Geometers Abandoned 2,000 Year-Old Math. This Million-Dollar Problem was Born - Hodge Conjecture

The Hodge Conjecture is one of the deepest problems in analytic geometry and one of the seven Millennium Prize Problems worth a million dollars, offered by the Clay Mathematical Institute in 2000. It consists of drawing shapes known topological cycles on special surfaces called projective manifolds, and proposes that similar shapes known as algebraic cycles can be used to model them, provided that the rotation number is zero.
The Hodge Conjecture has been looked over by numerous mathematicians, some of them so famous that it sounds almost like name-dropping: Alexander Grothendieck, Micheal Atiyah, John Tate, and John Nash, among others. It was proposed by William Hodge in 1948 in his book Theory and Application of Harmonic Integrals and gained fame once he gave a seminar on it in Cambridge during the International Congress of Mathematicians in 1950.
In this video I attempt to give an intuitive picture of the Hodge Conjecture and also a small overview of Algebraic Geometry in general.
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Music Credits:
Inspired by Kevin MacLeod
Link: incompetech.filmmusic.io/song...
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Wind Of The Rainforest Preview by Kevin MacLeod
Link: incompetech.filmmusic.io/song...
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Music: www.purple-planet.com
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Image Credits:
Klein Bottle: Tttrung / CC BY-SA (creativecommons.org/licenses/b...)
Zinc replacement Copper reaction: Qiang Fu / CC BY (creativecommons.org/licenses/...)
Pause Button: Fabián Alexis / CC BY-SA (creativecommons.org/licenses/...)
Projective Space: Original: Mark.Howison at English Wikipedia This version: CheChe / CC BY-SA (creativecommons.org/licenses/...)
Hirzebruch: Konrad Jacobs / CC BY-SA 2.0 DE (creativecommons.org/licenses/...)
Hodge's House:Stephencdickson / CC BY-SA (creativecommons.org/licenses/...)
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Sources and Citations:
1) John Forbes Nash, Michael Th. Rassias, Open Problems in Mathematics.
2) Popular lecture on Hodge Conjecture by Dan Freed (University of Texas). web.ma.utexas.edu/users/dafr/....
3) Algebraic Topology, Allen Hatcher. pi.math.cornell.edu/~hatcher/A...
4) The Conversation, Arun Ram. theconversation.com/millenniu...
5) Wikipedia contributors, 'Hodge conjecture', Wikipedia, The Free Encyclopedia. en.wikipedia.org/wiki/Hodge_c...
6) user40276 (physics.stackexchange.com/use..., How algebraic geometry and motives appears in physics?, URL (version: 2013-11-07): physics.stackexchange.com/q/8...
7) Fitzcarraldo (mathoverflow.net/users/12420/..., Why is the Hodge Conjecture so important?, URL (version: 2011-02-03): mathoverflow.net/q/54197
8) Grothendieck, A. (1969), "Hodge's general conjecture is false for trivial reasons", Topology, 8 (3): 299-303
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Background:
No Copyright Motion Graphics
Motion Graphics provided by www.youtubestock.com
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Chapters:
0:00 The Magic of Coordinate Geometry
1:56 Intro
2:06 The End of Euclid
4:01 Poncelet's Infinite Space
6:14 Algebraic Cycles
9:22 Manifolds?
11:50 Framing the Hodge Conjecture
13:06 Recent Developments and Final Thoughts
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Finally, special thanks to my younger brother, the cameraman.

[@deepstariaenigmatica2601]

Please, do a series on Millenium prize problems. These videos are really educational!

[@kinertia4238]

I am, actually. I've already made videos on a few of them, and I'm actually purposely leaving Riemann and P vs NP for last - both of them already have exceptional videos here on KZbin. But I have one about the BSD Conjecture in the works. Look forward to it early next month!

[@deepstariaenigmatica2601]

@@kinertia4238 Thanks for these videos, man. Most of the videos of these topics are pure gibberish. You make it far more comprehensible. Keep up the good work. It's true that there are good videos on both of them but my unsolicited advice would be to present those videos in a way that is educationally unique to only your video. 👍🏽

[@aaronflores1106]

This problem is actually the hardest of them to understand so its totally possible at this point for him to do it.

[@charlesrosenbauer3135]

@@kinertia4238 When you do P v NP, it's definitely worth discussing the Baker-Gill-Solovay theorem, as well as the theorems about natural proofs and algebraization. Most of what has been proven about P v NP is that the proof techniques mathematicians try to use keep being shown to be incapable of solving it. I've seen videos on BGS, but the natural proof and algebraization theorems are much less well-known.

