these long, detailed numberphile videos are rare but they're always the best

Edward’s a master of mathematical storytelling. Great video, great author.

This guy is my favorite professor in Numberphile

I love Edward. Thanks for having him back on. He has the mindset that most esoteric subjects in math, like what a local system is, or a Drinfeld module, can be explained simply. I think this is a fantastic frame and a necessary precondition to indeed explain simply ;).

This is one of the great numberphile videos. Exactly why I will be a patron for as long as Brady keeps making them

Possibly the best Numberphile video yet. I love this longer format where experts discuss huge topics.

The Langlands program is absolutely fascinating! I’m so glad Brady gave it an entire hour

Frenkel is an extraordinary communicator and a joy to listen to. His passion for the material really comes through and you can feel that it's rubbing off on Brady.

An 1 hour Numberphile video? All for it :)

Wow an hour long with Edward Frenkel! What a treat!

This is awesome. As a recreational math guy that loves to tinker and try to understand these complex topics, these videos are invaluable!

Edvard is an incredibly nice and down to earth guy. Listening to him almost makes me regret giving up mathematics

What always surprises me most is how spot-on Brady is in all of his work. I've been called knowledgeable over a wide variety of fields myself, but I don't think I've ever been quite *that* incisive. For example the idea that there might be other kinds of correspondencies/homomorphisms/functors between fields of mathematics really *did* have to be put in, while I would have missed that one, evenwhile being reasonably well educated and interested in math myself. Obviously I'll be combing through Edward's book forthwith, and an hour-long with a mathematician (also a pedagogue) of his pedigree is always a treat. But since these videos are about science education and outreach, as an ardent follower, I think Brady's role in getting the thing done might be a bit understated.

This was brilliant! Please have more of these long, detailed videos on difficult topics. Edward Frenkel is a great explainer!

I was only watching Professor Frenkel's video on the Reimann Zeta Function the other day! I'm happy to see a documentary-length video with him as the subject matter expert!

Are we not going to appreciate how he effortlessly slid his books into the conversation? Apart from being a mathematical genius, he's also a marketing genius.

We need more of these kinds of lectures covering different fields, their introductions, programs etc !

Watched the beginning of the Abel Prize lecture about Langlands just to realize the amazing effort that Brady is putting into the graphics. Here in this video, the graphics is so complimentary to the story. Wonderful work.

Edward Frenkel is someone we need in every school.

I have two points to mention... 1. I find it mesmerising that Prof. Frenkel is able to not only make eye contact with Brady but also to stare directly into his camera lens to truly connect with the audience at large... a masterclass indeed! 2. I saw the BBC Horizon documentary on Wiles' feat (circa 1993) and the visuals have always struck me... a cross between a facetted torus and a weird cathedral-esque 4-d pan of columns... this always confused me yet I see now, thanks to Prof. Frenkel's simple description of what an elliptic curve and a modular form really is, that it is really so simple a concept to grasp (an example of how, sometimes, a popular documentary using flashy imagery can be misleading(?)) Thankyou for this indepth exploration... I've learned soooooo many things🥰

What a brilliant way to concretely elucidate an esoteric topic. He teaches it in a way a bright child could understand, with an unbridled and infectious enthusiasm. 10/10

Brady always does a great job of bringing the importance of these topics to the surface with the right questions!

omg I read Dr. Frenkel's book "Love and Math" a few years ago!! it's one of my favorites, he has an incredible life story. So cool getting to hear him discuss the langland program!

The hour-long Numberphile deep-dives are rare, but also really nice when they come out. I think the rarity makes it even better, as it means I'm really going to sit here and just listen along for the whole thing rather than hop around between 2 or 3 of them.

