2023's Biggest Breakthroughs in Math
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- เผยแพร่เมื่อ 21 ธ.ค. 2023
- Quanta Magazine’s mathematics coverage in 2023 included landmark results in Ramsey theory and a remarkably simple aperiodic tile capped a year of mathematical delight and discovery.
Read about more math breakthroughs from this year at Quanta Magazine: www.quantamagazine.org/the-bi...
00:05 Ramsey Numbers
One of the biggest mathematical discoveries of the past year was in graph theory where the proof of a new, tighter upper bound to Ramsey numbers. These numbers measure the size that graphs must reach before inevitably containing structures called cliques. The discovery, announced in March, was the first advance of its type since 1935.
- Original story with links to research papers can be found here: www.quantamagazine.org/after-...
06:21 Aperiodic Monotile
The most attention-getting result of the year was the discovery of a new kind of tile that covers the plane but only in a pattern that never repeats. A two-tile combination that does this has been known since the 1970s, but the single tile, discovered by a hobbyist named David Smith and announced in March, has been a sensation.
CORRECTION: In the video, the image presented as the 'turtle' tile is in fact a rotated 'spectre' tile. To see the correct version of the turtle tile, you can visit Dave Smith's webpage: hedraweb.wordpress.com/2023/0...
- Original story with links to research papers can be found here: www.quantamagazine.org/hobbyi...
- Build your own aperiodic tiling patterns with Kaplan's online tool: cs.uwaterloo.ca/~csk/hat/h7h8...
14:20 Three Arithmetic Progressions
Two computer scientists, Zander Kelley and Raghu Meka, stunned mathematicians with news of an out-of-left-field breakthrough on an old combinatorics question: How many integers can you throw into a bucket while making sure that no three of them form an evenly spaced progression? Kelley and Meka smashed a long-standing upper bound on the number of integers smaller than some cap N that could be put in the bucket without creating such a pattern.
- Original story with links to research papers can be found here: www.quantamagazine.org/surpri...
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Crazy that a tiling enthusiast just found the right Einstein tiles.
Einstein was the biggest scientific fraud in history.
shows the neglect of so called institutions on real research in modern society
would have been crazier if a tiling hater found them
@@Wakssbm You sure about that? You are completely ignoring the intuition of the one making the discovery, he's not randomly stumbling around. Of course, the discovery only matters with the proof, and the knowledge gained by it (ie that infinite shapes work). Institutions frequently pay people to mess around trying to solve problems, they use their intuition to generate ideas worth further pursuit. In this case that initial idea was created by an outsider, who might not have the expertise to pursue it further, but he found the right people for the job.
@@helloicanseeu2 it's amazing how you are able to ignore all of the discoveries that are made every fucking year by researchers in institutions in order to preserve your idea.
This year, one solution to a famous problem was achieved by an outsider, great.
What about the other two big discoveries that the video mentioned? What about all of the minor progresses in math that were not so important but gave their contribution? Even more, what about all of the major discoveries in chemistry, physics, engineering, biology, that were made from people working in institutions?
I'm just so curious to hear from you why this single (and fantastic) example of success is able to overshadow the rest of the world
Smith’s story is an inspiration to all of us math hobbyists outside of a formal academic environment!
And a reminder for academics to engage more often with math and science enthusiasts.
@@rarelycomments I agree. It annoys me when I always read on Twitter/X: "Postdoc/PhD candidate offer for ..." whereas many "hobby researchers", who only hold a master's and are interested in research but choose not to do a PhD (which probably is the right decision) could do just as good or even better, bringing some nonacademic experience to the table. I saw some researches in maths (operations research), where the procedure could have been speeded up much faster let alone they had hired a compsci student who is good in maths to make calculations, visualizations etc. as they were trying to "predict" and guess whether a certain property holds for some large values, instead of hiring a CS student to write code for them showing results from simulations that their idea does not hold for big values.
@@rarelycommentsyou'd be surprised to hear about how many bullshit emails academic people receive. The above is a very singular case that will never be the norm.
@@lolmanthecat I know, I get them.
It's easy to say all non-academics are cranks and conspiracy theorists, but the reality is that there are plenty of very smart people doing very cool stuff in their free time.
Unfortunately those people tend not to jump and shout about their work, as they question themselves and their work. Unlike the crazy people.
