Manifolds 1 | Introduction and Topology

[@brightsideofmaths]

Please do the quiz to check if you have understood the topic in this video: thebrightsideofmathematics.com/courses/manifolds/overview/

[@zachchairez4568]

NOW A MANIFOLDS SERIES?!? We’re not worthy! Thank you thank you thank you 🙏

[@NewDeal1917]

00:00 Intro 0:21 Course overview and intuition for manifolds 2:55 Metric spaces, open balls and neighborhood 4:51 Topology definition 7:41 Example. Indiscrete and discrete topology

[@harjotsingh8530]

Thanks! I think it would be hard to argue against this being the best math channel on youtube

[@brightsideofmaths]

Thank you very much :) I am really glad that I can help!

[@profjonb6944]

I've been studying this for a year independently. So excited to follow along and check my understanding!

[@narfwhals7843]

Thank you so much. I've been diving down the differential geometry rabbit hole and couldn't, for the life of me, make sense of the "open sets" that kept popping up. We define them ourselves when building the space! Suddenly a lot of things make sense...

[@ChaineYTXF]

I had the exact same issue. In some courses this is not well explained

[@lucaug10]

Wow, that was fast hahaha, so happy to see that the Manifolds course has already begun!

[@goblin5003]

Agreed!

[@tensorfeld295]

Her is an incomplete list of manifold-books: 'An Introduction to Manifolds' by Loring W. Tu (his aim is to calculate de Rham Cohomology of Manifolds) 'Vector Analysis' by Klaus Jänich (also available in German) 'Introduction to Smooth Manifolds', by John Lee

[@byronvega8298]

I'll add three of my favorites to this. 1) A geometric approach to differential forms by David Bachman 2) Introduction to tensor analysis and the Calculus of moving surfaces by Pavel Grinfeld 3) Manifolds, Tensors and Forms an introduction for Mathematicians and physicists by Paul Renteln

[@jansniezynski7348]

As a physicist specializing in general relativity I can't wait to see your videos to review my understanding! Are you planning some deeper dive in differential geometry?

[@brightsideofmaths]

Thank you very much :) That's the plan!

[@njitnom]

@@brightsideofmaths YEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAH LETSGOOOO BABYYYYYYYYYYYYYYY INTO THE ABYSS OF GEOMETRYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

[@botondkalocsai5322]

I think manifold focused topology education is very much missing from the worldwide acknowledged physicist education.

[@hyperduality2838]

Positive curvature is dual to negative curvature -- Gauss, Riemann geometry. Curvature or gravitation is therefore dual, gravitational energy is dual. Ellipsoids are dual to hyperboloids -- linear algebra, matrices -- Gilbert Strang. Gravitation is equivalent or dual (isomorphic) to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. You can start by watching this about mathematics:- kzbin.info/www/bejne/d6jFi5SKn72iY9U Deductive reasoning (analytic, rational) is dual to inductive reasoning (synthetic, empirical) -- Immanuel Kant. 'A priori' (before measurement, mathematics) is dual to 'a posteriori' (after measurement, physics) -- Immanuel Kant. Concepts are dual to percepts -- the mind duality of Immanuel Kant. Here are some physicists talking about duality (start at 1hour 12 minutes):- kzbin.info/www/bejne/Z17EgZmladChm80 and also kzbin.info/www/bejne/i3XQiGNqjKeEr6M Supremum (minimization) is dual to infimum (maximization) synthesizes the Riemann integral:- kzbin.info/www/bejne/qmmrmWppfd2DZ9E Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic. Integration (convergence, syntropy) is dual to differentiation (divergence, entropy). From a converging, convex (lens) or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics. According to the 2nd law of thermodynamics all observers have a syntropic perspective. My syntropy is your entropy and your syntropy is my entropy -- duality. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! "Always two there are" -- Yoda.

[@angelmendez-rivera351]

@@botondkalocsai5322I agree! I am glad to see others who can see this viewpoint. I have been saying this for years.

[@johnartzi5693]

I literally just started learning manifolds this is amazing

[@therealjordiano]

Thanks so much for this video series, really nice explanations and made easier to follow than a lotta notes i've encountered

[@brightsideofmaths]

Great to hear! And thanks for your support :)

[@felixliebl8324]

Danke sehr für diesen grandiosen Kurs! Ich hoffe doch, Du hast (oder bekommst bald) einen Lehrstuhl - wobei, wenn ich es mir recht überlege: Mit Videos erreichst Du wohl weit mehr angehende Mathematiker und Interessierte als jeder Dozent im Hörsaal! Mach einfach weiter so :)

[@StratosFair]

Yayyyyy, we got the series on manifolds, I'm excited for this one !

