The Million Dollar Equations

The Million Dollar Equations - with Tom Crawford 81,892 viewsSep 17, 2020 The Royal Institution 1.23M subscribers In the year 2000 it was announced that seven of the biggest unsolved problems in mathematics would each be given a $1million prize. Only one has been solved. Watch the Q&A: https://youtu.be/AFc7kGfLSIc Subscribe for regular science videos: http://bit.ly/RiSubscRibe The seven million dollar equations are: the Riemann hypothesis, Navier-Stokes equations, P vs NP, the Poincare conjecture, Yang-Mills mass-gap hypothesis, the Birch and Swinnerton-Dyer conjecture, and the Hodge conjecture. In this talk, explains four of them. Chapters: 00:00 - Introduction 05:30 - The seven Millennium Prize problems 09:00 - The Riemann hypothesis 30:41 - P vs NP 46:16 - Poincare conjecture 58:57 - Navier-Stokes equations Tom Crawford is a mathematician at St Edmund Hall at the University of Oxford where he teaches maths to the first and second year undergraduates and visiting students. Tom completed his PhD in applied maths at the University of Cambridge in 2016, where he conducted experiments looking at where river water goes when it enters the ocean. Tom’s website: https://tomrocksmaths.com/ Tom’s YouTube channel: https://www.youtube.com/tomrocksmaths Tom on Numberphile: https://www.youtube.com/playlist?list... --- A very special thank you to our Patreon supporters who help make these videos happen, especially: Adam Leos, Alan Latteri, Andrew Downing, Andrew McGhee, Andrew Weir, Anonymous, Christina Baum, Dave Ostler, David Crowner, David Lindo, David Schick, Fairleigh McGill, Frances Dunne, Gou Ranon, Greg Nagel, Jan Všetíček, Jeffrey Schweitzer, Joe Godenzi, jonas.app, Kellas Lowery, Lasse T. Stendan, Martin Steed, Matt Townsend, Michelle J. 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Chapters Introduction 0:00 The seven Millennium Prize problems 5:30 The Riemann hypothesis 9:00 P vs NP 30:41 Poincare conjecture 46:16 Navier-Stokes equations 58:57 Buy The Royal Institution merchandise $14.77 Teemill $24.62 Teemill $24.62 Teemill $49.23 Teemill $18.46 Teemill MX$73.74 Teemill 235 Comments rongmaw lin Add a comment... The Royal Institution Pinned by The Royal Institution The Royal Institution 1 year ago We have something a little different for you today. The brilliant @Tom Rocks Maths ran an interactive livestream about the Millennium Maths Problems, getting the audience to pick which problem he would explain. It was an excellent night and we really enjoyed the format. What do you think? Did you catch it live? Should we do more interactive livestreams? Should we get Tom back to explain the last three problems? Let us know in the comments! 27 Marcin Szyniszewski Marcin Szyniszewski 1 year ago I wish Tom would also go through all the other problems! Great stuff! 21 The Royal Institution David Terr David Terr 1 year ago Very good introductory video on the Millenium Problems! About 6 months ago I decided to try to tackle the BSD conjecture (problem #6). I gave up after a few months though, after getting about a quarter of the way through Silverman and Tate. Although this is supposed to be an undergraduate-level math text and I have a PhD in algebraic number theory, I quickly became overwhelmed! Just trying to understand the proof of Mordell's theorem is bad enough! For many years I've been fascinated by continued fractions. I was hoping there might be a way to use them in solving the BSD conjecture, which no one has tried before. If by some miracle I do end up solving this problem, my proof will most likely involve very different methods than are currently being applied. In any case, I love math and I'm fascinated by the Millenium Problems, or at least those I understand! 