Topology was barely born in the late 19th century, but that didn't stop Henri Poincaré from making what is essentially the first conjecture ever in the subject. And it wasn't any ordinary conjecture - it took a hundred years of mathematical development to solve it using ideas so novel that they were worth at least a million dollars.

Here I talk about the Poincaré Conjecture by introducing fundamental topological concepts like homeomorphisms and homology, and eventually talk about its solution through Ricci Flow.

Note: At 12:25, it's 3-Manifold, not 3D Manifold. I don't know what possessed me to say 3D.

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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...
License: http://creativecommons.org/licenses/b...

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

The Forest and the Trees by Kevin MacLeod
Link: https://incompetech.filmmusic.io/song...
License: http://creativecommons.org/licenses/b...

Music: https://www.purple-planet.com

http://www.bensound.com/royalty-free-...

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Image Credits:

Tesseract, 24-cell and Tetrahedral Compound: Robert Webb's Stella software is the creator of this image. A link to the website: http://www.software3d.com/Stella.php.CC BY-SA 3.0 (https://creativecommons.org/licenses/..., via Wikimedia Commons
Klien Bottle Homology: Steelpillow, CC BY-SA 4.0 (https://creativecommons.org/licenses/..., via Wikimedia Commons
John Stallings Photograph: George M. Bergman, CC BY-SA 4.0 (https://creativecommons.org/licenses/..., via Wikimedia Commons
Saddle Shape: Nicoguaro, CC BY 3.0 (https://creativecommons.org/licenses/..., via Wikimedia Commons

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Sources and Citations:

1) https://people.math.osu.edu/fiedorowi...
2) https://www.math.unl.edu/~mbrittenham...
3) Evelyn Lamb, 'What We Talk about When We Talk about Holes,' Scientific American (https://blogs.scientificamerican.com/...)
4) http://infoshako.sk.tsukuba.ac.jp/~ha...
5) enry (https://math.stackexchange.com/users/..., Why is $S_1$ said to "enclose" a 1-dimensional hole if the region it encloses is a (2-dimensional) disk?, URL (version: 2017-06-06): https://math.stackexchange.com/q/2311769
6) Wikipedia contributors, "Homology (mathematics)," Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/w/index.php?...

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Chapters:

00:00 - The Euler Characteristic
03:00 - Intro
03:10 - What Even is a Homeomorphism?
05:07 - The Euler Characteristic, Reprise
07:50 - Introducing Homology
09:20 - The need for a Fundamental Group
11:55 - The Conjecture
13:50 - MORE Dimensions
15:05 - Deforming Surfaces with Ricci Flow
17:02 - Ricci Flow 2.0
18:24 - The Million Dollars