Why Vector Bundles

Vector bundles are important mathematical objects arising in geometry and topology and include the Mobius band and tangent bundles as examples. In this video, we give an informal introduction to the topic aimed at giving a feel as to what these objects are in the topological case, and how how they help us understand topology.
There aren't many good references for the material in this playlist. Most treatments are much more advanced. These include Atiyah's "K-theory".

[@nguyendangson1989]

Best video for motivation to learn vector bundles. If you only read their definitions, you cannot understand why they have to do that. Don't know why, definitely you cannot learn them. I just hope all abstract mathematical definitions might be explained like that.

[@alainmarx3924]

Your videos are marvelous. Deep content explained as simple as possible (but not simpler) using example. I am watching the whole serie.

[@irshadayoob3720]

That was great!!!! Please make some formal lecture series on these geometric objects. Thanks!

[@MKWKezer]

Some of the comments here are hilarious, lol. Nobody can define all terms, take some basic classes/read some books to understand them. Great video, a lot of gems in here for sure! It surely would have taken me half a year to read somewhere that a suitable category of vector bundles is equivalent to a suitable category of finitely generated projective modules, if it were not for this video.

[@xenmaster0]

'Take some basic classes.' I have a degree in physics, and that doesn't change the fact that mathematics has become do complex that even specialists in one subfield can't understand another subfield. Your sneering comment is hilarious. As Chen Nin-Yang remarked, "There are two kinds of mathematics books: ones where you get lost after the first, and the ones where you get lost after the first paragraph." Why don't you tell Nobel laureate Yang to 'take some basic classes'?

[@MKWKezer]

@@xenmaster0 Hmm, I'm sorry if I offended you but I didn't direct the comment at you and apologize if I did offend you. You are right that mathematics is becoming more and more complex and that experts cannot understand research topics in other mathematical topics (without spending some considerable amount of time). However, there are also points where I need to disagree with you. - This is a video on pure mathematics, so it is logical that you don't know some of the terms taught in basic graduate math classes. (That is of course not a problem.) - Some comments are only made to give additional intuition, there is no need to understand every part - they are merely instructive for people who know about these things. Fractional ideals e.g. belong to algebraic number theory. Modules are taught in abstract algebra and well, you need to spend some time to appreciate projective modules, so just giving a definition would be no use. (Projective modules are absolutely not easy, at least at first.) For anyone knowing these, it is a very cool fact to give an equivalence of categories to certain projective modules, but it is not the end of the world if they don't. - You say "As the video goes on, more and more undefined terms enter. It soon becomes clear that nothing can be defined in higher mathematics, since each terms requires a seeming infinity of other terms to define it." Let me wholeheartedly disagree. On the other hand, it is simply true that mathematics is extremely difficult and requires an intense amount of time to understand. Of course, you need to put in a lot of time to gain an ok understanding of all of these intuitions. In elementary school, there was no way for the student to appreciate vector spaces - in the same fashion one has to study enough to appreciate these things. - I don't really know your Nobel laureate but this phrase seems to be only half joking. Math books (and I'm sure so are physics books) are difficult, so yes, you will get lost if you don't persevere and spend an enourmous amount of time. Being difficult is also different for everyone, what is difficult for Yang surely doesn't mean the same thing for you and I. And well, I'm sure Yang knows what modules are - if not, then I would really advise him to take some basic classes...

[@valerianmp]

The "undefined term" is basic knowledge for anyone who already take graduate-level algebra course, which is kinda reasonable background to assume anyway for anyone learning this

[@tim-701cca]

This video mainly provides the motivation of "why study vector bundles". So informal defs and facts are used and he indeed defines some terms formally in his other video. It is always assumed audience to have some math background. That's why he doesn't define all the terms.

[@sam-fb1ds]

thanks Daniel Chan, the presentation of so much knowledge was super!

[@VasilevArtem-g4u]

Thank you. A great intro to the topic

[@melodymolander3981]

This was an excellent overview and helped me so much! Thank you!!

[@jammasound]

Very nice, explained the topic very well

[@irmaksaglam3782]

Wonderful!! Thank you very much.

[@user_2793]

pardon my french, but holy fucking shit was this such a pleasure to watch

[@hyperduality2838]

Duality is two equivalent description of the same thing! Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung. Subgroups are dual to subfields -- the Galois correspondence. "Always two there are" -- Yoda.