[@brendawilliams8062]

@@aaronflores1106it’s a mind bender. I once the arrows take to being diagonals then which way is the big bang

[@uhbayhue]

I've said this before, but I'm honestly baffled by your level of understanding at such a young age. You inspire me man!

[@kinertia4238]

Thanks for your support! I really appreciate it.

[@MrAlRats]

@Mohammed Mesum Hussain 091 From his moustache, I would guess that he is 14.

[@Grizzly01]

@Mohammed Mesum Hussain 091 57

[@alexk.7064]

The video is fascinating, but one of the things that stuck with me is how long it took the world of maths to start using graphs for studying equations (and functions in general). Without graphs I don't think I would have much of an intuitive understanding of even basic concepts of equations. Well done Descartes.

[@santerisatama5409]

Graphs are good, coordinate systems with real line metric not.

[@swapnilshrivastava116]

Please never stop making these videos.. You are amazing.

[@swapnilshrivastava116]

I loved mathematics ever since I first got my eyes on algebra and conic sections. Its been a long time since I have completed my education but I am still fascinated by mathematics. Can you suggest some video library where I can watch and learn about everything there is in the field of mathematics? it's a treasure for me to find

[@kinertia4238]

Numberphile! Numberphile is your friend. They have thousands of videos featuring professional mathematicians explaining interesting concepts. You can also check out 3Blue1Brown, or if you want to LEARN math then you can watch MIT OCW videos. Start with this video and go down the rabbit hole: kzbin.info/www/bejne/rV6sZ4uKi7-Lrdk

[@NoobMaster69Patil-vd6ht]

I am feeling like i literally came to my home despite being in home now. So smooth and lovable talk!! I'm having hard time in university that i am not feeling attached to its classes, teachers and atmosphere at all. I want this kind of atmosphere to study that i get through your videos. 💕

[@colinbrash]

So glad I found your channel, this is amazing stuff. You explain complex mathematical ideas and problems in an intuitive way, without sacrificing the important pieces. And you also tell a story, something that is not often done or done well by mathematicians. Thank you for this channel, I hope it gains a lot more attention. This is beautiful work, and I can see it being a huge inspiration for many to five deeper into the mysteries of mathematics.

[@benjaminangel5601]

Such a great video! You’ve put together the best explanation I’ve seen on the hodge conjecture

[@oofusmcdoofus]

You never let me expectations down. Amazing work.

[@shreyasjv4877]

Cool vid! Note to everyone: Michael Atiyah passed away last year and is a brilliant mathematician. Everybody should read about him!

[@shreyasjv4877]

en.m.wikipedia.org/wiki/Michael_Atiyah

[@kalebmark2908]

I’m floored at the quality of these videos. Amazing!

[@alexey5805]

I rarely leave comments but your work is amazing. Thanks for video!

[@ShivamPandey-oe3cj]

Amazing video.Keep posting videos.Where are you from and what are you studying currently and from where? What is the topic of the next video?

[@brendawilliams8062]

Brilliantly presented Thankyou.

[@nerkulec]

This is so good! Please keep it up!

[@domc3743]

i love your videos so much, never stop

[@jonnywilliams5326]

Fantastic video 🎉 Thanks for sharing.

[@F.E.Terman]

Looks interesting. But Please, please provide subtitles. Or turn down that very loud foreground 'music'. Preferably both. 😮 I'd certainly try again if you did. Thanks!

[@marcosainte6114]

HUMBLY SPEAKING...I ABSOLUTELY LOVE YOU.

[@abhisheksoni9774]

Awesome and inspiring 👏 Keep it up

[@randyhelzerman]

Wonderful intro to the problem. Let see if I can't make myself a million dollars *chuckle*

[@pabloraindogarcia8107]

Lmao

[@naveen.v4734]

Your videos are so good , you earned yourself a subscriber

[@user-qg8nl6sg5g]

Please, put the subtitles as I have hearing problems thank you in advance

[@ExhaustedPenguin]

You have a talent for explaining

[@oofusmcdoofus]

Youre back! Nice!

[@AllanKobelansky]

Seriously good presentation. 2.75k subscribers ... c’mon people. Show some love.

[@redaabakhti768]

thank you so much man keep up the quality content

[@kinertia4238]

Thanks, will do!

