The Most Difficult Math Problem You've Never Heard Of - Birch and Swinnerton-Dyer Conjecture

[@beplus22]

As a professional mathematician, I should say that this video is very visually interesting, but contains many mistakes: 1) 0:00 Elliptic Curve Cryptography (ECC) is actually WORSE than RSA when it comes to being broken by quantum computers. One of the proposed crypto-schemes that is believed to be resistant to quantum attacks is SIDH, which is also based on elliptic curves, but it is not widely implemented. 2) 3:50 There is some confusion here. What we learn at school is that there is no formula giving all of the solutions of general polynomial equations in terms of radicals, when the degree of these polynomials is greater or equal to 5. For polynomials of degree less than 5 there is a formula just like the quadratic one we learned at school, that gives us all of the solutions, but if you are only interested in rational solutions, then you have to actually check which solutions are rational. The Rational Root Theorem provides us with an algorithm for finding RATIONAL (and only rational) solutions of polynomial equations in any degree, though one should note that this algorithm requires factoring integers. 3) 6:23 What is written in the screen is NOT an elliptic curve, it is a CUBIC CURVE. Elliptic curves are special cases of cubic curves, of the form " y^2 = x^3 +ax +b " over the complex numbers (they have more general forms over different fields). 4) 8:40 There are NO elliptic curves with genus greater than 2. In fact, in algebraic-geometry, elliptic curves are DEFINED as algebraic curves of genus equal to 1 (+ some other conditions). This is because over the complex numbers an elliptic curve is EQUIVALENT to a torus (a doughnut, which has 1 hole and therefore genus equal to 1). 5) 8:40 There is also confusion regarding Mordell's conjecture. Mordell had already proved that the rational points of an elliptic curve form a "finitely generated abelian group", which is what you explain at 10:05 in simpler terms. This is called Mordell's Theorem, or more generally Mordell-Weil Theorem. Since elliptic curves cover the "genus = 1 case", Mordell thought about algebraic curves of genus greater than one, and conjectured that they all have only finitely many rational solutions. This was known as the Mordell Conjecture and is now Faltings Theorem, after Faltings proved it. General comments: at 17:33 it should be noted that this is a "weak" form of what is now known as the Birch Swinnerton-Dyer Conjecture, but it is correctly stated. Another comment that I would like to make is that elliptic curves are interesting and well-known precisely because their rational solutions (or points) form a group: that means you can "add" points, in a rather interesting geometrical way, and also reverse this process back, only going from rational point to rational point. I thought this would be explained at 10:38 , but unfortunately it was not. Some of the pronunciations of the names and historical remarks were also off, but that would be nitpicking :)

[@kinertia4238]

Really appreciate your comment. I'm currently only a first-year undergrad, so my formal training in this field is somewhere between limited and non-existent. I have pinned the comment to the top so that people do not get misled by any information provided in the video.

[@beplus22]

@@kinertia4238 Thanks, it is still a nice video :). It is indeed very hard to talk about the BSD conjecture to a laymen audience, maybe you could ask to other people in your uni what are the main properties of each object being introduced. For instance, elliptic curves form an abelian group overthe complex numbers, that is what makes them so interesting. They are part of a much more general class called "Abelian Varieties", which all share this property. The BSD conjecture can also be extended to these abelian varieties!

[@pussiestroker]

@@johandh2o No. beplus22 actually cleared up a lot of confusion that I had about what was presented.

[@nachiketakumar9645]

@@kinertia4238 are you IITian

[@deltalima6703]

@@kinertia4238 That makes perfect sense. I wondered who is the kid and how he gets so much of it right. The video was quite good, thanks for making it. The pace was just a bit too quick, I would hit pause too slowly to see stuff, could edit out a second or two less on each cut (I dont mean scribble on a blackboard and waste my time). Your use of the word "integral" as having to do with integers may be grammatically correct, but man its painful when it means infinite infitesmal sums to me. Great video, thanks!

[@Aleph0]

This is so beautiful! I love how you don't shy away from showing us the real math. It's a really great service, as most sources found online are either aimed at seasoned elliptic curve veterans, or watered-down popular culture renditions of these topics. Your channel hits the middle ground perfectly: it's both rigorous and accessible. Keep doing what you're doing; your videos are phenomenal.