@@lolmanthecat i have some innovative new insights to the Collatz Conjecture...
The second tile that I found (10-kite) was correctly named the turtle but was shown in the video as the spectre (at least one other person noticed this). Really, just thrilled the story got covered, thank you all.
Brilliant goalkeeping! Oh, and well done on the tiling thing too!
Are you David Smith?
thank you for your hard work David!!
Really cool work! So inspiring to realize there are still problems out there that don’t require formal math training to tackle 🙂
@@robertroach9157 Yes. Whilst watching the video I'm thinking, do I really sound and look like that.
Smith did not have luck. That is genius. He did it three times.
I think the first step was brute force and passion. Then, though he had a head start, he kept playing around with what he had found, till he made more discoveries.
@@jeremylakeman the spirit of a mathematician
I believe I made my own luck by giving myself a chance (or perhaps I was the chosen one?). I wouldn't have got far with it though without the help of Craig, Chaim and Joseph (they did all the hard work). Thank you for your comment.
@@JellyMonster1 If this account really relates to you from the video, i just want to say beautiful job there. Of course the mathematicians did the hard work of proving that it is infact aperiodic, but i guess without rigorous mathematic education this really would not be possible. But i have the feeling that the more you follow the academic way of mathematics, the more you tend to lose the creativity and spirit of just playing around with ideas without the heavy theory-machinery... and hence i think it's so great to see a person who did exactly that. Would really not say it was the 'easy' part (especially finding three such patterns), both parts were hard, but the nature of these parts is entirely different.
for real, the mathematician didn't accept Smith was the real genius
Smith is a god damn beast putting out aperiodic monotiles one after another.
No he isn't
@@talananiyiyaya8912why?
@talananiyiyaya8912 you just jealous
What have you contributed to the field of maths? @@talananiyiyaya8912
@@talananiyiyaya8912you don't belong here
It is incredible that Paul Erdos had a hand in all these initial discoveries. What an incredible Mathematician and human.
I notice that also! Amazing man.
well yeah, theyre all combi problems
That dude is everywhere
Erdos was involved with aperiodic tiling? He isn't mentioned in Grunbaum & Shepherd's Tilings and Patterns.
Love the story of Dave smith. He’s three totally independent and super fast reacted discoveries are definitely NOT good luck, but a really deep insight about symmetry and patterns, which is built throughout his life being a puzzle enthusiast. Professional mathematicians may have good skills proving and generalizing stuff, but he deserved a recognition of creative originality. That tells us that mathematics can be down in not only one way.
I love how the huge improvement in the Ramsey bound is from 4 to 3.997.
I know this is huge, but I still had to laugh when reading it.
MY CHIHUAHUA IS NAMED RAMSEY && THAT MAKES IT SO MUCH FUNNIER 2 ME💯😭😭😭
@@waff6ixnot funny
@@chicagobricks1008 humour is subjective
@@galactoman5503 but cringe is universal
@@chicagobricks1008 IF U KNEW RAMSEY IRL IT WOULD MAKE U LAUGH 2💯🤣
"Tiling enthusiast". I absolutely love that description.
Yes, on hearing that, the sort of person who springs to mind is someone who likes sticking tiles on bathroom walls.
Love these reviews. I genuinely get more excited about Quanta's annual reviews than I ever was for the Nobel. Fantastic to see the bleeding edge of humanity's advancements.
you probably recognise Nobel prize as being corrupted or subverted. recently the physics prize was awarded to a Japanese man for his model, well he should have got the prize relating to his model rather than the prize relating to physics. it is a total farce and it makes me deeply sad. when I was a student 50 years ago the quality of the sciences, mathematics was so much higher. we were taught the trivium which is logic, grammar, and rhetoric, and quite simply a Nobel prize in physics awarded for a computer model would not be accepted by even a single student.
@@ayyleeuz4892 I assume you mean Dr. Syukuro Manabe? I don't know what you mean by "should have got the prize relating to his model rather than the prize relating to physics". Isn't atmospheric physics also physics? Why isn't inventing a reliable physical model of the climate physics?
Why does the appraisal of one thing have to necessitate a bashing of another? Can’t two things coexist?