6 reykjavikingur reykjavikingur 1 year ago Thanks for explaining the less popularly known "million-dollar" problems, especially the Poincare Conjecture. Also, though I appreciated your coverage of P vs NP, it still seems like I understand that problem less every time I learn about it. 1 the deeliciousplum the deeliciousplum 1 year ago π = 4∫ 🍩²☕️ Wonderful and inspiring talk. I so regret not having spent a greater amount of time and focus on maths. Sigh. Yet, it is a joy to explore what people have discovered and what they are working on. Thank you for making and sharing this vid. 8 The Royal Institution Fresh Actv Fresh Actv 1 year ago Cool usage of algorithms to get a whole generation into STEM learning ✌🏻 4 PhenixOrbitAll PhenixOrbitAll 6 months ago Your love for Navier-Stokes is contagious :) Thx for showing some beautiful real life applications of them. Great video 👍 Mitchell Ireland Mitchell Ireland 1 year ago Nice work mate! With the Navier-Stokes equation: pardon my lack of understanding but how do you put a number/value on the 'random' input of the turbulence? Would you need to identify all the variables at that point in time and give a value to each. 2 Oogway Master Oogway Master 1 year ago I ve always dreamed with solving one of the list especially navier-stokes. I love fluids theory and study. 4 Szczepan Szczepan 1 year ago Thank you, Tom! Finally, someone explained to me P vs NP :D. Please also to rest of the problems. You have a gift to put them into simple words. 5 The Royal Institution David Acer David Acer 3 months ago I can solve Tom's version of the Riemann Hypothesis easily - the function given is NEVER equal to zero, for any complex number s where that function is defined. The Riemann Zeta function is actually the analytic continuation of the function he gave. Also, "finding a value of s with real part not equal to a half would earn you a million dollars" - OK, I give you any of the trivial zeros. Where's my million dollars? Also, 1 is not prime. The details really matter! LaserFur LaserFur 1 year ago (edited) 36:00 I would argue that checking the traveling salesman is also difficult to check. So you have a route, but how can you tell that is the best route? The code in my application takes shortcuts to reduce the processing time so it's a given that it might not be the fastest route. But how would I write code to test that it is the fastest? edit: I would also like to note that you can't even say what percentage between the best and the worst a route is without solving for best and worst. 1 Fwiffo Fwiffo 1 year ago Feel empathy for the tattoo artist that had to draw a series of permanent, straight, parallel lines on a person's skin. 38 Jon Wesick Jon Wesick 1 year ago Where can I find more info on the Yang-Mills problem? Alberto Bec Alberto Bec 11 months ago "Adding a 3rd dimension makes mazes easier" Tell me you've never played Zelda BOTW without telling me you've never played Zelda BOTW ann onn ann onn 1 year ago I have discovered a truly marvelous proof of this, which this comment is too narrow to contain. 78 Evolved Copper Evolved Copper 1 year ago I'm glad he covered Navier-Stokes. My closing thought above all the knowledge and information is I dig his cash money shirt 5 Oleg Grenrus Oleg Grenrus 1 year ago I'm sad to see that Riemann's hypothesis, P=NP and Poincare's conjecture are the three problems always presented, there is so much content on them! But the others get barely a fraction of attention. I think Clay Institute could increase the stake on the other four problems, so they get more exposure! 1 Wesley Deng Wesley Deng 1 year ago Good luck. This is the hardest way to make $1 million. 😆 84 David Wilkie David Wilkie 1 month ago Excellent Teaching presentation, must be well worth something? MömpfLP MömpfLP 1 year ago You could also say 1>1/2, 2>1/2, 3>1/2, 4>1/2, ... and therefore ζ(-1) > ∞ which leads to the result of ζ(-1) = ∞ But because we know that ζ(-1) = -1/12 could there be a non infinite number as a result of ζ(1) as well? Maybe we can't deal with infinity like that. 2 Bill Todd Bill Todd 1 year ago (edited) If one were to plot the sum of all positive integers (zeta of -1) it would zoom upwards but since it is supposed to reach, eventually, -1/12 . At what point does it cross zero? 1 John Lewis John Lewis 1 year ago You'd deserve a million if you understood the question let alone answer it. Jorge Amaral Jorge Amaral 1 year ago Further to my previous comment, as a joke: The traveling salesman problem is not difficult to solve, we just have slow computers :D 5 Milorad Menjic Milorad Menjic 1 year ago 25:01 but that is kind of logical proof not strictly mathematical because it is not possible to really add or do anything with infinite set because you will never come to the end. 2 The AI Epiphany The AI Epiphany 1 month ago (edited) 29:10 mathematician's equivalent of an overflow. 