The Biggest Project in Modern Mathematics
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- เผยแพร่เมื่อ 16 พ.ค. 2024
- In a 1967 letter to the number theorist André Weil, a 30-year-old mathematician named Robert Langlands outlined striking conjectures that predicted a correspondence between two objects from completely different fields of math. The Langlands program was born. Today, it's one of the most ambitious mathematical feats ever attempted. Its symmetries imply deep, powerful and beautiful connections between the most important branches of mathematics. Many mathematicians agree that it has the potential to solve some of math's most intractable problems, in time, becoming a kind of “grand unified theory of mathematics," as the mathematician Edward Frenkel has described it. In a new video explainer, Rutgers University mathematician Alex Kontorovich takes us on a journey through the continents of mathematics to learn about the awe-inspiring symmetries at the heart of the Langlands program, including how Andrew Wiles solved Fermat's Last Theorem.
Read more at Quanta Magazine: www.quantamagazine.org/what-i...
00:00 A map of the mathematical world
00:25 The land of Number Theory"
00:39 The continent of Harmonic Analysis
01:20 A bridge: the Langlands Program
01:46 Robert Langlands' conjectures link the two worlds
02:40 Ramanujan Discriminant Function
03:00 Modular Forms
04:36 Pierre Deligne's proof of Ramanujan's conjecture
04:47 Functoriality
05:03 Pierre De Fermat's Last Theorem
06:13 Andrew Wiles builds a bridge
06:30 Elliptic curves
07:07 Modular arithmetic
08:56 Infinite power series
09:20 Taniyama - Shimura - Weil conjecture
10:40 Frey's counterexample to Frey's last theorem
11:30 Wiles' proof of Fermat's Last Theorem
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Quanta Magazine is an editorially independent publication supported by the Simons Foundation www.simonsfoundation.org/ - วิทยาศาสตร์และเทคโนโลยี
Give a raise to whoever the artist of this video is. They have done such a good job at creating visual support to make it easier to understand. Amazing job!
Seconded.
Thirded!!
fourthed
Fifthed!
seventhed
I didn't know mathematicians had their own program of a grand unification just like physicists do.
Thank you for the video!
It's a pretty significant overstatement to say that the Langlands program is a theory of grand unification. But! It does make a good story :D, and the use of "bridge building" as a method of problem solving is fundamental to many areas of modern mathematics, at least at this moment.
@@Cris-kt9df exactly my words. building bridges is something mathematics and practically every other stream of science achieves to do, and it all falls under that one umbrella of the grand unified theory of everything
Mathematicians don't have their own grand unification like physicists (If we exclude axiomatic systems in mathematical logic, such as ZFC, which is a well-established basis of unification for mathematics). The whole "Langlands is grand unification for mathematicians" is just rhetoric used by science popularizers because the public is somehow familiar with the struggles of particle physics.
The video says that ''Langlands program may reveal the deepest symmetries between many different continents, a kind of grand unified theory of the mathematical world...''. I didn't mean that Langlands program itself was a grand unification theory but that the idea of grand unification exists within Mathematics itself just like it does in physics. The reason why this was surprising to me is cause in physics, for example, a grand unification sprung from quantitization of General Relativity does not seem possible so scientists come up with new theories and modifications to be able to achieve that whereas Langlands, as far as I understand, is motivated to reveal something we don't know about different fields of Mathematics; are there any further connections between them, if so what are these connections?
Whereas physicists are motivated to come up a theory that describes the right symmetries of nature in high energies and large scales, mathematicians in this context would be motivated to uncover all the bridges between different fields. Thus, a grand unified mathematics would be the one where all different ''continents'' are connected.
@@Casreor I think my comment was more directed at the language in the video--which in the end was a very very nice piece of media. If you found it interesting and thought provoking, then that's fantastic :D. I don't mean to rain on anyone's parade.
Now, I will take a risk and mention some kinds of physics which I don't understand. As far as I can tell, the Langlands program seems more akin to, say, AdS/CFT correspondences, or mirror symmetry. Or, more directly, there's a paper by Kapustin and Witten which frames (a version of) Langlands duality as an "electro-magnetic duality". So it seemed to me that these kinds of comparisons are more appropriate, rather than to grand unification. But that's really in the weeds. Have a good day!
