What is...the Hodge conjecture?

Goal.
I would like to tell you a bit about my favorite theorems, ideas or concepts in mathematics and why I like them so much.
This time.
What is...the Hodge conjecture? Or: Topology and algebraic geometry.
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Slides.
www.dtubbenhauer.com/youtube.html
TeX files for the presentation.
github.com/dtubbenhauer/My-Te...
Thumbnail.
en.wikipedia.org/wiki/Orienta...
en.wikipedia.org/wiki/Kummer_...
Main discussion.
en.wikipedia.org/wiki/Hodge_c...
mathoverflow.net/questions/54...
burttotaro.wordpress.com/2012...
webusers.imj-prg.fr/~claire.v...
www.claymath.org/millennium/h...
www.redalyc.org/journal/5117/...
www.theguardian.com/science/b...
en.wikipedia.org/wiki/Millenn...
Background material.
en.wikipedia.org/wiki/Project...
en.wikipedia.org/wiki/Complex...
en.wikipedia.org/wiki/Homolog...)
en.wikipedia.org/wiki/Hodge_t...
en.wikipedia.org/wiki/Hodge_t...
en.wikipedia.org/wiki/Hodge_s...
en.wikipedia.org/wiki/De_Rham...
en.wikipedia.org/wiki/Weil_co...
en.wikipedia.org/wiki/Poincar...
en.wikipedia.org/wiki/Fundame...
en.wikipedia.org/wiki/Algebra...
Computer talk.
doc.sagemath.org/html/en/refe...
doc.sagemath.org/html/en/refe...
Pictures used.
en.wikipedia.org/wiki/Complex...
Picture from • What is...the cohomolo...
en.wikipedia.org/wiki/Kummer_...
en.wikipedia.org/wiki/Complex...
www.researchgate.net/publicat...
KZbin and co.
claymath.msri.org/hodgeconjecture.mov
• The Hodge Conjecture (...
• The Hodge Conjecture (...
• Geometers Abandoned 2,...
#algebra
#geometry
#mathematics

[@xyzct]

Oh my God, I absolutely LOVE your _favorites_ videos. These overviews are just so enjoyable and illuminating for a mathematics enthusiast like myself. Thank you so much!

[@VisualMath]

Thank you so much for the comment, that is very much appreciated 😘 I am glad that you are a math enthusiast just like me 🤣

[@JAYMOAP]

Your channel is criminally underrated mate

[@VisualMath]

Well, its a math channel :-)

[@lowerbound4803]

Sunday random scroll is quite good. Thanks😊😊

[@VisualMath]

🤗

[@user-vn7iy8uo4s]

Please sir i am a grade 11 sciences student in cameroon i have a suggestions on thé proof of thé Twins prime conjecture but i do not know what to dp

[@VisualMath]

I only have one word of advise: many people have worked on this and were hitting a wall. So its likely that your argument contains a flaw, double and triple check that first. If you then insist, then go for computer verification which might be the only way to convince people. In any case, good luck. P.S.: I go by they/them so "Sir" is not all that much appreciated 😅

[@RepTheoAndFriends]

Already back from Los Angeles? That's fast.

[@VisualMath]

I had an 3 hour delayed shitty flight, but I am already back 🥲 The slides were prepared in advance 🤫 But now I am curious: how did you know that I was visiting LA?

[@RepTheoAndFriends]

@@VisualMath I participated to some small degree online and watched your talk. I really liked your and Victors talk

[@VisualMath]

@@RepTheoAndFriends I see, excellent 😁

[@mastershooter64]

Pure mathematics is a pathway to many abilities some consider to be unnatural.

[@VisualMath]

@@mastershooter64 🤣

[@devrimturker]

I thought , it is trying to find a relationship between , " floppy" manifolds and " internal structured" algebraic varieties :)

[@VisualMath]

Jup, that is a way of saying it 😄

[@JAYMOAP]

🔥

[@VisualMath]

Thanks, you seem to have fun ;-)

[@JAYMOAP]

@@VisualMath definitely haha. Your channel is outstanding 👌

[@galletas67]

Im in algebra 1 wtf is a manifold

[@VisualMath]

You do know what a manifold is 😁: you just haven’t come across the name. They are usually introduced quite late. E.g. I first met manifolds in physics (even in classical mechanics), because we were always taking integrals over some very strange objects. Turns out these objects are called manifolds 🤔 Anyway, manifold = a geometric object so that every point has a neighborhood which is a disc (a n-dimensional disc for higher dimensions; here n=2). For example, your pair of pants is a manifold, and a patch on it would be an example of a disc neighborhood. The idea is essentially: A) Discs are easy. B) Manifold = we patch them together from discs. So, locally easy, but interesting global behavior. (Sometimes you read Eucledian space instead of disc but the idea is the same.) 1d examples are lines and circles. 2d examples are sphere, a torus=surface of a donut, etc. 3d examples would be a golf ball, a 3d sphere, etc. And sorry, the Hodge conjecture is so tremendously difficult to explain...😥

[@stevelam5898]

You mean on a classical or quantum computer?

[@VisualMath]

I was thinking plain binary here 😂 I am not sure if anyone has studied the Hodge conjecture with an eye on quantum computing. I would like to know now ☺