@ayyleeuz4892 Just because you can't keep up with modern research doesn't mean it has gone down in quality. Modelling natural phenomena is literally the point of physics. What are you even talking about.
The distinction is that Nobel prizes are given for research that has been proven to be important for the field, and as a result, is usuaully decades old from the initial discovery and is more akin to a life-time achievement award. If you want cutting edge science, look for things like the new horizons breakthrough prize in physics (given to 'junior' researchers), etc.
Dave is literally a genius in his own geeky passion for tiles and that’s amazing ❤
I like the term 'in his own geeky passion' and yes, it is amazing. Thank you so much, DS.
Props to David Smith for making several discoveries . I can only imagine the thousands of hours he put into his tile hobby, and how he found something a mathematician , or a computer scientist couldn’t find . Genius !
The tile section was absolutely nuts lol
THE ONE PIECE IS REAL
@@IN-pr3lwLMFAO
@@IN-pr3lw mom said it's my turn to use the neuron
Props to David Smith. This shows how members of the public, even those who don't have professional scientific training, can still contribute to knowledge if they have the will. We can all learn something from him about where to focus our attentions in life, towards things that move us forwards as a species, even a little bit, and away from the vapid materialism that we're told will fully satisfy us.
Thanks for highlighting the contributions of an amateur mathematician. There are many and they can have important ideas too. Researching something outside the regular systems doesn't make you a crank.
Pleased to discover I'm not a crank :)
I like that there are so many computer scientists involved in these math breakthroughs :D
At the top, they're both the same 😅
It’s because combinatorics is easier to explain and often uses computers
@@leeroyjenkins0 Well, they are computer scientists, not pure mathematicians lol
@@leeroyjenkins0combinatorics isn’t any easier than other areas of maths, it just tends to be easier to explain it to non experts.
Pure maths has a lot of prerequisite knowledge, even after finishing an undergrad you’ll still not be able to understand most of what’s happening at a research level
@@DadicekCz theoretical computer science is effectively a branch of mathematics. There's a reason why the P vs NP problem is one of the Millennium Prize Problems.
I heard about the Einstein tile discovery a while back and thought "wow what a lucky guy". Now I understand it's not luck, he understands it in some way clearly to have done it three times.
Smith is such an inspiration. It's incredible that in a world with over 8 billion people and advanced technology, there is still room for passionate amateurs to make their mark on the academic world.
Julian Sahasrabudhe lectured my cohort this year for linear algebra! He's a very fun lecturer, didn't realise he was also doing such important research, but I'm glad for it because it probably means his job is nice and safe and he gets to lecture more courses haha
wow that is so cool. Now I'm beginning to wonder who (of my current and past lecturers) have done really cool stuff
I was just thinking: I’d rarely see programes like this in the old days of tv only. It’s such a gift to be able to watch this!
The study on the Ramsey number actually perfectly fits with the inquiries scientists had about the human microbiome in the "Neuroscience and Biology discoveries of 2023". It can help us better interpret the relationships between the thousands of microbes, and give meaning to certain combinations of these microorganisms
2 of the 3 were studied by Erdös. Speaks how prolific the guy was.
Yes, I also noticed Paul Erdos's repeated appearance .........wow!
Thats how it is for a lot of academic people back in the day.
I know we should respect the dude, but all i can remember about him is that my higschool math teacher referred to him as pal. As they were school mates
this video is biased toward combinatorics, which happens to be where Erdos specialized
i had no idea who that was before this video
I literally wait and look forward to these 4 year in review videos put out by quanta each year! Amazing quality and so fascinating!
Probably one of the best math news videos I've ever seen. Congrats guys! and thank you to the researchers for such amazing discoveries!
The more math we reveal to ourselves, the more advancements and scientific discoveries we'll find. So exciting
Do you know some recent examples of maths expansion leading to technological breakthrough?
@@ayy2193 google is 2 steps away
@@ayy2193 many people think that advancement in comouter hardware is what helped computers become much faster in the last decade, but actually it is mostly advancements in combinatorical algorithms
Math is the abstract thing that is used to describe already discovered things. It's not the other way around.