Being depicted as an engineer must be a mathematician's worst nightmare
Only if they a childish enough to encourage such gatekeeping
😅 So true
@@sepg5084 Well when you look at the fact that engineers do a lot of rounding and mathematicians love precise numbers you can see why mathematicians wouldn't want to be depicted as engineers
I am a coward. I wasted my life.
As a Mathematician I can confirm this.
The visual of Ramanujan writing in a slate is an authentic touch!
Context: Ramanujan was born to a poor Indian family and did not have money to purchase papers(which was expensive at that time) and he always worked on slates.
Writing on slates is more satisfying than slamming your hand on keyboard.
@@vinitrout3679 True. Solving on paper(not PC) is so much better
Do mathematics handwritten, publish paper on computer = peace
Alex Kontorovich is such a great narrator for any math related videos, its genuinely SO fun to watch!
YES I LOVE HIS VOICE!!!
I just realised now it is Kontoroviches voice :o
awesome
is it related to... that Kontorovich?
@@6884 by "that" if you mean Alex Kontorovich, then yes
and the Langlands program is not directly related to Alex, he just narrates math related topics like these in a comprehensive and easy to digest way
This was a wonderful explanation and video. I also love that we’re still puzzling things Ramanujan and Fermat thought about hundred(s) of years ago.
Last time I was this early to a verified reply.
didnt expect to see you here
@@bagochips1208 He's more open minded than a neurosurgery patient
@@fragileomniscience7647 Hahaha
They also didn't have smartphones and technology to distract them. A lot of those kinds of thoughts happen when the mind is quiet.
Alex Kontorovich (guy who voices this video) was my calculus professor in college. Very talented man and incredible teacher.
fellow rutgers student! Regrettably i never got to take number theory with him
@@bencardwell5545 Yeah, he was great, I wish I was able to take one of his other courses as well
Yo fr tho he was the best teacher.
I would watch an infinite playlist of this content. As an amateur math enthusiast with a somewhat undergrad level of understanding, this stuff is fascinating and beautiful.
Stuff like this makes me want to pursue a degree in Mathematics, however I don’t trust our school system to teach it properly. It’s very sad to me. Math is very visual but I was only taught the rules, not what we’re actually trying to accomplish with our proofs and equations. I wish I knew better way to fill in the gaps.
Some sort of infinite series??
@@orangenostril As the parts of the infinite series go into smaller and smaller detail, it will become integral to our understanding of the bigger picture of modern mathematics.
@@randomirrelevant1788 I'd highly suggest online sources like Brilliant so you can do it at your pace whenever you want with plenty of visuals and examples.
Or Khan Academy if you'd rather not spend money. (don't tell Brilliant I said that)
@@randomirrelevant1788 That's how school maths is. On college/university, it's a whole different story. You have to prove pretty much everything
I’ve always struggled to understand how Wiles proof worked - this is the best explanation I’ve heard!
One noteworthy point in this context is that Wiles did not prove the whole of the Taniyama-Shimura-Weil conjecture. He "only" proved it for semistable elliptic curves, which the curve one obtains from a^p+b^p=c^p happens to be. So this was enough to imply Fermat's Last Theorem.
The full conjecture was shown later by former students of Wiles', in 2001 or so.
@@lonestarr1490 That student is basically in Wile's shadow then because you don't even seem to remember their name.
@@whannabi at keast the guy narating the video said his full name , so we can search him up
@@whannabi Which is fair, since Wiles is the one who proved the most famous unsolved problem in mathematics.
@@ricobarth Along with Taylor who closed the gaps in Wiles’ proof.
This is a stunning piece of math. It almost feels like art, it's so poetic.
What a beautiful profile picture you have...
@@bidyo1365 thanks 😊
Fitting for the trash bin of modern day egos, sadly enough. :/
@@awesomedata8973 who shit in your coffee
Elliptic curves are dual to modular forms.
Duality creates reality!
I am a professor of applied mathematics. I have been trying to understand the basics behind the proof of Fermat's Last Theorem and this is the first explanation I have seen that makes sense to me. Kudos to Alex and the creators of this video. The graphics is amazing as well.
🍷👍
How are you a professor and not know this
He said he is a professor of applied math. Math is currently so varied that no one can learn many branches of math at the same time. The last universalist was Henri Poincare.@@fatmilf1498
@@fatmilf1498Because maths is a huge subject and a specialist in one branch doesn't necessarily know much about another advanced field. Especially because Fermat's Last Theorem, and the maths behind the proof (modular forms, elliptic curves and other bits) don't fall under "applied mathematics"
This is a truly beautiful video, from the design to the script, everything is on point and the overall product looks amazing, thank you for inspiring while informing.