[@nachiketakumar9645]

Please make more videos on theoretical physics and Mathematics

[@nishashekhawat6347]

Awesome work man

[@martinstubs6203]

"Two lines intersecting at infinity" would imply that infinity was a number, which it isn't. So you can just forget about all this.

[@jayantapaulpersonal2]

You are nice teacher........ Wishing more subscribers..... ❤

[@simonjoseph6938]

this chap is a genius at explaining

[@elmastrolazo]

Really good video!!!!

[@stevelam5898]

Nice video pal.

[@Gdhillon333]

Try to add subtitles if possible. Your content is great

[@matti1610]

Awesome videos!

[@2009joseastorga]

Great video dude

[@vybhav345]

Good going brother. Can me make more videos about other Millenium problems?

[@rumansaha6910]

I love your vedios...can you say me which subject I need to study to understand the Hodge conjunct problem...I searched it also in Google but I couldnot find the answer.....plz help me...plz plz🙏🏻🙏🏻🙏🏻 Thank you.. love you... and plz plz

[@josephoyek6574]

Take a shot everytime he says *TOPOLOGICAL*

[@harriehausenman8623]

great video! 🤗

[@RCrosbyLyles]

Nice work. Rock on!

[@Krishnafied]

You are brilliant!

[@dylanparker130]

really interesting!

[@BleachWizz]

Awesome video

[@reinerwilhelms-tricarico344]

Even though I understood every word, I still don't have the foggiest idea what the Hodge conjecture is.

[@KaliFissure]

Great video. If the universe is infinite and yet contained/constrained does that mean that parallel lines ONLY touch at infinity but not before? That they converge (at both ends) but that the convergence is infinitesimal and so it is only at infinity where they actually meet.

[@epicmorphism2240]

the universe isnt infinite

[@imaduddinalawiy3426]

please also include subtitles, great contents anyways 👍

[@faruhjmishenkopetrovich4011]

I wish you good luck!

[@rohitchatterjee2327]

Nice video guy

[@JeanSarfati]

Thank you.

[@stipepavic843]

niceee, subbed!

[@mueezadam8438]

Subbed!

[@gekkkoincroe]

I watch your videoes but i understand nothing , i like it that way i don't want it to be overly simplified,.

[@steveneiffel8227]

I think it is a misunderstanding that connecting math fields is "just" something fancy. Often breakthroughs follow insights gained from different fields. So having the right number of analysis tools then is fundamental for breakthroughs.

[@dannywatts6614]

can you repeat the part of the stuff where you said all about the things?

[@marioabrantes4932]

Very good

[@anamarijavego6688]

Hi, I have a small suggestion I would like to give you. your content is amazing, but perhaps when you're standing outside we can hear part of the wind blowing and some nature sounds, which makes your voice less clear. My suggestion would be to perhaps have some small microphone attached to your shirt..? Would make it way clearer I think

[@kinertia4238]

Thanks for commenting! I was aware of this problem, so I've changed the location and microphone in all my new videos. I won't be at the same place.

[@alistairaccordion]

What i would like to know is who is working on this at the moment. I guess it a step at at time, but I do wonder if someone like Peter Scholze might work on this, as his field is algebraic geometry. Be exciting to know.

[@franklinlingga5491]

a lot of people are working on this, since 2000. But, it is not easy. It is more than puzzle, it is the formation of the world to be understood if human can solve it

[@reinerwilhelms-tricarico344]

Very nice and interesting. You talk a little bit too fast though for my taste. In fact, you could have spent more time on the more complicated 2nd half. Now I need an ordered list of books or articles that i can traverse in one run to really understand all of this - without getting stuck in the first rabbit hole ;-) Thanks for the sources and citations. That may be handy.

[@matiasdanieltrapagliamansi3109]

thanks !

[@KStarGamer_]

At 8:16 Isn't that the absolute value of x^3 - 8 not the modulus?

[@kinertia4238]

Yes, you're right. In India, at least, where I live, we often use the terms 'modulus' and 'absolute value' interchangeably (for the quantity represented as |x|). I'll make sure to use the more universal terminology from next time.

[@Grizzly01]

@@kinertia4238 Don't sweat it. 'Modulus' and 'absolute value' are synonymous terms in lots of places.

[@saiyan_blood__8492]

Very good ending...😀

[@KaliFissure]

The inversion/eversion of the circle is the best model for our universe.