Perfect balance for me of assumed knowledge, math ability, and introduction to new concepts. A real pleasure to watch :)

For a basic overview of what "representations of Galois groups" means, I'll break it up into the two parts. Galois groups, and representations. Galois groups are the groups of symmetries of field extensions. That is, if you have one field contained in another field (fields basically being nice systems of number-like things with all the nice properties), the Galois group tells you all the symmetries (automorphisms) of this field extension; all the ways you can transform the bigger field in a way that keeps the smaller field completely fixed, but also where the larger field retains exactly the same structure. The simplest example of this that everyone will be able to understand is the Galois group of the complex numbers over the real numbers. There is the trivial "identity" automorphism; you just keep every complex number the same. Then there is also complex conjugation: you can swap i and -i, and swap all the other complex numbers accordingly, and the complex numbers will behave exactly the same (the structure is preserved). And furthermore, this doesn't affect the real numbers at all; they are fixed under complex conjugation. It turns out that these are the only possibilities. These symmetries form one of the most trivial groups, called Z/2Z or C2; the cyclic group of order 2. So the Galois group of the field extension of C over R is isomorphic to Z/2Z. Representations of groups are, as the name suggests, ways that you can represent the structure of a group. Specifically, it's the ways that the structure can be represented in terms of linear algebra. At a very basic level, we are looking for all the different and interesting ways that we can choose a vector space, and a set of linear transformations (matrices, basically), so that each element of the group is associated with a linear transformation, and the linear transformations interact in the same way that the elements of the group interact. It's a little bit more than that though, because there are endless ways you can make the vector space way bigger than it needs to be for the given group. So really it's more interesting to ask about irreducible representations; ones where all of the dimensions of the vector space are inseparably mixed together by the group's representation, and so it can't be split into two smaller representations acting independently. It turns out that the complete list of irreducible representations is extremely interesting; if you just look at the traces of all of the linear transformations (gathering up linear transformations that come from the same conjugacy class of the group, which are basically the same as each other but viewed in a different basis, so have the same trace), you get a table of numbers with conjugacy classes in one direction and irreducible representations in the other, called the character table, that has amazing properties. Firstly it's square; there are exactly as many irreducible representations as there are conjugacy classes in the group. Secondly, with the correct weighting by size of conjugacy classes, this table's rows and columns are all orthogonal to eath other. That's just the beginning; there are so many cool things about the character table, but I digress. A simple but nontrivial example might be the symmetric group S3. It has 6 elements, usually described as the permutations of 3 symbols. These are collected into 3 conjugacy classes; a class with just the identity, the class of transpositions (2-cycles), of which there are 3, and the class of 3-cycles, of which there are 2. There are also, of course, 3 irreducible representations. There's the 1D trivial representation, where every group element is mapped to the 1D identity transformation (1). There's the, again 1D, sign representation, where every even permutation (identity and the 3-cycles) is mapped to 1, and every odd permutation (the 2-cycles) is mapped to -1. And finally there's the 2D representation that corresponds to the symmetries of an equilateral triangle in 2D space, where the identity maps to the identity, the 2-cycles correspond to the 3 reflectional symmetries, and the 3-cycles correspond to the clockwise and anticlockwise rotational symmetries. The character table in this case is quite simple, so it won't look so interesting. But you can look up the Schur orthogonality relations, and check them for yourself.

What I like of professor Frenkel is that he is not only presenting his story to Brady, but is actually seeking contact with us, the viewers by looking at the camera i.s.o. only Brady.

This is the most exciting video I have ever watched on Numberphile.

Beside his extraordinary explanation on Langlands Program, I studied mathematical education and was stunned by how he introduced the idea of number and negativity using floss, and of course topology as well.

I see Frenkel, I clickel.

Brady, I love these long form videos with great communicators. (Ed’s chat about string theory immediately comes to mind). Also, while speaking about great communicators, I appreciate YOU so much for the questions and insights you have. So many times you blurt out the exact thing that I am thinking! So long story short, thank you for all you do!

It's amazing to see Edward Frenkel taking time to explain in so detail.