[@kinertia4238]

Glad you enjoy it!

[@aliasgeranees8893]

aha aleph0 and kinertia....both amazing channels...may be in "near" future you guys could collab.....you know as soon as one of you reaches 1 million subscriber mile stone

[@FlippityFloppittyFlom]

Caption: null!

[@King_Imani]

Greatness knows great greatness

[@samuelbevins247]

I have a strong feeling this channel is about to explode

[@ramyakumar4356]

Ur right

[@jinjunliu2401]

aw heck yeah it is!

[@cycklist]

Sadly it didn't 🙁

[@artsmith1347]

@@cycklist Apparently not so far. His explanations and the illustrations are great.

[@mustafa-damm]

You have a conjecture l

[@johnchessant3012]

18:56 I've heard it joked that Bhargava should receive $625,000 for that paper (62.5% of the $1 million prize)

[@x_gosie]

This dude, deserve more subscriber! Keep up the good work, we all learning from you! Love you🙏☺️

[@kinertia4238]

I appreciate that!

[@mariahamilton5305]

Bryan Birch briefly taught me when I was a first year student - as I write this, he's still alive and well and helping to price up 2nd hand maths and computing books, in his early 90s!

[@johnoldland7841]

As a first year undergraduate in the bar of St Catharines college, I was introduced by a friend who was very drunk to Swinnerton-Dyer who was then the master of Catz. My drunk friend came into the bar with SD and she said "This is Sir Peter Swinnerton Dyer; this is , eh, John. "

[@amirabbasasgari6183]

I have been looking for a gentle introductory overview to the conjecture for a while and this was a decently intuitive and visual one. Even if there have been some minor mistakes as some have mentioned (I'm not expert enough to comment on those), I still really appreciate your work to demonstrate connections of these many math areas in a pretty understandable manner with minimum background required. Great job!

[@bhavya5413]

Hi bro I'm your 67th subscriber and checks for your channel everyday for a new video. I love your channel , I appreciate 👍👍👍

[@henriklovold]

Assistant professor of compsci here - I approve of your video. Great job, and keep more coming, I'm looking forward to it :)

[@rishitiwari8644]

The clarity of your content delivery is remarkable! Keep up the good work!

[@AnitaSV]

0:57 ECC is equally susceptible to quantum shor’s algorithm. Any hidden sub group can be solved by quantum computers. You need something like lattice based techniques to be resistant to quantum computers.

[@anamarijavego6688]

the quality of your videos is outstandingly good

[@kalebmark2908]

I... LOVE this video. How do you not have more subscribers??

[@mikeschmit7125]

One of the most underappreciated channels ever.

[@5wplush243]

I have a few doubts that aren’t clear to understand in the vid:- 1. How do elliptical curves have genus’? You explained Mordell’s conjecture but this wasn’t clear. I thought only 3d figures (‘holes’) could have a genus… 2. Why are only primes studied in 14:05 on the table?

[@5wplush243]

Otherwise great vid!

[@kinertia4238]

These are very good questions - in fact I was hoping that people wouldn't catch the first one (I had the exact same doubt), since that's a topic that deserves a video of its own! 1) Mathematically, a 'genus' has several different definitions depending on the structures you're studying. In topology it is the number of holes, but in algebraic geometry the genus is nothing but a defined invariant of a one-dimensional variety on a field. That means that if you draw differential forms on the curve then their dimension would be one (it's possible since the curve is a manifold). The two concepts are subtly related - the topological definition was what inspired Mordell - but you'd need a good amount of grounding in topology before you can really 'get' it (I don't have formal instruction in topology either, my doubts are usually cleared on Stack Exchange). 2) It's a standard practice. Finding solutions modulo primes is a lot easier than other numbers due to a variety of reasons. For simple examples you can look at Fermat's Little theorem, Wilson's theorem, or cubic reciprocity. In case you're asking why this works with only the primes, then, well, that's the million dollar problem.

