A Quick Intro to Fiber Bundles (Hopf Fibration)

Fiber bundles are useful and interesting mathematical structures, with applications in quantum mechanics and other areas of math and physics, but sadly they are tragically underappreciated. In this video, I'd like to spread awareness about the beauty of fiber bundles.

#math #physics #quantum #topology #philosophy #trippy #science #geometry #Hopf

[logankennedy]

I’m an algebraic topology Ph.D. Student and I study homotopy and symplectic homologous groups of hyper spheres, so I’ve used the hopf fibration a lot. It’s really cool to see it in such a beautiful and visual way though!

[WindmillEntertainmentGames]

that ending monologue was really beautiful. great video

[iamnotalive]

the "monologoue" at the end gave me goosebumps

[zray]

Fiber bundles are the bane of my existence as a physicist. Judging by the amount of research I see using that concept, I would never call them underappreciated.

[a.l.]

10:43 Your end monologue is fantastic. It reminds me of Heidegger's approach to philosophy. Thank you.

[konstantinrebrov]

I feel like a wizard every time I watch your videos, and understanding what you're talking about.

[AA-gldr]

I love this and I am so thankful that you exist and decided to teach the world.

[InZiDes]

Fiber bundles (and the generalization) are everywhere in Algebraic Geometry. Is vastly appreciated.

[holomagic]

I think your channel is fantastic and I echo your closing sentiment. That understanding of mathematics is the way I try to get my students to understand it.

[swamihuman]

- WAXING PHILOSOPHICAL...
--- Math, not of space and time, but only passing through briefly.
--- We get a glimpse, then it's gone - yet it's always there!
--- Esoteric, enigmatic, arcane - perhaps only existing in the realm of Plato's "Forms".
--- Whatever the case, enjoy it while, and how you can...
- [Thx for the glimpse, Richard.]

[arnavshukla]

That monologue, just wow. I'm out of words to describe how elegantly this video was produced. Thanks Richard <3.

[mayabenowitz]

Most underappreciated piece of mathematics and arguably one of the *most* important pieces of mathematical physics!

[anthonybrakus]

I'm a self taught mathematics enthusiast. I think that the book that introduced me to fiber bundles and the Hopf fibration specifically was Dr. Roger Penrose's Road to Reality. It taught everything we know in physics starting with the Greeks to present with all the math. Nothing left out, what is normally covered in 10 books and 15 courses, all in one tome. That book is very comprehensive with almost no examples or deeper explanation. (Love and hate that about the book). It nearly broke me, especially trying to conceptualize high dimension complex spinorial tensors and what they mean geometrically. This vid was nice. Just enough info to get a student, new to the subject, into a fine pickle.😂
JK, I do kind of feel that profound confusion is the only way to prepare your mind for fiber bundles. Another related concept that is wonderfully criptic is the way fiber bundles are used in algebraic topology... The fundamental group, homotopy, homology, hole chasing in higher dimensional spaces. That sort of fun stuff. Anyway, thanks again 🙏🏾

[jeanklein]

The monologue in the end was very pretty

[yaotianji]

I can’t fully understand everything since I am only a third year math major student, but it is great to develop some intuition at this time, and thanks for providing such a beautiful video.

[Achrononmaster]

Much appreciated. The awesomeness of the universe shouldn't trouble the mathematician once they understand Gödel. Mathematics is a precise way to understand the world but incomplete. Poetry (or religion, philosophy, mysticism) can be a way to completely understand the world but very imprecise. Knowing what uncertainty and imprecision means is the appropriate humility.

[thinkhardthinkharder]

What a beautiful closing articulation of platonic ideals!

[leonstenutz]

❤ "... and yet there it is, on your screen, look at it go..." ❤

Galileo, Newton, Erdós, Leibnitz, Plato, Pythagoras ... they and others would all be mesmerized, delighted ... 😊

[ELECTRHERMIT]

I think we are moments away from someone creating a deep learning model of hopf fibration states and solving QCD and Singularity mysteries and then gravity. Something about the fibration being made with one circle rings true with me personally.

[V.squared]

I love the ending monologue. thank you.