I've heard it all many times before... Something clicked today... I'm forced to blame you, whomever you are Mr. Frenkel, Thank you.

Edward is so sympathetic and gifted. I just cannot help but adore him.

I am myself mathematician (from Paris) and I am happy to discover how enthusiastic was Edward Frenkel when he speaks his magic mathematics. I am going to buy his book and I hope to understand better from him, because he is also very pedagogic ! True chance for his colleagues to have him with them ! Last thing, I remember Edward Witten (another Edward !), who proves that the "Geometric Langland program" can be interpreted as a Mirror Symmetry, ..., Electrifying !

Fantastic as always. Edward is a rock star mathematician.

Ayyyyy, been a while since we've seen Frenkel! One of my favorites to listen to. His other book Love and Math is great, too.

Many thanks for such a fascinating in depth introduction to the Langland's program. Edward's enthusiasm is contagious. Can't wait for the next instalment!

I read this guys profile on wikipedia, he finished his phd in 1 year in harvard at age or 24? What a genius

Best video in a long time :) These long form videos are always like a nice present!

Thanks for having Dr. Frenkel back again. It was interesting learning about the correspondence between elliptic curves and modular forms with a detailed example. Would love to see more videos like this!

Professor Frenkel always has something interesting and then presents it with great enthusiasm. Excellent... and I now have to go back to Ken Ribet's video!

This video is amazing. When I saw the image, I immediately thought of a Smith Chart for a Vector Network Analyzer.

This was great. Can I propose a part 2 of this? Going into more depth on Galois groups?? would defo be up for that!

Amazing video. Didn't see how long it was when I clicked play but was enthralled to the end. Great job.

This 'magic' appearing in numbers has always fascinated me. Thanks for showing me one more of these 'miracles'. Wow, wow and wow.

The most exciting thing in Mathematics, explained by the best mathematics explainer on the planet. Absolutely brilliant, Numberphile hits another home run. Thank you thank you thank you!

Fantastic! Really the first time I could have such a deep understanding of this fascinating Langland's programme! Thanks for doing this video, and of course to the brilliant Edward Frenkel, and giving the required time to make us understand! Please continue doing this on this fascinating programme, or similar math mysteries! I think that 100k views in 3 days is just the sign that the public is also catching on this and wants to know! This is so important to make maths being understood to as many people as possible, as it is so impossible to grasp such level of maths for so many people, even with some years of maths in college, as compared to physics where people can really catch up much more easily with things, because of our general intuitive grasp with real things around us.

I would love to see a similar Numberphile video on Curry-Howard isomorphism (correspondence between logic and type theory) or Homotopy Type Theory (correspondence between topology and type theory/category theory).

I'm not even an amateur in math, just wrote a couple heuristic algorithms for a modified TSP problem and that was my limit. But this video was fascinating and inspirational.

Great video, the connection between the two things was very well-motivated by Prof. Edward. Loved it 🙂

Ed Frenkel is always fascinating. Thanks to him and Brady for this superb video. Makes you feel going back to Pr. Frenkel’s great book!

And this is lecture 1 of his course. Keep up for the next 3 months. It was glorious, but so much information.

This was definitely one of my favorite numberphile videos, great interview and great speaker! Thanks both!!

You know, I was just thinking a week ago when this guy would return. I loved his video about the whole -1/12 controversy - really put it in a new light for me.

I thought I will watch the first few minutes and tune out. I almost did not blink for an hour, and I would've listened to him for another hour. Amazing topic.

i forgot where i heard this, maybe another numberphile video, but the math guy said "there is a life before and after knowing about generating functions" because they are that powerful.

... his expression at 23:57 ... absolutely portrays his passion and drive ... very infectious

Fascinating subject, clearly explained, infectious enthusiasm. Kudos.