[@nicolasbourbaki9393]

1. The genus of an algebraic curve is an invariant that arises by playing with some weird formal sums, called divisors. But i don't know exactly how that relates to topology. To be fair i didn't even knew they genus is already a concept in topology. But one thing that might be interesting is, that the genus g is equal to 1/2*(d-1)*(d-2) where d is the degree of the algebraic curve. Because of that, the curve of every Polynomial of Degree 3 has a genus of one and is, by definition a eliptic curve. Curves of genus 0 are Isomorphic to the projective Plan of dimension 1. These are Curves defined by a Polynomial of Degree 1 or 2. And now mordel conjecture (or falting theorem) says that every Curve of genus 2 or higher have only finitely many rational points. These are exactly the curves defined over a Polynomial of degree 4 or higher. 2. Studying Solution of Polynomial over modulo Primes has one Major advantage. The whole number modulo a number is a field again if and only if, that number is a prime. The converse is quit simple, because if 0 =! n=pq, and p,q are both not n then p*q is 0 modulo n although p and q are not 0. This is impossible in a field, thus Z/nZ (= whole number modulo n) is not a field. The other direction is basically Z/pZ is a finite Integral Domain and this makes it already a Field. Because the multiplication with an element a of Z/pZ as a function has to be invertible. Basically because it is an injective ring homomorphism and an injective function from a finite space to another finite space with the same cardinality is surjective. Therefore the Image contains 1 and the pre Image is the inverse of a. Now study of Solutions Modulo a Prime can be embedded in a rich Theory of the study of Solutions of Polyomials over a finite Field. One of the Most Important Theorem in this field are the Weil Conjectures. They give an easy Description of the Solutions of a Polynomial over a finite Field. And basically you can apply the same Techniques you used on Curves over the Rational Numbers now on Algebraic Curves over finite fields. You can define a genus. take the union of two algebraic curves, define the function field of two of an algebraic curve and so on. Of cause not every thing works the same but most things do. And you can do all this, mainly because Polynomial rings over a field do have nice Properties. For example you have a prime factorization. They are Integral and so on. All really nice things to work with.

[@poproporpo]

@@nicolasbourbaki9393 Yes it's the man himself

[@theflaggeddragon9472]

To answer your first question, and Im somewhat bewildered this wasn't addressed in the video, the set of _complex_ solutions to a non singular Weierstrass equation y^2=4x^3+g_2x+g_3 defines a 1-dimensional complex curve which is in turn a 2-dimensional real surface which, when appropriately embedded in the complex projective plane is genuinely the surface a donut, i.e. a surface of genus 1, or surface with one "hole". The characteristic shape of the real solutions, an oval and an arc stretching infinitely up and down, are the slice through a donut, and the arc connects to itself at a certain point at infinity.

[@keving958]

What a great video! I wish I had seen this 15 years ago when I was learning this stuff in college. But, I can't be the only one who chuckled at the end when he referred to BSD as the "cherry on top" while flashing a picture of pastries with strawberries on top.

[@King_Imani]

1 of the best KZbin Maths videos I have watched. This is truly a great video on maths and the title is click bait with 5000IQ

[@PiAndAHalf]

here before this channel blows up, keep it up!

[@PartyFunCrazy]

Just found your channel, loved the video! I'm so excited to see where this goes for you! I wish you well, keep going!

[@therealAQ]

I will always appreciate a good trip on BSD

[@sairithvickg6663]

Great Work Bro !! I hope your channel gets 1 million + subscribers

[@helojoeywala6622]

Super good video! I didn't think it would be this well made and edited but it blew my mind! Good job and good luck to you in the future.

[@colinbrash]

Wow, this video is absolute gold!!

[@dogbiscuituk]

There is a formula for the roots of a quartic.

[@pierrecurie]

Bring some radicals, and you can get the roots of a quintic too (pun intended).

[@jakoblenke3012]

@@pierrecurie A formula for the roots of a quintic is impossible

[@pierrecurie]

@@jakoblenke3012 en.wikipedia.org/wiki/Bring_radical

[@OwenMcKinley]

Your explanations and content are incredible! Supremely well done. Hope to continue to see more neat math videos from you in the future.

[@dylanparker130]

really enjoy your videos - even though I'm sure a great deal of planning goes into them, I like the way it sounds as though you're just thinking out loud at times

[@kinertia4238]

Glad you like them!