34:33 laughs in Terrence Tao. 42:45 I don't think that's the reason they cap it - it's simply because out of 8 billion people on Earth you were probably the 1st person that "needed" that piece of computation hahah. 45:30 I think if somebody was to find a solution to TSP that is P-time, that would be an amazing step forward as you can map many NP problems to TSP (if not all?) and thus you've pulled a whole class of NP-class problems into the P-class. 56:50 I don't think those 2 are the best stories we have - such people don't care about (immediate) fame nor about time as they work for eternity. I think it's the credit assignment in the academy at play here and his deep moral/ethical foundations. His proof leveraged Ricci flow and that whole theory came from Hamilton for whom he had huge respect, but they (Fields medal/millennium prize boards) didn't want to give any credit to Hamilton despite Grisha's request. That one seems far more plausible to me personally. Peter Gerdes Peter Gerdes 1 year ago Sorry to nitpick (love the videos here) but it's kinda misleading to say that our internet security depends on us having a bad understanding of prime numbers. That suggests that this security will break when we get a sufficently detailed understanding of prime numbers. And I understand why you said it that way (and I couldn't have done a talk like this as well as you...even presenting my own theorems I make lots of mistakes) but it's totally possible, indeed arguably likely, that the computational complexity of factorization is simply large (ohh and RSA is getting rarer these days and Diffie-Hellman/EC are getting more popular). John Tavers John Tavers 1 year ago the amount of work you'd have to do to solve any of these problems would be worth way more than 1 Million dollars. there are certainly easier ways to make that money. 3 Cycklist Cycklist 1 year ago I thought Perelman turned down the awards because he thought it was unfair for him to be considered the solver of Poincare's conjecture when he was 'just' building on the comprehensive work done by so many before him. jorge Valdivia jorge Valdivia 1 year ago You are super cool man thank you for this presentation 2 Ryan Murray Ryan Murray 1 year ago Fantastic video. Just followed Tom after watching this. 1 Edgar Carpenter Edgar Carpenter 1 year ago The example of the traveling salesman problem would not work, because the continental U.S.A. is about 3000 miles across, and the salesman wants to cross the country twice. So it would be 100 cities in under 7 or 8 or 9 thousand miles, not 4 thousand miles. British people are usually vague about how big the U.S. is! Edward Strinden Edward Strinden 1 year ago Do a part 2 with the other 3. 3 Ngahuia Ashby Ngahuia Ashby 9 months ago creating a numeral counting sysem were 3 can multiply into every other number at a higher point Viktor Mellgren Viktor Mellgren 1 year ago The one for google maps is not TSP. The destinations are in order, so its just an computation of n, not n! 3 Niranjan Hanasoge Niranjan Hanasoge 1 year ago 45:20 Whoa ... if I were to discover a polynomial-time algorithm to solve the Traveling Salesman Problem, I wouldn't win the Millennium Prize, but it would help you to win it? What kind of a scam is that! 😂 6 Tahmid T Tahmid T 1 year ago Well that was somewhat odd. I didn’t expect a mathematician to claim that 1+2+3... to be equal to -1/12, specially without at least mentioning analytical continuation. 17 Alpha Omega Alpha Omega 4 months ago If you add 1/2 forever; it will always equal 50% of infinity. The rule being the count can never be larger than one. The other side would not go beyond 66% of infinity. How could these numbers ever reach infinity? The real infinity is able to encompass that imaginary number given to it plus add all the previous numbers before it. Like: 100 plus all the previous whole numbers is equal to 550. However, true infinity can handle adding those decimal points - extending it further. 