[@IgorVladK]

I think it would be helpful to dwell some on the definition of "open set" in the absence of metric. It seems like a delicate point to me, worth digging a bit deeper into.

[@saturdaysequalsyouth]

I agree. I'd even add that many words and terms shared by Topology (and abstract math in general) and common every day use need to be expounded on in great detail to avoid confusion and misunderstanding.

[@MatthewDaly]

One elementary example of a topology that cannot be a metric space is called the cofinite topology. Here, let X be any infinite set, and take T={U⊆X | U is empty or X\U is finite}. A little intro set theory should convince you that all of the topological axioms are satisfied, and yet there is no distance function on X that would generate T as the open sets.

[@jalepezo]

U know the math gets real when the teacher has a german accent

[@grantorino2325]

Indeed! (Just make sure to keep him safely away from his stupid sister, DeeDee.) 👱🏻‍♀️

[@MrOvipare]

Oh I've been waiting for this series! Cristal clear introduction!

[@brightsideofmaths]

Glad you like it!

[@flooreijkelboom1693]

It's here!!!! Amazing :D

[@ativjoshi1049]

Just when I was about to catch up to Functional Analysis😪. PS: A big thank you for providing free access to such high quality videos.

[@brightsideofmaths]

Glad you like them! :)

[@thomasyoung398]

I am going to learn manifolds this coming semester and your video helped me a lot. Thank you🥰

[@brightsideofmaths]

Glad I could help!

[@khalidmohammadjama]

New age new series good luck 👍

[@miriamstudyaccount8735]

Bell notifications are on! Very excited for this series, keep it up

[@batmanrobin6711]

Personal note (sorry about that): Informally, a topology describes how elements of a set relate spatially to each other. The same set can have different topologies. For instance, the real line, the complex plane, and the Cantor set can be thought of as the same set with different topologies. Formally, let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if: Both the empty set and X are elements of τ. Any union of elements of τ is an element of τ. Any intersection of finitely many elements of τ is an element of τ. If τ is a topology on X, then the pair (X, τ) is called a topological space. The notation Xτ may be used to denote a set X endowed with the particular topology τ. By definition, every topology is a π-system. The members of τ are called open sets in X. A subset of X is said to be closed if its complement is in τ (that is, its complement is open). A subset of X may be open, closed, both (a clopen set), or neither. The empty set and X itself are always both closed and open. An open subset of X which contains a point x is called an open neighborhood of x.

[@tensorfeld295]

Are you planning to describe vector fields and ordinary differential equations on manifolds as well? If so, you could combine vector fields and differential forms to get general tensor fields.

[@HelloWorlds__JTS]

It could be confusing at (5:45) when you say that for T to be open, the entire set X must be in T. This seems to contradict the requirement that all subsets of T be open, because obviously some elements in X must be boundary points. The resolution of this confusion is that, by definition, a topology includes the entire set X AND all open subsets of X. The entire set X isn't open, but it obviously must be included in the definition, according to what it means to be a topology on X.

[@brightsideofmaths]

I don't understand exactly what you mean. The whole set X cannot have any boundary points because there is no "outside".

[@HelloWorlds__JTS]

@@brightsideofmaths thanks for the clarification. I was confused about what it means for a set to be open and/or/nor closed -- I thought it had to do with the boundary. But then it helped to realize, motivated by your comment above and by your functional analysis video on open and closed sets (kzbin.info/www/bejne/iIrXdmNvmq2Yp6s), that a boundary only has meaning for subsets, and to be open or closed doesn't require consideration of a boundary at all. After this realization, I also came to understand why the entire set X, and the empty set, are each both closed and open.

[@brightsideofmaths]

@@HelloWorlds__JTS Great :) I am glad that my videos can help you!

[@punditgi]

Always wondered about manifolds. Thanks for these videos! 😃

[@brightsideofmaths]

Happy to help!

