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 namedropping: 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.



Music Credits:

Inspired by Kevin MacLeod
Link: https://incompetech.filmmusic.io/song...
License: http://creativecommons.org/licenses/b...

Wind Of The Rainforest Preview by Kevin MacLeod
Link: https://incompetech.filmmusic.io/song...
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Music: https://www.purpleplanet.com

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Image Credits:

Klein Bottle: Tttrung / CC BYSA (http://creativecommons.org/licenses/b...)
Zinc replacement Copper reaction: Qiang Fu / CC BY (https://creativecommons.org/licenses/...)
Pause Button: Fabián Alexis / CC BYSA (https://creativecommons.org/licenses/...)
Projective Space: Original: Mark.Howison at English Wikipedia This version: CheChe / CC BYSA (https://creativecommons.org/licenses/...)
Hirzebruch: Konrad Jacobs / CC BYSA 2.0 DE (https://creativecommons.org/licenses/...)
Hodge's House:Stephencdickson / CC BYSA (https://creativecommons.org/licenses/...)



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). https://web.ma.utexas.edu/users/dafr/....

3) Algebraic Topology, Allen Hatcher. http://pi.math.cornell.edu/~hatcher/A...

4) The Conversation, Arun Ram. https://theconversation.com/millenniu...

5) Wikipedia contributors, 'Hodge conjecture', Wikipedia, The Free Encyclopedia. https://en.wikipedia.org/wiki/Hodge_c...

6) user40276 (https://physics.stackexchange.com/use..., How algebraic geometry and motives appears in physics?, URL (version: 20131107): https://physics.stackexchange.com/q/8...

7) Fitzcarraldo (https://mathoverflow.net/users/12420/..., Why is the Hodge Conjecture so important?, URL (version: 20110203): https://mathoverflow.net/q/54197

8) Grothendieck, A. (1969), "Hodge's general conjecture is false for trivial reasons", Topology, 8 (3): 299–303



Background:

No Copyright Motion Graphics
Motion Graphics provided by https://www.youtubestock.com
YouTube Channel: https://goo.gl/aayJRf



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



Finally, special thanks to my younger brother, the cameraman.