Quanta Magazine - 2022-06-01
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: https://www.quantamagazine.org/what-is-the-langlands-program-20220531/ 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 - VISIT our Website: https://www.quantamagazine.org - LIKE us on Facebook: https://www.facebook.com/QuantaNews - FOLLOW us Twitter: https://twitter.com/QuantaMagazine Quanta Magazine is an editorially independent publication supported by the Simons Foundation https://www.simonsfoundation.org/
Being depicted as an engineer must be a mathematician's worst nightmare
Only if they a childish enough to encourage such gatekeeping
😅 So true
@Sep G 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.
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
@Bag O'chips He's more open minded than a neurosurgery patient
@Fragile Omniscience 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.
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.
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
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??
@Orange Nostril 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.
@Random Irrelevant 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)
@Random Irrelevant That's how school maths is. On college/university, it's a whole different story. You have to prove pretty much everything
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 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!
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
@Ben Cardwell Yeah, he was great, I wish I was able to take one of his other courses as well
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?
@sankalp shrivastava Yeah, this is beyond amazing. Savants with no disabilities, just abilities.
@Awwabien TG I don't think so looking at his profile
Quanta is creating a bridge between cutting edge math and the public. We need more of these.
This! 💖
Exactly 👍 & Hilarious 🤣
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.
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
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?
@it's 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
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).
❤💕💖
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!
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..."
@Thomas Bevan 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
Honestly I did not expect such high quality in all aspects, cought me off guard. The way how all aspects of communitcation work together is facinating. The audio, the grafics, the writing and last but not least; the explaining. It all works so harmonicly together
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 🙏
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.
@Lone Starr That student is basically in Wile's shadow then because you don't even seem to remember their name.
@w花b at keast the guy narating the video said his full name , so we can search him up
@w花b 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.
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"
This is an absolutely wonderful explanation of the connection between these two once disparate fields of math
I'm not a math guy, but this video was excellent. Beautiful visuals, great explanation, and captivating flow. Wonderful job!
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.
Thank you for explaining it so clearly without oversimplifying! Great storytelling!
The hidden beauty of math never fails to astound me. This video was great. Keep it up
Just what is so beautiful about math?
@Paul Coy it's intangible
It helps us quantify and understand our beautiful world, of course.
God's work reveals itself in many forms...
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!!
With all the genuises that have come and gone thru the world of elite mathematics, I find it surprising that there are still many areas as yet unknown. Amazing.
This is a beautiful presentation, explained in a simple manner. Whoever made the script and the animation needs to get recognized a lot.
🍷👍
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.
Thank you for explaining it so clearly without oversimplifying! Great storytelling!. Being depicted as an engineer must be a mathematician's worst nightmare.
This is a stunning piece of math. It almost feels like art, it's so poetic.
What a beautiful profile picture you have...
@Bidyo thanks 😊
Fitting for the trash bin of modern day egos, sadly enough. :/
@Awesome Data who shit in your coffee
Elliptic curves are dual to modular forms.
Duality creates reality!
Incredible visual representation and art in this video and it demonstrates such a deep understanding to be able to convey these concepts so well.
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.
An absolute gem of a video, from math content to explanation, from artistic graphics to rhythm. Pure awe !
I understood the fundamentals of this based largely on the animation. Good work! 😊
I absolutely adore this video. Interesting topic, unbelievable beautiful animation and great narration. Please do more videos like this!
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 YouTube 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
Well done. I believe this video will be very helpful in the course of designing to teach secondary school students basic concepts in "advanced" math either for enjoyment or to help them further pursue mathematics.
I still fondly remember when I first studied number theory and modular arithmetics at university, really opens a new perspective on numbers and mathematics in general
What a consummately excellent video. The premise, art, geographic analogy, and insight. Thank you and keep it up!!
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.
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.
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.
@Akash Choudhary 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
@E A sports[offline] you didn't miss it. It didn't even pass close to you
@Akash Choudhary 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*
Loved the analogy, really helps show the sort of "fields" there are within math and this intriguing relationship!
I have always been intrigued by Fermat's last theorem. This video is the first one to let a layman like me understand how Wiles did the proof. Amazing!
Excellent explanation. Never seen anything close to making this extremely complex proof being explained in a relatively accessible way.
This video makes me want to be a mathematician. Videos like this that show the beauty of math should be shown to any student that is skeptical.
Fantastic! I struggled some times to understand but looking at the real maths is really hard.. I'm only a physicist, admiring modern Maths from afar.. thanks a lot!
A suggestion: I'd love to understand other modern advances in maths (am I alone? I doubt..), like the work of Fields medalists, put in simple words like in this amazing video.. for example, what are Peter Sholze perfectoïds, etc..
Making maths understandable by many people (to all seems out or reach) is, I think, very important. Physics can be much more easily understood, as it relates to some reality people can grasp, and simple analogies can eventually been made, helping further.. Maths is so mysterious to so many because of the high level of expertise to understand the basic objects at hand.. In this video, you can understand as long as you have a somewhat basic knowledge in maths with arithmetic, functions, series, and things like this.. even if all other things remain mysterious.. at least, you understand what people have tried to do, which is maybe the basic curiosity many people have..
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 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.
Joel Cooke - 2022-06-01
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!
Arc125 - 2022-06-02
Seconded.
B J - 2022-06-02
Thirded!!
happy - 2022-06-02
fourthed
Nemanja Vuksanovic - 2022-06-02
Fifthed!
Kinjal Basu - 2022-06-02
seventhed