[@josephmarshall2030]

Thanks friend your very bright, do you attend MIT?😊

[@frankthommessen1382]

Mathematicians at Dunkin Doughnuts be like

[@dharmendrasen8522]

Kya ye video mujhe hindi me mil skta hai

[@5wplush243]

Idk how but your editing seems lacklustre in the transitions. You shd work on fine-tuning it. Moreover, what’s the pt abt grothendick’s( I apologise for the misspelling) work in string theory doing here. I can’t seem to figure out the correlation. Care to elaborate? And while your vid is brilliant you might wanna be a bit careful when dealing with topology and other topics laymen may not understand. I can freely say I did not understand it at first glance. That’s the part you have to go real slow at if you want everybody to understand! A final request: could you please point to some papers and your sources to give additional reading material? Thx so much and keep up the good work

[@5wplush243]

Apologies. Just realised you did put your sources. Any reading material( links or books)?

[@kinertia4238]

A few of the sources are textbooks (Hatcher's Algebraic Topology, for example, but they are not meant for the layperson); you can also try to read the John Nash book. IIRC then Matt Parker's book 'Things to make and do in the fourth dimension' also has a good chapter on topology.

[@kinertia4238]

I agree with the editing part - I realized that the camera had turned on a bit too late a few times, but that was after I had finished recording, and was unable to fix it because of the Corona lockdown. I added the Grothendieck part because I had mentioned the fact that the Hodge conjecture was part of his Theory of Motives which helps in modelling the behaviour of strings (it's the Physics Stack Exchange source). As for your last point, I'll admit I had a great amount of difficulty in this particular topic because I honestly believe there is no simple explanation which can be given for the Hodge Conjecture, it's just too abstract. I will be covering topology in a future video (it's fifth or sixth in the line so you can expect it by June-ish) so I'll try to better it with more intuitive supporting animations. Thanks for the feedback! I'll keep this comment in mind while shooting the next video.

[@alecapin]

14:26 Hail Hugh Mongus!

[@joyboricua3721]

@0:59 "Out of nothing he created a strange new universe"... I disagree with this statement. I'm already about to drift away from this video; but I'm glad I didn't. It's a good video, regardless.

[@harshchuphal3889]

Liked it.

[@Moreoverover]

Aee Poncelet's ideas not debated? I personally do not believe the lines meet at infinity, since the intersection stops exactly when it reaches 90°.

[@kinertia4238]

It's not a conclusion, it's a definition. Mathematicians define parallel lines to meet at infinity under certain circumstances - it's not always an assumption you keep, though. For instance, parallel lines do not meet in R^2, the Euclidean plane. RP^2 is an extention of the Euclidean plane where such lines are _defined_ to meet at Infinity. You could philosophically argue that such a construction is not 'truly' possible, but if that's the case then the fifth postulate is the least of your worries - CP^2 is 4-dimensional, for instance, which is not true in the real world because it has only 3 dimensions.

[@Moreoverover]

@@kinertia4238 Well the world is 4 dimensional as Einstein showed, but the world could also be a 2D projection. Why is it useful to define such spaces when they have such shaky philosophical underpinnings (quantum mech. is pretty shaky too so maybe it doesn't matter?)? Has there been any real world benefit to this RP^2 space?

[@KaliFissure]

Also the natural torus, without any added values, is the so called “degenerate” torus which has apparently a flat surface but the grid density changes. Remind you of a universe? Visibly flat but the manifold (space) varies in density (gravity). There are also singularities at each pole where grid becomes solid. There is interior “sphere” with identical but inverted character (antimatter universe) . What physicists get wrong is that a neutron star is aggregation of points, plural, trying to become singular. Thus white holes will not be huge spewing maws but singular points (neutrons) diffuse and dispersed across the surface of the universe, emerging at lowest energy locations, expanding from point to larger than point. Neutrons emerging and decaying into hydrogen. From volume to larger volume.

[@marcosainte6114]

The sudden change in pattern-YOU ALL KNOW I EXIST.

[@EricMartinez-dg2lu]

Kinda like inverting on a plan, one side 360° negative - or + positive; what we have is a self licking icecream cone ○>

[@omargaber3122]

Please add automatic translation.

[@santerisatama5409]

Geometers did not abandon continuous geometry. Formalists and their axiomatic set theory did.

[@SoulReap199]

Great video although I'm too dumb to understand this

[@alejandrodeharo9509]

add subtitles, please

[@maciej12345678]

go faster 12:16 if you go faster on most complicated topic you are BRAIN but slow on basic yeah that right

[@arjun_garva]

Where were you ?

[@kinertia4238]

I just graduated high school, so I was in 12th grade the previous year - had no time at all to make videos.