The hidden beauty of math never fails to astound me. This video was great. Keep it up
Just what is so beautiful about math?
@@paulcoy5201 it's intangible
@@paulcoy5201 That by just fiddling with numbers you can probe the universe and discover fundamental truths of its inner workings and underlying laws, plus if it weren't for algebra, geometry, calculus and all that, you wouldn't have all the fancy tools and knowledge that make today's society possible, you might not even have cohesive agrarian societies since you'd be too busy fighting your neighbour over the alleged size of their plot.
Also humans generally find beauty in order emerging out of chaos and finding patterns in seemingly random collections of information, solving puzzles. There is no shortage of beautiful, fun and/or useful things to find in math
It helps us quantify and understand our beautiful world, of course.
God's work reveals itself in many forms...
Dig deeper into the Langlands program at Quanta Magazine. You can explore all of our past coverage of developments in the Langlands program here: www.quantamagazine.org/tag/langlands-program/
gigachad math like this
What a consummately excellent video. The premise, art, geographic analogy, and insight. Thank you and keep it up!!
I remember Andrew Wiles explaining in an interview how he solved Fermat's Last Theorem. Obviously he didn't go into detail, but it was all very abstract, and one of the things that stuck with me was him saying that if he could solve Taniyama-Shimura, he would get Fermat for free. I've been wondering how that would technically work, and I'm happy I've stumbled across this video that explains it so well!
Amazing work, and special compliments to the animation team.
It should be noted that this is only an explanation of the arithmetic Langlands Correspondence for so-called global number fields (such as the field of rational numbers Q); in fact, Fermat's Last Theorem which Wiles proved (or rather, the Modularity Theorem which implies it) is a special case of this version of the Langlands Correspondence (for what is known as the reductive group GL(2) of invertible 2x2 matrices). There are various analogues of the Correspondence, such as the Langlands Correspondence for global function fields, the local Langlands Correspondences, and the geometric and quantum Langlands Correspondences, and each can be viewed as a toy model that might help us probe the original arithmetic correspondence, which hopefully will help us understand things like the zeta functions and distributions of primes. There are also many other parallel systems of results and conjectures, such as Langlands Functoriality and Duality, which are too complicated for a TH-cam comment, but are arguably even more important than the Langlands Correspondence itself. In fact, the Langlands Correspondence and Langlands Duality should be viewed as two big important lemmas that supports the conjecturally unifying result that is Langlands Functoriality.
Hmmm yes this is math talk
That sure is a lot bigbwords
well noted mr. wizard sir
Quanta, you've done it again. Stunning visuals, engaging and vivid explanations, and an overarching scope to the it all up. I love what you all breng out to into the world, thanks so much!
THIS IS SIMPLY ONE OF THE BESTEST VIDEOS I HAVE EVER SEEN IN MY LIFE. THIS CHANGED MY ADDED TO MY PERSPECTIVE TOWARDS MATHS, THIS MADE MATHS SO MUCH MORE AMAZING TO ME. THANK YOU SO MUCH
Quanta is creating a bridge between cutting edge math and the public. We need more of these.
This! 💖
Exactly 👍 & Hilarious 🤣
I am amazed by how much I missed in schools I never bothered with maths I always thought it was just boring but now that I’ve seen all this I truly appreciate maths and it’s beauty
Almost none of this is taught in schools unless you take Maths at undergraduate level or higher. So you didn't technically miss it.
@@akashchoudhary8162 no,that is not what I meant what I meant was that I missed the beauty of maths because always we were thought to solve only in a particular way and the teachers would get visibly annoyed if I asked them a doubt
@@easports2618 you didn't miss it. It didn't even pass close to you
@@akashchoudhary8162 that’s the problem though, schools don’t teach to think in math only to apply it right away. To some scenarios I don’t understand. We must teach the what, why, and how numbers function instead of memorizing formulas.
its*
It bring me so much joy to know so many others care about math and science.
What an excellent video!. I've wanted to understand the basics of the Fermat proof for years, but this is the first explanation that makes sense.
