المدة الزمنية 20:18
The Most Difficult Math Problem You've Never Heard Of - Birch and Swinnerton-Dyer Conjecture
تم نشره في 2020/07/15
The Birch and Swinnerton-Dyer Conjecture is a millennium prize problem, one of the famed seven placed by the Clay Mathematical Institute in the year 2000. As the only number-theoretic problem in the list apart from the Riemann Hypothesis, the BSD Conjecture has been haunting mathematicians ever since its conception in the mid-20th century, and relates to several different concepts and applications, including Elliptic Curve Cryptography. The BSD Conjecture is essentially an attempt to determine rational solutions of elliptic curves through a variety of different approaches, including but not limited to L-functions, statistics, complex analysis and rank determination. In this video I attempt to explain the BSD Conjecture in layman terms without shying away from the inner beauty of the equations. Slight error at 14:14, the equation should be y^3, not y^2. ------------------------------------------------------------------------- Music Credits: Special thanks to E7VN (https://open.spotify.com/artist/6erp85MigVPLLohx5xPxwT) for creating some great original soundtracks. Inspired by Kevin MacLeod Link: https://incompetech.filmmusic.io/song/3918-inspired/ License: http://creativecommons.org/licenses/by/4.0/ Wind Of The Rainforest Preview by Kevin MacLeod Link: https://incompetech.filmmusic.io/song/5729-wind-of-the-rainforest-preview License: http://creativecommons.org/licenses/by/4.0/ Music: https://www.purple-planet.com http://www.bensound.com/royalty-free-music -------------------------------------------------------------------------- Sources and Citations: 1) Gunter Harder and Don Zagier, 'The Conjecture of Birch and Swinnerton-Dyer' (https://people.mpim-bonn.mpg.de/zagier/files/tex/BSDwHarder/fulltext.pdf) 2) Popular Talk by Manjul Bhargava in the Clay Mathematics Institute, 2016 (/watch/)gD6bzIWQbg2Q6 3) Notes on Bhargava's talk (https://www.claymath.org/sites/default/files/bhargava.pdf) 4) Wikipedia contributors, "Birch and Swinnerton-Dyer conjecture," Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/w/index.php?title=Birch_and_Swinnerton-Dyer_conjecture&oldid=948783592 5) Stewart, Ian, 'Visions of Infinity: The Great Mathematical Problems' (Basic Books, 05-Mar-2013) -------------------------------------------------------------------------- Chapters 0:00 Introduction to Elliptic Curves 1:40 Intro 1:50 What is Number Theory? 3:27 Types of Diophantine Equations 6:22 Solutions to Elliptic Curves 11:50 Clock Arithmetic 13:13 Analyzing Solutions using Modular Arithmetic 15:45 The Conjecture 17:45 What's been done? 19:03 Conclusion
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مقاطع الفيديو ذات الصلة على The Most Difficult Math Problem You've Never Heard Of - Birch and Swinnerton-Dyer Conjecture:
1) 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) 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) 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) 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) 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 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 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 , but unfortunately it was not.
Some of the pronunciations of the names and historical remarks were also off, but that would be nitpicking :) ....وسعت 415
Good job and good luck to you in the future. 5
If I could improve would think, it would be the audio quality.
Keep it up! 3
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 on the table? ....وسعت 21
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 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!!
by the way I Fail
because a number n=pq (in RSA original notation) is just the
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
Great video, btw! Thank you for making this.
Way ahead of you buddy.
But nice video