0 to 1 will always be out of reach. MIRAV PRAJAPAT MIRAV PRAJAPAT 1 year ago Dude is way too advanced. Super cool. 2 Anil Raghu Anil Raghu 1 year ago Poincare conjecture and Banach Tarski have something to do with spheres, 1 Vitringur Vitringur 1 year ago Wouldn't it make sense to just say "I can make this number as big as I want" rather than saying infinity isn't a number and you can't get to infinity yet continuing to speak of infinity as a number and saying you can reach it. Just saying "infinity means that no matter how big the number is I can make it even bigger" isn't that complicated. Paolo Lioy Paolo Lioy 7 months ago (edited) prime numbers are just a chaotic progression fully predictable !! just a little bit more complex than usual non 'interactive' ones .. Beach&BoardFan Beach&BoardFan 1 year ago P NP traveling salesman, and NS are super fun! 1 jen sen jen sen 1 year ago what about emmergent programs (bee or ant colony) which solves the TSP in less than exponential time (polynomial from what i read, not sure though)? 1 John Salkeld John Salkeld 1 year ago Should say the analytic continuation of this formula Kris Pucci Kris Pucci 9 months ago The problem regarding the Travelling Salesman problem is that the 100! is a brute force method. What about an actual mathematical proof? Would that be faster? Wojtek Skaba Wojtek Skaba 1 year ago TSP is to find the shortest route and it s not in NP. Go Clip Go Clip 7 months ago I was just watching Mr. Beast Video where a man walks home taking half a million-dollar by playing a game. Why does the proof that could change the course of history worst just 1 million dollars? I kinda feel sad. David Wilkie David Wilkie 1 month ago (edited) I my own experience with Mathematical terminology, (not good), the feeling of division by zero or infinity implied smooth continuity of operation, (Sigma implies adding discrete quantization), so now that I have words for it, (BUT, take no man's word for it, at RI), then the Observable Eternity-now Interval interpretation of this situation is that the Absolute Zero Kelvin i-reflection containment vanishing-into-no-thing out of the operational picture-plane containment states, "says" n = Infinity-Singularity Reciproction-recirculation. Euler's Conjecture e-Pi-i sync-duration connectivity, roots 1-0-infinity probability indicates the exclusivity of zero-infinity positioning of trivial non-location "outside" primary superposition existence. This challenge question is a contradiction of terminology, and the required context for QM condensation modulation=> measured Physics. Maybe Mathematical Disproof Methodology Philosophy would accept that. Ie Disproof is operationally = Proof, it's just natural Mathematical reasoning by reverse process. ("Show your working", every Teacher says) Saying the "Real" part is real is typical Quantum Logic, but the tricky part is i-reflection containment states of primary superposition connection calculation fields of "dark" implication. Fun to imagine. One half is "of the real-time whole ", another version of observable physical manifestation of transverse trancendental e-Pi superposition condensation, here-now-forever. Superposition Singularity in Black-body Superspin Modulation is the Eternity-now ONE-INFINITY time-timing sync-duration recirculation operation Interval, WYSIWYG pulse-evolution differentiates integrated metastability condensation. "Physics is Everything" you can identify.., that much is true. So the Hypothesis is more of a Riddle based in word meaning or theory-conjecture than physical computing of AM-FM time-timing continuous Actuality. How do you get more mathematically rigorous than identification of the Observable, Absolute Limit? The "Prize" is knowing you are completely embedded in metastable Unity.., and the Uncertainty Principle. andy530i andy530i 1 year ago Just give it to Neil Ferguson - I'm sure he'll work it out. 2 Manoj Bhakar PCM Manoj Bhakar PCM 1 year ago 57:00 may be he know a flaw in solution and thats why his soul don't accept the prize until he solves that flaw too. but as he is more expert in that problem. he already knew that that flaw is far from revealing by normal mathematicians. so as a "first breakthrough", he published his solution. as you goes deeper in science and maths, it makes you more human. no wonder he has pure soul. Jorge Amaral Jorge Amaral 1 year ago (edited) Salesman problem POSSIBLE workaround: -Get computer to check distances between all cities (number is MUCH less than all routes) -Pick the shortest distances that cover all cities -Order them by proximity It should be MUCH faster than check all combinations of routes. Side note, for google note that there is more than one road between two cities, so with 10 cities goolgle is already checking many dozen alternatives an picking the shortest/time efficient Just my two cents, hope this answer gets to Tom! Cheers from Portugal! Evolved Copper Evolved Copper 1 year ago (edited) Was this guy on Numberphile? Okay it is tattoo man, as i thought 32 duggydo duggydo 1 year ago I knew this was a bad video when he claimed the sum of all positive integers was -1/12. Mathologer provides an actual explanation. Numberphile (Padilla) and Tom clearly don’t understand it. 8 Evolved Copper Evolved Copper 1 year ago Yes, factorials scream at us Otto Salmenkivi Otto Salmenkivi 1 year ago I was only a little frustrated to do 10! in my head, just took a few minutes. 