[@arjun_garva]

@@kinertia4238 keep the good work

[@deepstariaenigmatica2601]

@@kinertia4238 Wait, 12th grade is considered high school where you live??

[@benitomussolini823]

@@deepstariaenigmatica2601 That is fairly common in current and former British colonies.

[@deepstariaenigmatica2601]

@@benitomussolini823 So, this is generalized everywhere in the US?

[@justaguywhoisanidiot159]

This is similar to veritasium's video

[@SirFlickka]

−2(1a,i,−x)=(−2a,−2i,2x)

[@kenichimori8533]

Chess character King is zeta function. char_king

[@dcterr1]

Interesting, but I think I'll pass on trying to win $1M from proving this conjecture. Topology was never my strong point in math I'm afraid.

[@user-ff8qw2ry6u]

Egyptian mathematics. Sa 1:11

[@elchicotony8688]

Maldita sea ubiera escogido un tema más fácil de explicar en mi tarea ;v

[@Pete-Logos]

666!! (I've never been one of those people to yell out "1st" or whatever but it feels kind of cool being the 666-th like.) Damn...I'm one of those guys now, huh? 🙄

[@goedelite]

People who speak English with an accent should speak slowly, very slowly, and - if possible - be assisted by sub-titling.

[@wdobni]

its just bullchyt ... its like seriously studying the genetics and chromosomes and natural history of unicorns, imaginary beasts that don't exist. If you say that 2 parallel lines eventually meet somewhere in the far far distance then they weren't parallel lines....they were lines that were almost parallel.

[@wdobni]

there are only 3 dimensions of space.....you can study 8 or 11 dimensions if you want to, but until somebody demonstrates an object that actually exists in reality and which occupies more than 3 dimensions of space then all your studies are no better than ancient astrology, the study of nuances or objects or fairies or elves that don't exist except as delusions of the mind.

[@Hermetics]

Stop thinking in straight lines dude and consider curves as primordial, while angles hide numbers truly. Also consider the 0|0 as the observer of mathematics as magnetic polarity in the physical plane, then you can venture further into coloring the 10 single-digit numbers as uniting logic with the imagination (COMPLEX numbers for idiots).

[@dsm5d723]

The problem with classing shapes in topology with a hole number is that 3D read-write is a string-loop phenomenon, and writing and folding dynamically builds crystal structures with a predictable pattern of energy-matter reficiation geometrically. Problem: Wilczek had a 5D metric, or binary compressed cycle, and it was tossed for "irrational" eigenvectors, values and states. That was for Time Crystals in 2004, and he got a Nobel for being wrong.

[@marcosainte6114]

I AM GOD.

[@gekkkoincroe]

Are you Indian ?

[@gekkkoincroe]

I think there is a verb for throwing something out of window

[@gekkkoincroe]

Can't you tell?

[@gekkkoincroe]

No i forgot it

[@gekkkoincroe]

Could have just googled it ?

[@gekkkoincroe]

I'll forget about the video is i do that

[@dsm5d723]

I had the Ramanujan flash of Light, and with NO literate math training. Mental arithmetic for life I have always done quick in my head, abstractions and negations in integers were always irrational to me as a child prodigy gone wrong. Before I watch, and this is a good channel, topology to an ancient was this: The human mind has a geometry of sense inputs, the 7 Gates of Thebes. As light, sound smell, taste and tactile inputs "complexify" in the brain, consciousness is the result. The maps need inputs to render the territory when it is dynamic. For your own SPIRITUAL edification (you are too smart for your own good as is), please look at NJWildberger's stuff on Ancient mathematics. They had a more discreet and engineering based approach. I think the sexigesimal system of Old Babylon is the most elegant thing I have ever seen. Not better, but the mathematical poetry of a culture. From the Greek to make. Clay is about earning. Pay up or see the wrath of God exacted. From the ATP Bank. As the Descarte of the shopping cart, he did a very French thing. He snuck in the idea of thinking inside the Laplacian box. That is what I call Big Box Calculus. Leibniz had the ring-many worlds intuition, but log scaling to a sphere of gravitational influence was the baggage of a surrender monkey. Let Them Compute CHEESE! Then we can make the cake. Ahem, quantum insanity. Lensing, curvature and the physics problems of the real world are not in a computer. If I want, ECE is done in a few weeks. You could probably crack it with my help in a week. TALKING about what the math is saying is all that it takes. I am the Y2K Bug; Clay is not a mystery, it is a swindle. $6 million is pennies compared to what I am owed.

[@TheAAZSD]

Great video!