@@hr1100 most of quantum physics was firstly worked out through the mathematics and later verified, same with general relativity
Congratulations to David Smith, as an hobbyist myself this gives amazing inspiration to continue ones pursuit of knowledge.
smith is just amazing. bro found like so many of them in a row when people were struggling for years. and when they said his tile wasn't good enough, he went on an found ANOTHER ONE, which turned out to be an actual einstein tile
goddamn
What an incredible year. Looking forward to the next . 🎉🎉🎉
The only thing my brain can grasp here is that Mathematicians have great hair
The guy with the long, black, messy hair looks like he spends about 18 hours a day in front of a computer drinking Monster energy drinks.
No way scientists found the One Piece 😂
It might be the best TH-cam-video-idea to make the biggest breakthroughs in science in every year. Keep going
Bless you David, what an inspiration you are to amateur maths enthusiasts you are that an ordinary man can still make a new and wonderful discovery.
Thank you.
8:36
Knowledge. Fame. Scientific papers. The man who had studied everything in this world... Sir Roger Penrose. The words he said at his death spurred many to do research.
Roger Penrose: The one aperiodic tile? It exists! Go out there and find it!
Words he spoke drove countless men out to the field. And so men set sights on the Einstein Tile, in pursuit of their dreams of an aperiodic plane. The world has truly entered a Great Math Era!
The whole year has been bad news (wars, climate etc.) so I realise how much I love those annual sum up on sciences ! Ones of the few branches of human activities where humanity still progress toward a greater good. Thanks !
"climate" 😂😂😂
This content is great. So approachable but still pretty challenging.
"Tiling enthusiast". I absolutely love that description.. The tile section was absolutely nuts lol.
These people are so passionate about math, i love them. If we all had a passion for something positive like this, world would be a better place.
I love these videos from Quanta, they're great!
This series is so good. Thank you Quanta
Thanks for putting these together!
Read about even more 2023 breakthroughts in math at Quanta Magazine: www.quantamagazine.org/the-biggest-discoveries-in-math-in-2023-20231222/
CORRECTION: In the video, the image presented as the 'turtle' tile is in fact a rotated 'spectre' tile. To see the correct version of the turtle tile, you can visit Dave Smith's webpage: hedraweb.wordpress.com/2023/03/23/its-a-shape-jim-but-not-as-we-know-it/
Dave was on a roll❤🎉. Nice contribution
these are so interesting thank you!
All knowledge is equal. Your understanding of tiles may comes through years of academic training or years of puzzling but in the end, both are equally important. Imagine the scientific breakthroughs we could achieve if we dared to look outside of the academic box.
The second einstein Dave Smith discovered was, as you correctly say, the one that got called "turtle". However, the turtle is another shape made out of kites; 8 make a hat, 10 make a turtle. what you show at 11:58 is the "spectre" which you also show at 13:46.
Yeah, well spotted, I noticed that too, aka Dave S.
I don't even understand mathematics that much, but I just enjoy listening to them talk about it. ❤
Mathematics is the systematic application of quantity.
Quantity = amount = number.
A quantity is an abstract way of saying “something that is separate from something else.”
Example: two apples.
There are two *separate* objects in 3D space.
If you could magically fuse two apples together they would make one apple.
1.0 + 1.0 = 2.0
In the same way that…
0.5 + 0.5 = 1.0
🙂
Excellent video. The context for each highlighted breakthrough was really well done. Chapeu
While not featured in the video, only mentioned, I want to give a shoutout to Olof Sisask. He was the lecturer/teacher when I studied Combinatorics at Stockholm University. Without a doubt the best math teacher/lecturer I have ever had.
He was friendly, pedagogic, never made you feel stupid for asking a question and was really invested that everyone understood the subject matter that he was trying to teach us.
Just a great person.
By looking at paths of consecutive numbers in the collatz conjecture I found what I’ve called collatz triangles where vertical 1x2 rectangles have a number in the top half and it plus one in the bottom half and the rectangles are staggered so the bottom of a rectangle becomes the top of a rectangle to the right after applying the collatz function. Going to the left I multiply by two. The top rectangle’s top half is 2 less than a prime number and is odd so what works there is -1, 3, 5, 9 etc. The down left direction collection of rectangles, the left top side of the triangle has a pattern so the top of each of those are given by this
(2^k *t)+(2^(k+1) -2), where t is the top number in the triangle and k is the row number with k=0 being the top row of the triangle, or the row containing t. The right side of the triangle also has a pattern where the tops are previous top on upper row *3 +4. I used this to generate a formula of sorts with (t+2)(2^(n-1)-2) becomes 9^(upperbound((n-1)/2))*(9t +4.75)-0.5, where n is greater than or equal to 3. The triangle proves visually that up down up down, the fastest growth has to come down a bit before it could continue so it can’t go up infinitely with fastest growth.