3628800. I hope its correct though. Niranjan Hanasoge Niranjan Hanasoge 1 year ago No, no, no ... Google Maps is not solving the Traveling Salesman Problem (TSP) when you add multiple destinations to your driving directions. It takes you from destination to destination in the order that you specified. It does not reorder your destinations, as would be done in TSP. (Destination, location, waypoint, node ... you know what I mean.) Even when choosing the optimal path between two consecutive destinations, it is not doing TSP. Instead, it is using one or more of several highly tractable graph search algorithms. (How do I know this? I googled it.) To solve the TSP on Google Maps, you have to use the Google Maps API, Google OR-Tools, and/or third-party tools. Having a Google employee in the audience is probably not a bad idea. 2 Hamiltonian Path on dodecahedron Hamiltonian Path on dodecahedron 1 year ago (edited) 14:10 did he say 1 is prime ? 8 George Slater George Slater 1 year ago I found them quite easy to explain. The Riemann has the answer/connected to all else, know one solves them all. I’ve shared the reasonings/ratios in model form and the written word, even explained the written word, yet nobody cares Solar Crystal Solar Crystal 1 year ago 1 million dollars in 2000 is about 1.5 million dollars in 2020. The Clay institute is making out like bandits Abhay Sharma Abhay Sharma 1 year ago IT would've have been epic if he had stripped for every problem. 8 Vladimir Louis Vladimir Louis 4 months ago How much do you get for writing down the equations and your name? BlueRedAndYellow BlueRedAndYellow 1 year ago I got all the proofs but they're too long 1 Manoj Bhakar PCM Manoj Bhakar PCM 1 year ago Manoj Bhakar PCM P vs NP --------------- check means -- check with human mind easy means -- as fast as human mind. solution ----+++---- 1. human mind does also work by following rule of physics and maths. so does the computer. 2. when you "check", for example when you check salesman problem, your mind quickly check the problem either by a good algorithm (which we need to extract from our mind by understanding how it works) or by applying all possible routes. 3. same can be done with the computer, if we make a computer as fast as human mind and as inteligente as human mind. 4. so what you can check fast, can also be solved fast. cactus loves balloons cactus loves balloons 1 year ago I got 99 problems. 2 Kula Ndifor Kula Ndifor 5 months ago Perelman: Keep it, and let me be. KAĞAN NASUHBEYOĞLU KAĞAN NASUHBEYOĞLU 1 year ago 👍 felopian kid felopian kid 1 year ago "If i keep adding this i will get to infinity". isn't it a point of infinity that you won't get there Korou Khundrakpam Korou Khundrakpam 1 year ago Tom, I'm a huge fan of your tatoos. Come to India some day. I'd love to meet you. Peace. 3 Type 1 Mum Type 1 Mum 1 year ago I got an E in GCSE maths. Why am I here? 2 Givtolmaknecrup Givtolmaknecrup 1 year ago I don’t care about that million dollars. When the amount of hotdogs in a packet is the same as that of buns... problem solved Paul Jameson Paul Jameson 1 year ago I have lost the plot after 15 minutes of this, sorry, if I donate some money for number 5 poster would that help. Sword of the morning Sword of the morning 6 months ago As a 46 year old woman trying to listen. For the first hypothesis. The Reichman one. If your replacing any number for a number that isn’t a prime isn’t that the answer? Ela Le chat Ela Le chat 10 months ago good luck for the proof of zeta(-1)=-1/12 Tony Tony Tony Tony 1 year ago S =0 mike thek mike thek 1 year ago I'm still trying to collect the money for a clock that woks onboard ships . Mm 3 Enrico Enrico 1 year ago One?? It's not a prime!! 1 H N Nagarathna H N Nagarathna 1 year ago This guy has his own yt channel he's crazy 1