[@SphereofTime]

2:21 diff form, manifold, topology

[@danielantone6216]

you are great, this is the real Math

[@CristhianDebarros]

A question that arises to me, is to understand how a structure- $C^{\infty{},k}$ differentiable can be "stable" ? , Each structure $C^{\infty{} ,k}= \mathbb{P}^{4}$ 4-dim of manifolds , Since $B$ can be a Modl other than $A$ (or else the product tensor $A/B := A, B_{m} ) .... Donaldson-Thomas studied an idea of those models for a class of manifold conjectured by Calabi-Yau, in this general case $B_{m}$ is irreducible in A , and therefore $ A+ B_{m} := CY^{*} (X)$ proving how B and their respective differentiable-operators are stable "instances" of $A$ , in this case a compact of $CY$ is produced, if only if $A^{1}$ exists and is hypergeometric.... I think that this Differential-structure is "stable"

[@shirleymoon9934]

Thank you for the wonderful explanation :) I have a question however: let an element a ∈ the set X, than the set T= {∅, X, {a}} is a topology satisfying all the three conditions (if that's correct), so T is a topology in X. But isn't the singleton {a} a closed set in X?

[@shirleymoon9934]

But as the definition given, all elements in the topology should be open sets

[@shirleymoon9934]

one thing to add: X = the set of real numbers

[@zacharysmith4508]

This is probably pedantic but for your properties of a Topology @6:09 shouldn't 1-3 be written with subset notation since the null-set and underlying set X are sets and not elements? Sorry it's been three years since I've had to think about this. I'm not sure if writting these properties and how its being presented or what I'm saying they ought to be, matters.

[@brightsideofmaths]

They are sets and, on the same time, elements. Like subsets are elements of the power set.

[@zacharysmith4508]

@@brightsideofmaths I think my mind just wants to split hairs over something trivial. Anyway, thank you for the amazing material as always!

[@galileo3431]

2:37 "boundowy" 😂 Great video btw 🙏🏼

[@brightsideofmaths]

Keep watching for more crazy pronunciations of English words :D

[@sherifffruitfly]

good high level explanation of how topology comes from real analysis by abstracting out the "important" parts. following that idea one level further would be cool: an explicit explanation of why, while finite intersection closure is sufficient, for unions we need more: countable union closure.

[@erenoguz3297]

I have a question, why do we allow for only finite intersections. I have another question, why do we define topologies in terms of open sets and not closed sets or something else.

[@brightsideofmaths]

An equivalent definition with closed sets is also possible. Or even other ones. One just has to pick one that covers all properties one wants.

[@erenoguz3297]

@@brightsideofmaths Thanks for your response😊

[@lordlix6483]

Very clear! Are you planning to do a series on Category Theory as well?

[@zazinjozaza6193]

Did the term "open set" first come from the metric spaces or topologies?

[@brightsideofmaths]

You mean historically?

[@zazinjozaza6193]

@@brightsideofmaths Yes. It always seemed strange to me that elements of a topology were called open sets, so I am curious how and where the term originiated.

[@scollyer.tuition]

@@zazinjozaza6193 I think that the term came initially from metric spaces, the essential abstract characteristics of open sets were then agreed upon, and the term was retrofitted into topology to apply to any sets to which those characteristics applied.

[@botondkalocsai5322]

For the introduction, motivation part, searching the extrema of a function on a surface can be always done in the embedding euclidean space of the manifold with lagrange multipliers. The true motivation for the usage of topological spaces, manifolds in physics comes from general relativity and quantum field theory. In general relativity general relativity, it turns out that the vector space structure is insufficient to describe spacetime, and generalisation to pseudo-riemann manifold is needed, hence the requirement to generalise the differential calculus to manifolds. In QFT the great importance of the manifolds comes from continuous symmetries which are used to define physical quantities and described with Lie groups.

[@brightsideofmaths]

Sure! But I don't think that I should motivate the general concept immediately with advanced modern physics theories :)

[@utof]

1:18 i dont understand. Why can't we use calculus as usual? Can you provide an example how this falls apart? (Calc 1 student)

[@brightsideofmaths]

Calculus as usual would mean that you have a function f: R to R. We have to change the domain on the left for our problems here. Also f: R^n to R is not enough because the constraints are not included then.

[@utof]

@@brightsideofmaths Thank you for the answer! but what constraints are you talking about?

[@brightsideofmaths]

@@utof The constraints given by the sphere, for example. If you want a good overview, maybe the wikipedia article about "Lagrange multiplier" can help you.

[@utof]

@@brightsideofmaths Thanks a lot!!!!

[@agamanbanerjee9048]

I'm so excited, I can't thank you enough for this. :)

[@bobbybannerjee5156]

An excellent introduction to the basics of manifolds. May I know the name of the software/app with the help of which you are writing ✍️ on your computer 💻 screen?