I loved the video, it was very well explained! Good job. I found a small typo: at 11:40 one should read y^2 = x(x- a^p)(x + b^p) for the Frey's elliptic curve.
Yeah lol, noticed the powers move down there too
I’ve see some people on TH-cam trying really hard to explain taniyama-shimura and why it’s related to Fermats last theorem, but you just went there and did it. Bravo
Wow, I can't imagine how much work, effort and time was put into making this video. Both the animation and script are perfect!
Thank you! This perfectly illustrates the beauty I've been seeing in my head for years but so often struggle to convey to my friends!
Srinivasa Ramanujan is a fucking baller. Dude's almost entirely self-taught and made so many advancements to mathematics in his short life. Whichever y'all know, put them in the comments. I would love to know what you guys think of this man.
You indian?
🍷👍
Fr. It seems that Ramanujan was addicted to infinite series and prime numbers. I love his work on infinite series for π that converge incredibly fast and are still used today to calculate π digits up to trillion decimal places. Surprisingly, most of his works lacked proofs, only conjectures, like how tf did he arrive at those complicated results?
@@sankalp2520 Yeah, this is beyond amazing. Savants with no disabilities, just abilities.
@@awwabientg4845 I don't think so looking at his profile
As a math student videos like this motivates me to keep on studying and research about grand topics like the Langlands Pogram. You are a really great channel for math begginers.
I love this video it’s a masterpiece even tho I don’t really understand what’s going on . I am still at the beginning of my journey in mathematics but I think it’s really exciting to connect everything together and the illustration is amazing.
This is by far the BEST video on Fermat's last theorem. Thank you!
I have no words to say how great these videos are, I watched this in June and was hardly able to understand, and after 3 months of checking a lot of number theory and modular functions videos, I am able to understand a little more now, I will come back again once I learn some more.
Thank you for your effort. I've been curious about the proof of Fermat's last theorem for a long time. You makes it easy to be understood by normal people. Thank you!!
Truly awesome video. And such a beautiful and simple explanation of how Fermat's got proved as well!
I can’t believe how good this is! Please make more overviews of giant math concepts. I would love an intuitive explanation of the sporadic finite groups, and the monster group / monstrous moonshine theory and how it relates to Lie algebra and the E8 manifold.
I would like to see this too, with plenty of explanation of the intermediate steps. All too often I see " Group theory is the study of symmetries. Here are all the ways you can rotate a triangle and it remains the same. Got that? Well onto the Monster Group..."
@@ingolifs now you can ask chatgpt and it wont be bored of providing as many intermediate steps you would like. everyone has their own personal tutor now
@@hayekianman in my experience chatgpt is terrible at maths
I'm not a math guy, but this video was excellent. Beautiful visuals, great explanation, and captivating flow. Wonderful job!
Amazing video, I loved the visuals and very nicely explained!
Thank you for explaining it so clearly without oversimplifying! Great storytelling!
Beautiful. This video is a work of art. The visuals add such a charm to the explanation, truly mesmerizing how well they help illustrate this complex subject.
Thank you for explaining Wiles' proof of Fermat!
It's by far the best explanation I've seen.
Excellent explanation. Never seen anything close to making this extremely complex proof being explained in a relatively accessible way.
I would love to see videos on the contributions of Grothendieck. He seems to have been a world-historical genius, but I don’t really understand his contributions.
Deligne, mentioned in this video, was a student of Grothendieck.
Basically he gave new struture that are abstraction of Algebraic and geometric structure. His genius was unparalleled as he broke all the ancient laws of mathematics and create way of thinking that have more çomplex ways of navigation and intuition.
He created mathematical tower heigh above the contents to see mathematics far about normal range.
Mathematicians are still grappling with his work. Unfortunately, it would be a bit difficult to convey the spirit of his contributions to a lay audience, because his style of thinking was extremely abstract. He always looked for the "right level of abstraction" in which to see a problem, and it turns out that that level is often pretty high. For example, who would've thought that the right way to understand shapes defined by polynomials involves category theory?
whats the connection between abstration in maths, and abstraction in computer science?
@@rv706 Excellect but I think ""the next level of deeper abstraction" might convey be a bettter approximation of his work.
Michael.
Incredible visual representation and art in this video and it demonstrates such a deep understanding to be able to convey these concepts so well.
Loved the analogy, really helps show the sort of "fields" there are within math and this intriguing relationship!