Edward Frenkel can explain the most complex mathematical ideas in the simplest possible way which can be understood by anybody. This is a sign of highest inteligence not seen very often even among smartest people. And let's think for a moment that he does it in a language foreign to him. Which he started to use only as an adult.

How I wish I could have had a math teacher like Edward Frenkel.

Love these longer form interviews!

amazing! The topic, the energy, all of it!

Thanks for giving this an hour! Edward is one of my favourite speakers. Having some grasp of the Fermat proof is on my bucket list, and this takes me a couple of steps closer, esp in the elliptic curve/modular forms correspondence.

I like that this guy is not afraid to really explain it!

39:46 Brady's reaction when the proof finally ends "Ok, nice" 😂

This guy is a legend, love his way to tell things

This is hands down one of my favorite Numberphile videos

Edward has true insight. He is one of my favorite mathematician of this channel.

Man, I always love videos with Edward!

Thanks for this entertaining lecture!!! Professor Frenkel has a very interesting way of presenting things! What a topic!

What an enthusiasm! It is a pleasure to listen to him.

These types of videos are always so so interesting and my favourite

One of the best videos on Numberphile, thank you Professor Frenkel!

I love Frenkel. at 40:12 Bless his heart for thinking a non-mathematician will enjoy reading about elliptic curves, even if it is Ash & Gross's treatment.

The alien melody from Spielberg's/Williams's "Close encounters of the third kind" in the fragment about harmonic analysis... instantly recognized it, the movie is so deeply engraved in my memory...

Thanks for an insightful introduction to the langland program

Everybody should have a teacher like Edward Frenkel!

one of the best NumberPhiles ever, I've already ordered Frenkel's book "Love and Math" 🙂 !!!

Excellent storytelling and interview, thanks!

What a beautiful episode!

I love his work and his humanity.

Professor Frenkel is a superb teacher. Thank you.

amazing video Brady, and thank you Professor Frenkel. It really feels like 'maths youtube' is smothering me in langlands-like content and I love it! Peakmath, Zetamath, and now this, I'm feeling smarter than ever!

Fun video! I love Frenkel's enthusaism, I'll probably read some of his stuff

I had a high school math teacher who was a great mathematician but a terrible, terrible teacher. The guy in the video is just the best, a famous mathematician who is also a fantastic teacher. Wish these kind of people were more common! (They're super common on Numberphile of course, but harder to find in the wild)

Brady & Edward, you've outdone yourselves!! :) Thank you for posting this

love these longer/more in-depth videos

One of the best videos I've seen on your channel, and I've seen a lot! :)

amazing. Thanks to your channel and to E. Frankel.

How on Earth did those three mathematicians come up with that harmonic series? It feels like magic that it "just works" for that counting function. I'd also be interested to hear whether Professor Frenkel thinks Riemann might be solved in this way by translating it to some other domain of mathematics and treating it as a different problem?

This was a fun video. It took me a few sessions to fully appreciate the magic behind it.

I love Dr. Frenkel. I got his autobiography novel over 10 years ago thanks to Numberphile and it got me motivated to go back to school and get my graduate degree. I read his book, got inspired, started to self learn, and then enrolled when I realized my mind was still able to enjoy the whole process of learning difficult topics and theorems to solve complex problems. Whoever is fimling this, please stop zooming in, panning to erronous locations, auto focus/zoom harshly, because it looks like a POV of the blair witch project or an amateur adult film.

I like the fact that after so many years finally this topic is getting popular!

Amazing video! Thanks to both of you!

wow, amazing video. So well explained! Edward is very charismatic.

For "tunnels below surface", look closer at continued fractions, especially in the Stern-Brocot context which provides exact arithmetic visioned by Gosper. I conjecture that the elementary proof of FLT can be found there.;)

15:00 - spot on - equations are more than solutions

This is cool as 😎 A great explanation, with detail, examples, & overview / summary of significance If there is a Numberphile follow up in 100 years that would be awesome !

More of these long form vids please. : ). That was cool