[@r1a933]

Bro took the whole explanation to another level 💯

[@andreasburger4038]

Amazing stuff! Awesome work with the animations and the explanation. If I could improve would think, it would be the audio quality. Keep it up!

[@dcterr1]

Very good explanation of the BSD conjecture as well as elliptic curves in geheral!

[@SatyarthShankar]

Your content quality has been a constant (at excellent). Your accent on the other hand has been a variable.

[@davidschaaf5126]

Complicated topics beautifully explained

[@Kuribohdudalala]

Friend did his undergrad thesis on cryptographic applications of graphs, so this was really cool to see!

[@harriehausenman8623]

Great content, really fine production. Thanks!

[@rishiraj8738]

Great work bhai. I really appreciate it.

[@HanLe-px8ko]

love the graphics in your videos!

[@ND62511]

A bit of a correction on your explanation of Fermat’s Last Theorem; the theorem states that there are no NON-TRIVIAL INTEGER solutions to the equation a^n + b^n = c^n where n > 2. It’s really easy to get solutions to the equation if a, b & c are allowed to be real numbers. Infinitely many, actually.

[@TheHzh82]

ECC is just as vulnerable as RSA with a quantum computer.

[@durnsidh6483]

I know, it's strange that he didn't mention that it's breakable in the same way as RSA considering that is something that every technical source mentions. Maybe he was referring to SIDH and said the wrong thing but that isn't widely used and has even more problems.

[@TheHzh82]

@@durnsidh6483 Yes, SIDH is a completely different algorithm to ECC

[@jordan7828]

🤣

[@AllanKobelansky]

I Liked and Subscribed. Looking forward to more, compelling content. Very well done.

[@roman111117]

Crazy how I had never heard of elliptic curves outside of orbits in our solar system until yesterday. I argued it was pronounced elliptical curves and was proven wrong lol. And today you link it to computer science and cryptography. Love this.

[@deltalima6703]

Ellipses are something else. Hopefully astronomy stays clean (LGM002 = "little green men" lol!) and doesnt get messy like math and physics.

[@Mrpallekuling]

Very interesting video with an introduction to many complex math areas. Well done!

[@carlobenedetti2407]

Best video on this topic on KZbin

[@mr.maccaman2]

11:13 Anyone else catch that scream in the background?

[@hikingpete]

I'm sorry to say, the video introduction is very problematic. While Elliptic Curve Cryptography has its strengths, it's just as vulnerable to quantum computing as RSA, and in fact may end up falling sooner. If you're interested in post-quantum cryptography, there's a lot of options under development. The most widely deployed option I'm aware of is NTRUPrime.

[@kinertia4238]

Thank you for catching that, unfortunately it was only after I posted the video that I discovered that my source was faulty - it was a document from PGP which turned out to be heavily biased. I'll try to avoid such errors in the future.

[@carterthaxton]

64 GB = 512 billion bits. Dividing that by 576 bits gives about 1 billion times, not half a trillion (which would be 500 billion). Still very remarkable, but just keeping you honest. ;) Great video, btw! Thank you for making this.

[@dcterr1]

Very good, enlightening video! I noticed you made a mistake though. You presented the equation y^2 = x^2 + 5 as an elliptic curve. I think the exponent of x should be 3, not 2.

[@kinertia4238]

Oof, thanks for catching that! I'll add a disclaimer in the description correcting it.

[@Zibeline759]

The same quantum algorithm to break RSA can be used to break ECC. So ECC is not the cryptography of the future. It's the cryptography of the present, soon to be replaced.

[@redfullpack]

unlike paranormal mysteries which are largely Humbug Drivel, mysteries and enigmas from Mathematics _do educate the mind_ by the way I Fail miserably in mathematics subjects throughout school from kindergarten to college

[@marchevka22x]

Great intro to an interesting subject.

[@dcterr1]

If elliptic curves of rank greater than 1 are so rare, why do we care about them? In particular, do we need to use them to make a secure cryptosystem? And is BSD important in designing such cryptosystems?

[@paulisaac3489]

maybe I'm weird, but I like your math videos, not sure why they have so few views.

[@StephenBlackstone]

ECC and RSA are both examples of the same thing "the hidden subgroup problem" - both are breakable by versions of Shor's algortihm..