The almost miraculous achievement this channel and Alex make by explaining incredibly complex concepts simply enough to intellectually engage both neophytes and seasoned individuals . Whilst also creating a curiosity which is priceless. Bravo 👏. Thank you 🙏
Videos like these should be collected to create a modern school to teach our next gen.
There is a lot to understand and catchup very quickly as humanity progress, and these quick explanation and visualization really helps to get the basics and motivation for advance.
Thank you and your whole team for the efforts.
Great description of the connections! Love the video and explanation style!
Fascinating! Albeit I do not have the knowledge of any of those complicated subjects, I still sit through all 13 minutes to watch this!
Great video, thank you!
Until now I was aware that langland's program relates number theory with representation theory and that Ramanujan was a Number Theorist. We live to learn every day !
There are so many talented people out there creating incredible visuals and narratives that sometimes, I fail to see how insanely good their work is.
I think this video is amazing. I don't think I understood the whole point it's trying to make, but the visual support helped a lot.
Thank you for your work.
The visuals in this video are so amazing, congrats to the team who created this!
Beautiful done video. Thank you, Quanta!
That was great, i would love to see more modern math problems explained historically and simply lile this.
cool animations! do you plan to cover "L-functions, motives, trace formulas, Galois representations, class field theory", which you mention that you omitted?
What an insanely cool video. Excellent delivery and explanation with amazing visuals to support it.
I absolutely adore this video. Interesting topic, unbelievable beautiful animation and great narration. Please do more videos like this!
I loved this; it's a fascinating summary even for the math dunces like myself. I especially enjoyed it because it gives a follow-up to a particular favorite old bit of TV documentary I watched years ago: a PBS NOVA episode called "The Proof" about Andrew Wiles and Fermat's Last Theorem. It's actually quite touching. Highly recommended for anyone who enjoyed this (and can track it down).
❤💕💖
Maths is the most beautiful subject.
Super well made, truly inspiring!
Absolutely amazing video! Thank you so much for this!
This made me cry! Math is just so beautiful, almost poetic❤
it makes me cry as well , but mostly when its related to physics and astro physics .
An absolute gem of a video, from math content to explanation, from artistic graphics to rhythm. Pure awe !
The video was exceptionally BEAUTIFUL!
Absolutely fantastic video. Well done, Alex!!
This is a beautiful presentation, explained in a simple manner. Whoever made the script and the animation needs to get recognized a lot.
🍷👍
This is an absolutely stunning video, well done to the team that made it!
This is actually the best Mathematics video I've ever seen. Congrats!
These videos are so well made. Thank you!
Quanta Mag's videos simply do not miss. They're so unparalleled in their ability to explain complex topics in such a friendly, engaging way!!
I wish if such quality of videos can be made for our fundamental curriculum. Say for class 1 to 10th.
This problem needs to be solved only once and then the whole world can make use of it. No need for fancy tech startups or any thing.
These kind of beautifully drawn and curiously narrated videos can do wonders for children learning new things.
Alex, that was a brilliant video. You knocked it out of the park with the one describing the Reimann Hypothesis, and yet somehow you may have even topped yourself here. Well done
Artwork and storytelling are amazing men keep it up, you guys are gonna hit a million soon
This single video got me more excited about Mathematics than any other I've ever watched. Well done!
Danm this was such a nice video that it almost made Weil's proof idea seem 'obvious'/intuitive, now I really need to see his proof of the Taniyama-Shimura conjecture!
Check out Anthony Vasaturo's YT channel where he is uploading videos daily on Wiles' proof.
Thank you for putting together this very illuminating video on such complex topics in math. Simply amazing and beautiful.
The visual, voiceover and music are really amazing... I learned something new as well!
One thing I always found so cool was how *Andrew Wiles'* work built on top of *André Weil's* work lol... coincidence, I think not
Not a coincidence. Weil (and, later, that giant of modern mathematics that was Alexander Grothendieck) worked on the foundations of algebraic geometry and extended it so vastly that number theory itself could be expressed in geometric terms. This is called arithmetic geometry. That's where "elliptic curves over the rationals", the main theme of the Fermat-Wiles theorem, live.