[@brightsideofmaths]

In the description is a link for my website where you can find the information :)

[@de_oScar]

First example (from 7:50) It's my first real attempt at understanding topology and I don't understand how that fact ("all nontrivial subsets" of T={∅, X} are not open) follows / doesn't contradict "the elements of T are called open sets", and sadly "there are not many choices for the union and the intersection" to me is not trivial nor "simple". I roughly understand what "neighborhood of a point" is, if that helps anyone who'd kindly bother to explain. I don't understand either of the examples, but (b) is so beyond me that my head hurts when I try to formulate an appropriate question.

[@brightsideofmaths]

If T={∅, X}, then ∅ and X are the only open sets. All under subsets of X are not open. That's it. This is part (a). If you have trouble with intersections and unions, my start learning mathematics series might help: https:/tbsom.de/s/slm

[@de_oScar]

@@brightsideofmaths Thanks for a quick reply. How would no subsets of X be open, though? I assume we can almost freely choose X, like the set of reals or rationals.

[@brightsideofmaths]

@@de_oScar Yes, X is arbitrary but the notion "open" depends on the chosen topology.

[@d4rkmn643]

Why is a topology only defined as intersections of pairs of sets in T, while unions require an arbitrary amount of unions? Also, isnt it the same for it to be defined as if A,B are in T, then A U B is in T? Because if this the case we would be able to form arbitrary union the same way, right?

[@brightsideofmaths]

Finitely many intersections of open sets should stay open.

[@avadheshkumar1488]

Thank you Sir!

[@ChaineYTXF]

This looks to be a very nice series on manifolds. May I ask you what software you use? This would be perfect for me as a teacher😊

[@brightsideofmaths]

Thanks a lot! I have my tools listed here: tbsom.de/s/faq

[@atabaymahmudov8684]

Following this series

[@arijitpyne3435]

Please make public part 2, for it remains unavailable

[@brightsideofmaths]

Don't be too impatient :)

[@thedorantor]

So far, your lecture series on manifolds is brilliant and I already learned a lot from the uploaded videos! Do you perhaps have some kind of timeframe for me about when I can expect new videos to be uploaded and till how long or how many videos you will make for this course?

[@brightsideofmaths]

Thanks! I don't have a strict plan for uploads. Sorry.

[@thedorantor]

@@brightsideofmaths No problem! Thanks for making them!

[@batmanrobin6711]

Hi professor! Could you recommend a book very close to the structure of your course? A book that would be a good companion to your course...

[@brightsideofmaths]

Great suggestion! I put some books into the community channel. Take a look :)

[@mastershooter64]

awesomee!!! it's finally here! will you also be covering stuff like curvature of manifolds?

[@brightsideofmaths]

Yes, definitely! :)

[@mastershooter64]

@@brightsideofmaths thank you so much! how about stuff like the torsion tensor?

[@IlyasKhan-tc6pe]

Wow sir thanks keep it up

[@scollyer.tuition]

Very interested to see this. Are you planning to discuss Riemannian manifolds (eventually)?

[@brightsideofmaths]

Yeah, I definitely want to do that!

[@cristhiangalindo4800]

Well, I wanted to make an observation, when I was a doctoral student a question arose, how can we find a set of vector fields that are always tangent and finitely bounded? I thought 1- if I prove that a field-vectors in F is of the form F_{*}= F(x,y) is always tangent in \phi{}(x): F_{*}\to{} \mathcal{M } on the dual-base F_{*} , which is n integer (F\in{} n in the F-field of integers ) 2- if the basis is non-dual in \delta{} (x-y) "base-codual" then M is never a manifold with structure \phi{}(x) of a space of always tangent fields F_{*} . So if you want to prove that a vector-field is always tangent, use the idea of ​​the dual-bases of its "corresponding" vector space i

[@AbdulrahmanSOmar]

Do you have a textbook suggestion to read along with this course?

[@brightsideofmaths]

The books from Lee are very suited here.

[@javadnowroozi345]

I love your videos! You are amazing ❤

[@brightsideofmaths]

Thank you so much!!

[@chensun2427]

Thanks for the course

[@brightsideofmaths]

You are welcome :) And thanks for your support!

[@leventegyorgydeak1300]

How do we know that the elements of tau are open sets? Are they open by definition? Shouldnt tau satisfy an extra condition of it being an open set then? Although I am guessing this is what you meant, you even said that the elements of tau are open by definition, but you never explicitly said it or wrote it down, and I just want to make sure I understand

[@brightsideofmaths]

Yes, we call the elements of T "open sets". That's it.