[@RSLT]

👍 Just watched the video and I loved it! Hit that like button and subscribed to your channel. Can't wait for more amazing content like this! Keep up the great work! 👊😄

[@wilderuhl3450]

Let’s take a moment to appreciate Number theory. Way ahead of you buddy.

[@brendawilliams8062]

Well, lately 112 is a puzzler. I’m just weird that way

[@attilao]

The peafowl in the background is a nice touch.

[@sergiogarofoli573]

why quantum compute to factorize a number that is a miserable difference of squares? because a number n=pq (in RSA original notation) is just the reduced form of the n=[(p+q)/2]^2 - [(p-q)/2]^2 expanded expresion, and is a difference of squares. Just because the squares are hidden beyond invisible differences, doesn't mean they are not existing, you just cannot see them... That's how Sophie Germain made her identity... more than a sentury and half ago

[@tjongon5274]

Just wow, this channel is gold

[@Chris-lv1nz]

Excellent thank you ! I like the background music also :)

[@dr.rahulgupta7573]

Nice presentation of the topics in a beautiful manner. Thanks.DrRahul Rohtak Haryana India

[@based_king]

Good on you man :). Wishing you the best !

[@TheChicken313]

Hey, just wanted to start by saying that this was an amazing video! My primary academic focus is not math but I have a novice level foundation in university physics and calculus till the multivariate level, so I love more qualitative style videos like these that can help me appreciate issues in the field! I just had a few questions regarding the video that may allow me to appreciate the BSD conjecture even more. My primary confusion with regards to this video is the concept of "solutions". What first comes to my mind with regard to solutions in the context of polynomial functions is the first instance of this concept when you learn about exponential functions in the beginning of calculus. A "solution" with regard to an exponential function is of course a point or points where the function intercepts the x axis. Of course in these situations there are only finite or no "solutions". At 10:14 there is an image of genus = 0 resembling an exponential function which apparently has "infinite solutions"?? This suggests to me that I am misunderstanding what is meant here by a "solution". I have solved problems before involving solving for individual points on a elliptic curve, but again I am unaware what the idea of having infinite or finite "solutions" in this context. An explanation or example to help me better understand this idea from anyone would help me appreciate this problem infinitely more, and would be greatly appreciated!!

[@SmiteYaBgs]

How do you edit these videos? What software etcc and do you draw those graphs and write the text yourself?

[@ARBB1]

Extremely high quality.

[@honeyboiii]

Amaazing content dude. Keep it up!

[@01binaryboy]

Awesome

[@lazgheen]

Nice animations. Which programs are you using?

[@bobfish7699]

Fascinating - and well explained too

[@agamgoyal8918]

Great video! 👍

[@kinertia4238]

Glad you liked it!

[@zer-mela]

I like ur presentation style it's fun and engaging

[@DocSineBell]

Wow, great video! Nice work man

[@alex1507er]

At 14:14-15:42 there is a bad misprint in the equation in the top of the screen: it is of second degree!

[@kinertia4238]

Yes, someone has already pointed it out, thanks. It should be y^3.

[@jnoelcook]

Just came across your channel yesterday and love it. I hope you will keep it up! These are some of the best math videos I have seen! One thing I do need to ask though, as I had to replay it 3 times when I heard it. Can't remember at which minute in the video it was, but you said that Mordell proved that for any elliptic curve there are only an infinite number of rational points when s = 0. You might want to look into this. Don't you mean that there are only an infinite number of rational points when L(C,1)=0?

[@fizqialfairuz5744]

I was here before this channel become famous.

[@FF-mr9wv]

Is more simple to say: f(x,y) x,y belonging to Q(rationals) such f(x)=0, where fx is an L-function (as well the Riemann zeta function). Source: wikipedia.

[@shreyasjv4877]

12:18 Tell that to George Orwell! Great vid as always!

[@shreyasjv4877]

I was mind=blown at 15:41!

[@shreyasjv4877]

It's kinda intriguing how you can use statistics to know the rank of curves. Could you make a vid or share vids/notes on that topic in particular?

[@LeonardoGPN]

Keep making videos, your channel will really grow.

