/o a 5-term progression the set of 1,2,3,5,8,10,11,12,14,17,20?
... but the latter one contains a 7-term a.r.: 2,5,8,11,14,17,20.
The correct largest subset of [20] avoiding a 5-term ar. progression is 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19. It does have 16 members. For more information check Sloane's OEIS.
very interesting, unfortunately I didn understand any of them, especially not the last one 😭
Lol too real
Langlands program is like dark magic... you need so much background to even start understanding it
In the second discovery, the szemerdi problem, the video says for a 5 term progression not to appear in a pool of 20 numbers, the highest density possible is 80 percent i.e. 16 number. But with in 16 numbers in the set, there is one 5 terms progression of 3.
Or may they were talking about 5 terms' progression of one particular number i.e. 4 in the case
13:33 THE ONE PIECE IS REAAAAAAAALL
I saw those two at a math conference like 5 years ago
I hardly understand these problems, however it's cool to see now in Maths it's too complicated that most of the time you need cooperation to solve problems.
I wonder how advanced math will be in the 22nd century
Chapters would be very useful.
5:05 Now we can stack higher dimension cannon balls, cool!
It's crazy that I can use Fourie's transform for physics/engineering without ever understanding what it represented until now. Just goes to show how powerful these tools we have are! Similar to how you don't need to be a mechanic to drive a car, I don't need to be a mathematician to be able to work with frequency. Although, I often feel like a monkey with a screwdriver 😅
The first problem is by far not a stupid question. My mom once asked if there was any software that could pack a bunch of 2D figures into a fabric plane, since she needed to cut down some pieces of a fabric and wanted to waste the least amount of it. Now think about how much material and waste could be spared with a software programmed with such solution.
"Geometry is confusing" ~ That's the only thing that I understood
5:25 Randomness has to do with how the numbers are generated. It has nothing to do with what they are, or with what patterns (if any) there are in them.
could you please provide the names of the music pieces used in the video? they sound great
what are the implications of having solved the langlands program?
(genuinely curious - i have no idea about any of this)
It always amazes me how the brightest minds spend their time and energy at the most useless tasks...
I wonder if there is a connection between this fact and the state we find the world is at...
Why are the breakthroughs always in combinatorics, number theory, discrete math etc? I wish you would cover fields such as geometric analysis, probability etc
I always enjoy these videos. Well done! Thank you Quanta.
A correction:
>"Any wave can be broken down into an infinite sum of sine waves"
This is fundamentally wrong.
Some waves contain more information than others. Almost all waves, even at infinite resolution, can be expressed in just a few bits. The more complex the wave, the more bits it requires.
There is a limit to the information physical waves have since physical systems are fundamentally discrete (Planck's constant, quantum mechanics). There's only infinite information in some abstract, theoretical waves. For example, a wave that somehow encoded pi π would contain infinite information, although from the top of my head I couldn't detail a method to do this. Real waves, and all mathematically described waves that I can think of, have limited resolution.
Just because a wave is continuous does not mean it has infinite composite waves. A pure sine wave is an example.
The fact that Fourier decomposition is broken down into component waves also shows this. Each component wave does not have infinite information, otherwise it would be useless as a component. The whole utility of Fourier decomposition is that it breaks down complex waves into simpler, well-defined components with finite information content.
A wave with finite information content cannot be made up of an infinite number of independent components, as that would imply infinite information content.
Also, don't forget the Nyquist sampling theorem, which also says that for a given bandwidth, *all information* may be described using limited sampling/limited information. There is a strict limit to the information potentially in a given band.
As someone who works in information theory, I notice that mathematicians ignore information theory, and its consequences, all the time. It makes me suspicious that there's low understanding and application of information theory in wider mathematics.
It also explains why I experience resistance with some mathematicians when discussing the fundamentality of information theory, like the Church-Turing Thesis. Mathematicians don't typically think in terms of information even though it is fundamental to their work. I think there's a general attitude that since they work in abstract concepts, information theory is somehow an implementation from abstract to application and thus can be ignored. I suspect there's a lot of low hanging fruit in mathematics that could progress the science if information theory was taken seriously and applied.
I’m going to show no four numbers in a row in the Collatz conjecture combine before merging at 4,2,1 when I get more free time.
I swear every time I see videos with "the latest math research" type content, it always includes some "packing" problem. Why is this such a popular thing? I'm assuming it's a problem in industry or something.
E: Makes me wonder who are the "brokers" between mathematicians and those who identify a problem in industry that can be solved with math.
It holds importance in Information theory and error-correcting codes, as they said in the video
Compression of data is essentially a packing problem
The topics you see in math journalism aren’t very representative of what mathematicians care about. Sphere packing is one of those things that’s really easy to explain to a lay audience, so it gets covered a lot, same with tilings of the plane.
I would say that there are probably 10 topics or more that you could put in a video like this based on their significance in the subfield, it’s just that 90% of math research is incomprehensible to those without specialized training.
Not only industry, but as mentioned in the video packing can be quite fundamental in condensed matter physics and chemistry.
The E8-lattice (optimal unit ball packing in 8D) for example was recently used in a paper called QuIP# about a new LLM quantization algorithm, which are used for compressing a large model, so that it fits on smaller GPUs
The process is straightforward: thoroughly squash the oranges, then place them into any container of your choosing. Avoid overcomplicating a task that is inherently simple.
Amazing.
The close-packing sphere problem, as reported, is WRONG without mentioning the two distinct ways of close-packing.
Only one is pyramidal and grows around a center and repeats every third layer; the other has an identical density, repeats every second layer, and grows into a column along an axis. Connecting the centers of the spheres of close-packing spheres yields a close-packing tetrahedra-octahedral lattice.
When the octahedra only share edges, they create pyramidal close-packing, leaving tetrahedral voids, also known as A-B-C in crystallography, because the vertices repeat every third layer. When octahedra share faces stacking into columns, they leave voids of pairs of tetrahedra sharing faces, known as A-B in crystallography.
I learned this from a sculptor in an art class. I was always astonished at how sloppy mathematicians are compared to artists on this issue. A professor of mathematics friend who taught the sphere of equal diameters close-packing proof to university students had yet to realize there were two ways of stacking the spheres in 3D with equal density. It turns out the proof is not affected by which of the two ways the spheres are stacked (she said).
Guys 2025 might be the only perfect square we ever experience
Thought you would have also the proof of Busy Beaver number 5.
That is coming in 2025
Closest i may ever come to sheaves understanding
Is there a connection between Langlands and Category Theory?
I approve of the use of A Jillion to represent an arbitrarily large number.
Wonderful video. See you all in one year.
if you were to take a box , and place spheres in a pyramid fashion , that would still leave some space at the top of the box, wouldn't it?(provided we are letting them touch the walls, ofcourse)