If someone could find a way to know which triangle any number is in and its position they could cut down a bunch of steps like how dividing by 2^k skips a bunch and could speed the search and maybe give insight on which numbers, like 27, grow quicker and longer than others. My formula took into consideration a few more steps than just the triangle because every other row combines in a node, what I call the number after an odd, except the rows 0 and 1. It would be cool to see a computer applying this speed boost to faster check numbers going to a lower number, where it has been checked that it’s gone to 1.
11:36 That's not the Turtle; the Turtle is a second polykite on the continuum. You're showing the Spectre tilted (and reflected).
Dave here, yes I noticed that too, shame.
thank you so much for your science, hard work and generosity
Just want to give some props to all the people involved in making this video. Amazing job
At 10:20 there's an illustration of the pattern but there's a tile placed in a way that it breaks the pattern, you can see on the right side the monotile won't fit into the created gap.
That's correct but that image (which either Craig or myself coloured in) was a brute force computational result using Craig's Heesch number software. The more computer time given to it, the larger the pattern became.
8:38 ONE PIECE MENTIONED!!!
loved this video, so insightful & inspiring! love math
Wonderful video, awesome visualizations. Thank you so much.
It's bizarre that this kind of breakthroughs never reach national television
I was ready to be cynical, but the story of David Smith is inspiring!
Well done Dave! THE tile man.
Very smooth explanation. Thank you!
I just realized that the whole monotile thing has another joke: THE ONE PIECE IS REAL
commenting for engagement. these videos are important
Thank you for share!
Great Video! Are you doing one on Breakthroughs in Chemistry as well?
I believe that Smith's discoveries are not purely probabilistic luck. I like to think that his mind developed an affinity and understanding of tile patterns, such that coming up with a tile with a specific property became second nature for him-a computation that his mind performs implicitly.
I believe they say that pattern recognition is at the core of mathematics. Additionally, intuition is crucial for scientific thinking.
This also makes me think that perhaps mathematics is deeply ingrained in the fabric of our minds, and one might accidentally develop mathematical thinking. I am so happy for Smith; his hard work paid off. It did not seem like hard work because he enjoys what he is doing. His mind must certainly have something special about it.
8:38 Just heard we're gonne find the One Piece. I'm in.
Great job Dave!
Thank you so much for caring about these aspects of this universe
David Smith, the man.
The story about the aperiodic monotile with and without reflection is amazing! Imagine how AI will be applied in the search and well as bringing in more hobbyists!
Still remembered when seeing the original paper on the hat tile was out, the original paper was 89 pages long and I was an undergrad math student browsing in a Facebook meme group. I had no idea what that is back then but watching this happening in real time was just surreal.
Loved the tiling discovery. Curiosity at work. And, he did discover something.
I don’t think Smith just “stumbled” on the solution. You don’t just “stumble” on the right answer over, over, and over again. Smith is a tile genius, and shows that people in academia are not the exclusive source of discoveries. By working together, we can combine genius/innovation with accredited validations with scientific standards/peer review making the best out of all contributions regardless of educational background ❤
It is kind of sad that the most notable achievements in math are in graph theory, tiling, and arithmetics.
While I do understand that probably only people in number theory can comment on how important it may be, it still feels underwhelming.
We have results in weak formulation of mean curvature flow.
We have people pushing forward solutions to conservation laws with unbounded variations.
Maybe I am missing some dates? Or maybe something else, but it does feel underwhelming to see only this.
When i saw the title of the video I was also expecting results in geometric analysis or pde, algebraic geometry etc. I think that the reason they only showed these breakthroughs is because the problems are easy to explain to a general audience, the problems in other areas of math require more background to understand them, so they chose not to mention them.
I think what you have is called an opinion.
Absolutely love this channel, narrators got a great voice
i wait for these videos like its my own personal christmas gift, especially the ones on math
So glad they discovered the One Piece is real!