[@MrController12345]

Thanks for such video... Really appreciated

[@amandeep9930]

Please tell me which software do you use to make those animations

[@kinertia4238]

I use Adobe After Effects for the motion graphics and Illustrator for the designs

[@amandeep9930]

@@kinertia4238 thnx bro

[@mathunt1130]

Excellent video.

[@vameza1]

excelent video!!! Congratulations!!!

[@metalim]

why the heck is there drill sound in background?

[@IshanBanerjee]

Sir I have become a big fan of yours

[@IsraelJacobowich]

RSA has only been hacked when cybercriminals managed to steal confidential preliminary data. RSA is provably secure ( assuming factoring is as hard as we think ). moreover: while shore's algorithm does indeed solves factoring in poly-time on a quntum computer. the hardware required is still many years away. ( just a side note )

[@xyzain_1827]

Excellent video!

[@Bruno_Haible]

08:43 "... any elliptic curve with a genus > 2 ...". What are you talking about? Elliptic curves have genus 1.

[@kinertia4238]

Slip, I guess. I meant any 'curve' with genus > 2.

[@JoeJoeTater]

Yeah, it doesn't seem like a great idea to found post-quantum cryptography on math with big open questions. That just provides a big opportunity for people to break it. It's especially concerning since NIST and the NSA succeeded in sneaking vulnerabilities into older methods that used elliptic curves.

[@OuroborosVengeance]

Espero más contenido. Buen canal

[@lavneetjanagal]

Awesome video. Thank you.

[@TymexComputing]

17:17 how precisely is 1.03313660856 equal to 1.0013660856 and if not equal which one is L(1) for equals(. , . ) 5 plus(. , . ) 3rd_power(X) 3rd_power(Y) ? Thank you.

[@literallyjustayoutubecomme1591]

I see that another comment may have addressed this, but I feel you may have misinterpreted the rational root theorem, it only tells us how the rational roots look IF they exist.

[@miguelivan504]

Loved it. Thank you.

[@iamanidiotbut5523]

Random Indian kid talking about nerdy and math stuff? Pretty cool background behind you man. Very well made video mate.

[@jaeimp]

In "Slight error at 14:14, the equation should be y^3, not y^2." I think you meant to type "Slight error at 14:14, the equation should be x^3, not x^2."

[@ophello]

I thought Fermat was the one who drew the note in the margins...

[@hrvojeabraham5080]

Contrary to the statement in the video, all order 4 poly. eqs. have algebraic solutions in C. Order >= 5 don't. This is called Abel-Ruffini theorem.

[@nicolasbourbaki9393]

All rational Polynomials have by definition an algebraic solutions in C. Because C contains Q and C is algebraically closed. Abel-Ruffini Theorem just says there is no solution in radicals to general Polynomials of degree 5.

[@hrvojeabraham5080]

​@@nicolasbourbaki9393 Thank you for your comment. Do you agree the statement from the video "... you can only solve linear, quadratic and sometime cubic equations. You can't solve the equations where x is raised to the power higher than 3." is incorrect? *All* orders are *always* solvable, which is much less strict criteria. And both 3 and 4 are *always* solvable in radicals, >=5 are not.

[@hrvojeabraham5080]

@@nicolasbourbaki9393 Is it possible you mixed up the terms 'algebraic solution' and 'algebraic number'? Algebraic solution is a closed form solution, i.e. solution in radicals. Algebraic number is any number which is a solution of rational polynomial in C. en.wikipedia.org/wiki/Algebraic_solution

[@kinertia4238]

What I actually said is that 'In school *you're taught* that you can't solve equations where x is raised to a power higher than three.' In fact I actively refuted that statement during my discussion of the rational root theorem.

[@nicolasbourbaki9393]

@@hrvojeabraham5080 yes i mixed that up, thank you for the clarification:)

[@yashiAxen39]

extremely fascinating

[@PrintEngineering]

Make another one that goes into depth for the secp256k1 curve! y^2 = x^3 + 7 mod (2^256-2^32-977)

[@dylanrambow2704]

The name Weil is pronounced like "vay." en.wikipedia.org/wiki/Andr%C3%A9_Weil

[@fabiotp5068]

The algorithm has found you my friend

[@Jooolse]

Great content!

[@chevasit]

Very good!