[@Sumpydumpert]

Great video !

[@ガンヤック]

Great video🎉thank you❤❤

[@callmedeno]

You should get an award

[@Pygmygerbil88]

How are these differ with fiber bundles

[@madhub6531]

When Y= R and X= Circle, the tangent space to Y at p , T_pY= R and tangent space to X at p, T_pX= Straight line (Tangent line to circle at p). Then T_pY/T_p X is a line parallel to T_pX ? How can we identify this to a line perpendicular to T_pX ? Normal spaces N_pX are dual spaces of T_pX . But I could not get the identification in general.

[@xrhsthsuserxrhsths]

wow you are very good sir! Thank you!

[@MDNQ-ud1ty]

Would it not be better to say that "There are only an even number of zeros" vs "There are only an odd number of zeros" and then do the description of the classes of zero crossings describe an invariant of the manifold?

[@alipedram5720]

Thank you!

[@lyubomir.andreev]

Great adventure!

[@brendawilliams8062]

Thankyou

[@minchanjeong940]

This is fking awesome

[@MDNQ-ud1ty]

Meanwhile Mr. Best has 42874928795853242309872495 subscribers.

[@stilingiceland1403]

In computer instruction set architecture，there is a term called instruction bundle

[@xenmaster0]

Q: Why is V a set of tangent spaces, and not just a single space with a lot more dimensions? Q: What is a module? What is the difference twixt a projective module and an ordinary module? Q: Why is the set of all tangents to the circle called a "bundle" instead of a set? Q: The set of real vector spaces is supposed to have the same rank as the manifold, but isn't the set of tangents to the circle infinitely large, and therefore of different rank from the circle? Or does rank refer to the dimensionality of the vector space instead of its size?

[@bensculfor4630]

A: a vector bundle should be thought of as attaching a copy of the real vector space V = R^r to each point of a topological space. Hence, we have one tangent space at each point of the topological space. A: en.m.wikipedia.org/wiki/Module_(mathematics). Basically, it's like a vector space over a ring instead of a field. Projective and finitely-generated modules are defined there as well. A: we call it a "bundle" because it has extra structure on top of just being a set, in the same way that a group or vector space has extra structure on top of just being a set. A: as he says at the start of the video, the rank of the bundle is the dimension of the vector space.

[@xenmaster0]

Alas, many terms left undefined. Q: What is the difference twixt a topological space and an ordinary space? Is there one? Q: When you day "which varies continuously with p," what exactly varies with p? How does it vary? In magnitude? Phase? Direction? Order? Permutation? What v sub p = t dub p of x mean? In what sense does this equality operate? Shape? Set theory? Group theory? Abstract algebra? Category theory?

[@bensculfor4630]

A: "ordinary space" doesn't have a commonly accepted definition. A topological space is defined here: en.m.wikipedia.org/wiki/Topological_space A: the family of vector spaces vary. "Continuously" means that for a bundle of rank r on X, if you take an open set U in the space X, the bundle restricted to U is homeomorphic to U×V, where V is the r-dimensional real vector space. A: He defines it at the start of the video. V_p is the tangent space to X at p. They're equal as vector spaces.

[@xenmaster0]

Q: What is the difference twixt a finitely generated projective module and an infinitely generated projective module? What does the module project, and how? Q: What is a fractional ideal? As the video goes on, more and more undefined terms enter. It soon becomes clear that nothing can be defined in higher mathematics, since each terms requires a seeming infinity of other terms to define it. As the combinatorial explosion of terms and definitions increases without limit, memory gets overwhelmed, and it soon becomes clear that we can't live long enough merely to enumerate all the definitions, much less understand them.

[@bensculfor4630]

A: en.m.wikipedia.org/wiki/Finitely_generated_module. Basically, it means that every element of the module can be written as a linear combination of a fixed finite subset of the module. It's the module version of a finite-dimensional vector space. A: en.m.wikipedia.org/wiki/Fractional_ideal. Basically, an ideal which contains elements of the field of fractions, where the numerator is in an ideal. The number of definitions can be overwhelming when first learning a new mathematical topic, but it's clearly not an insurmountable task, since thousands of grad students and researchers can write textbooks and research papers (and youtube videos) about the topics. Just keep studying and you'll eventually get the hang of it. P.S. "Betwixt" sounds weird and archaic, "between" would be the more normal way of saying it.