Dave is the pure definition of genius. Like other great minds, he can feel and put in practice what no one has been able to glimpse or deduce.
What a wonderful production quality in a video!
6:25 - my FAVORITE part. One of my favorite discoveries of mankind EVER tbh. aperdiodic monotile let's GOOOOOOOOOOOO.
The "Turtle" tile in the video is a wrong one. it's the mirror of Spectre.
Oh. I was like "Did they rename Spectre into Turtle? How interesting!" 😅
Were these really the biggest breakthroughs? Or just the most easily explainable?
Biggest breakthroughs in easily explainable problems :)
Though I still cannot fathom what the solution is, I’m excited and happy to see the advancement in this, and how enthusiastic and diverse people are working on this! Thanks for putting together this video
can't believe how well made this video is.
All of them are combinatorial problems. Interesting
Maybe, Quanta is just biased towards such problems? 🤔 It's definitely easier to explain in a short video than something like modular forms.
@@solderbuff Yeah thats true . This stuff is much more relatable to normal people than other stuff
@@solderbuffI think it might be something like this. I think a common comment (that I have also had) on some of quanta's work is that they (understandably) will sensationalize things. As you mentioned it's a bit hard to get people interested in more abstract and subtle things. Even though it's not reviewed fully and isn't confirmed, I would have liked a mention of Per Enflo's possible solution to the invariant subspace problem. As a math student it's even hard to hear about big things in less accessible fields (I only heard about Enflo's recent work because a professor mentioned it in functional analysis), so I imagine it's even harder for them to get into it for the general public. Although imo this definitely doesn't mean they shouldn't try.
I was thinking the first one ought to be doable by a computer by brute force but then it dawned on me that the total number of graphs with 6 vertices is like 1.3x10^12. With v vertices there are ((v^2-v)/2)! graphs. At 18 (Ramsey number for 4) it's 2x10^269. So, yeah. That's gonna take a while.
You speak strange words magic man
@@pictzone Not magic, just curious.
Erdős had a quote that went "Suppose aliens invade the earth and threaten to obliterate it in a year's time unless human beings can find the Ramsey number for red five and blue five. We could marshal the world's best minds and fastest computers, and within a year we could probably calculate the value. If the aliens demanded the Ramsey number for red six and blue six, however, we would have no choice but to launch a preemptive attack." It really helps you learn about the scale of these things. Even the largest lower bound we have for R(5,5) has far more than atoms in the universe number of possible configurations to check lolol
I don't follow your logic. Your formula makes little sense to me honestly. The number of undirected graphs with v vertices should be 2^(v(v-1)/2). If you consider directed graphs too, it will become 3^(v(v-1)/2). How did you come up with that formula?
@@jaimelannister141 The number of possible edges is (v^2-v)/2 or v(v-1)/2. Now I realize I've over-counted looking back, because I took all possible combinations of these edges as the factorial of that value.
Unfortunately this means my numbers are too large.For v=3 my expression gives 6 but there are only 3 unique graphs after reflection/rotation. When v=4 there are 720 graphs including repetition, but only 15 unique.
So the correct expression is 2^((v^2-v)/2) - 1, and the number of unique graphs you'd have to brute-force check when v=18 is actually 10^46 and a smidge.
Great content, thx for explaining why these discoveries are relevant t
These have to be the best yearly recaps
Julian Sahasrabuddhe - the surname is an Indian (Marathi) surname literally meaning "one having thousand (sahastra) brains (buddhe)"
Apt for a mathematician 😅
So basically, the one piece is real
These videos are awesome!
Fascinating, makes me want to drop everything I'm doing and go discover something new.
The one piece is real!!!
I'm so stupid, I understood nothing in this video.
Great vid. Thanks.
Fantastic!🎉🎉🎉🙏♾👏🏻
guys the one piece is real
Let's go find it
Just when I'm about to conclude that math is boring and very hard, it's videos like these, who know how to teach properly, get me back to saying "math is interesting"
@@Tommy_007 wdym. I never told I am expecting others to teach everything. I had this random urge to study statistics and probability man. I confirmed nothing in my comments.
It's thrilling to see brilliant and impassioned people discuss their work on their chosen area of focus.
Great video! Well done!