@@rv706 yeah I’m aware of that. My comment was more of a joke on how similar their names are
@@williamdarko1142: Oh I see! Well, guess what, I was also about to write "Fermat-Weil" in my comment and then I corrected myself :D
I've been learning all this informally in recent years. When I first saw the name Andre' Weil, I thought "wait, that's NOT the guy who proved Fermat's Last Theorem, is it?" Indeed it's not, but maybe there's some (I say this semi-jokingly) Langlands Theorem of Mathematicians, tying together those who work in different-but-now-known-to-be-overlapping fields, maybe something like an Erdos Number.
This is not a coincidence because nothing is ever a coincidence
Good video, one small thing, portraying the bridge between Fermat’s last theorem and elliptic curves as something Wiles just dreamed up is unfair and inaccurate. Some earlier mathematicians established a proof that proving a special conjecture would prove Fermat’s last theorem, and it was Wiles who proved that conjecture.
Edit: I know this was touched on later in the vid. I wish it was not painted the way it was at the beginning. Also, it is not just the connection with elliptic curves but the Taniyama Shimura conjecture which gets painted over
They literally say all of that in the video.
Are you trying to tell us you didn't watch the video, or that you don't understand the content?
@@tinkeringtim7999 I did watch the video, & while I think they did clarify the connection between elliptic curves and Riemman wasn’t wiles, I wish they’d spent more time discussing the mathematicians who made the connection with the later conjectures. However I made this comment halfway through the video, & earlier in the video they had painted it as if Wiles himself came up with the conjecture
Amazing visualization and narration. Loved every bit of it!
This is an absolutely wonderful explanation of the connection between these two once disparate fields of math
We need this type of storytelling in our schools!
I study mathematics and this video gave a great explanation to the Wiles proof of Fermat’s last theorem. I love the visuals also, got me inspired to study more and to explore the vast landscape of mathematics :)
Simply the best math videos, this one and the other on Riemann Hypothesis. The content is very clear and entertaining. Music, animation, narration, creativity, everything is just amazing. Your videos help understand math, not just use formulas.
all this stuff is super interesting to me. Thank you to quanta magazine. all of your content is top quality
My admiration and respect to the graphic designer behind these unbelievable animations. The combination of creativity and thorough technical knowledge blend harmoniously in the representation of such intangible concepts. Total mastery of art and craft.
Quite pleasantly surprised that this highly abstract program is getting a quality visual exposition. We do'nt often see this.
Is English your native tongue?
the visuals are absolutely stunning. outstanding video!
Never knew about it, but happy now. Loved the presentation of this video.
Anyone who finds this interesting should check out 3Blue1Brown's recent video "Olmypiad level counting". It does a fantastic job of explaining a related problem.
I'm normally pretty averse to mathematical topics in favor of harder physics and biology, but this is super interesting and relatively easy to understand! I would love if there was a series about more "continents" or something similar about this "World of Mathematics"
I am just in awe with this video. Really, really good.
Most beautifully illustrated video I have ever seen!
I'm a bit annoyed by the name 'harmonic analysis' for Ramanujan's side. Ramanujan was arguably a number theorist, so it should be fair to call his continent as 'analytic number theory' whereas the other one should be called 'algebraic number theory'. I understand that this may sound less exciting to the public, but still much better than saying 'number theory had not much to connect with harmonic analysis', etc., with which Hardy, Littlewood, and Ramanujan himself would have strongly disagreed.
It's true that the connection of prime distributions to harmonic analysis is quite classical. Langlands reciprocity connects a certain kind of harmonic analysis (representations of reductive algebraic groups) to the "Galois side" of the Langlands bridge, so the terminology is accurate.
This makes me miss studying math so much… but damn was I rough at number theory 😅
I just picked up Smullyan again. If you are more of an algebra/logic type, you would love him. Just found out the set of all finite subsets of _N_ is countable and figured out two proofs over the weekend. So it has to be the set of all infinite subsets of _N_ that is not.
You can tell I'm such an amateur, but hey. FUN STUFF.
Upon a second rewatch, I got glimmers of understanding. Speaks to something about the power of revisiting. Thank you for this video.
This video makes my heart race. The idea that seemingly separate areas of mathematics are intimately connected is so tantalizing that it makes me smile.
It never ceases to amaze me how many unknown scientists send ground breaking ideas to the giants of their field change the face of science.
Incredible love from 🇮🇳India
This is a seriously impressive piece of math exposition and explanation. Bravo!
Very sastified with your explaination and visualization for such a complex problem in maths. Thanks so much!