[@leventegyorgydeak1300]

​@@brightsideofmaths If a closed set was part of T, it would also be called an open set? I don"t immediately see Why a closed set could not be a part of T based on these conditions. let X be R and T be P(R). T satisfies all the conditions: - {emptyset, R} are elements of P(R) - the intersection of any 2 real number sets will still only contain real numbers, so the intersection is also the element of P(R) - The union of all the possible subsets of T (which is P(R)) is R itself, which is an element of P(R) then take the [1;2] closed set. this is of course an element of P(R), so it is an element of T, but it is a closed set. Is this because this has nothing to do with the open set definition I learnt in real analysis, it is just simply called that? Great work by the way, love the videos!

[@brightsideofmaths]

First, closed is not the opposite of open. A set could be open and closed at the same time. Second, you are right. This here is a new notion/definition of "open" :)@@leventegyorgydeak1300

[@leventegyorgydeak1300]

@@brightsideofmaths Oh yes of course, what I meant is "not open" instead of "closed". But it is clear now, thanks for clarifying! And for about the 10th rewatch I think I also understood why it is called the same thing: it is an abstraction of the classical open-ness of a set right? The whole point is that the open-ness of the set can be defined with respect to a topology as we like.

[@adityagiri3600]

thank you for this! can you also do group theory in the future?

[@kim8u96]

Isn't the definition of topology the same as that of algebra in measure theory?

[@brightsideofmaths]

No, it isn't :)

[@Rupeshkumar-gr6nu]

Great explanations.

[@brightsideofmaths]

Glad you liked it

[@azeds]

I enjoy this work

[@rick4135]

Topology definition is very similar to a sigma algebra right???? It seems a sigma algebra can hold more sets, in example Borel signa algebra holds closed intervals and singletons. Can I say mesure theory generalize the concept of volume whereas topology generalize the concept of distance??? Thanks for this awesome material!!!!!!!!

[@brightsideofmaths]

Yeah, you could say that. However, I personally would rather say that topology generalises the concept of neighbourhoods.

[@rick4135]

@@brightsideofmaths Now that I think about… it us a better description to what I mentioned above. Thanks!!!!

[@ichkaodko7020]

jesus h. christ, manifold and topology is coming. yaaaaay.

[@aleksanderaksenov1363]

And how can we express the full powe set if sthe set is uncountable?

[@brightsideofmaths]

If X is not a finite set, the power will always be uncountable. We are used to infinite sets :)

[@ARBB1]

Great work!

[@brightsideofmaths]

Thank you! Cheers!

[@muthukamalan.m6316]

Hi sir, In order to understand UMAP i came across your manifold playlist, but I watched multivariate calculus and you earned my subscribe. could you help me herewhat topic should i need to learn UMAP

[@negarerfanian]

Thank youuuuuu!!!! sir, would you also do algebraic topology?

[@malawigw]

MANIFOLDS!!!

[@pinklady7184]

*WOWWW. YOO DA BEST AND BESTEST.* Thank you.

[@gustavomezzovilla7248]

Ah yes ... The amazing Calculus 6 course. Will u follow Analysis on Manifolds by munkres?

[@brightsideofmaths]

Sadly, I don't know this book.

[@gustavomezzovilla7248]

@@brightsideofmaths Its a really great book to achive the stokes theorem for general Manifolds, along the implicity and inverse function theorem's in the chapter 2.

[@aweebthatlovesmath4220]

Thank you!!

[@arghamazumder7718]

Thank you sir for this video. Can we expect a series of lectures on general topology Sir?

[@brightsideofmaths]

Yes, sure! However, not in a few weeks ;)

[@Jaylooker]

Discrete spaces have some applications in representation theory

[@anjishnu8643]

Any books at the intersection of deep learning and topology anyone might know of?

[@HelloWorlds__JTS]

If you don't already know about these, look into works by Bronstein et al., Welling et al., and Gunnar Carlsson et al. They and the respective groupoids associated with their subdisciplines are putting out probably the best learning resources. Carlsson is big on topological data analysis, and the others are doing work that relies on concepts from various areas of topology. There are many others...

[@יהודההרשקוביץ-ד5ר]

360-76?

[@brightsideofmaths]

What?

[@Hold_it]

Yay! 😊

[@samueldarenskiy6893]

so a topology is almost like sigma algebra w/o the complement criterion

[@brightsideofmaths]

Similar but different :)

[@aseelmathematics2778]

Great !!

[@moussa4031]

❤❤

[@fabiangn8022]

Gracia.s

[@narek323]

You sound like the German version of Isaac Arthur

[@brightsideofmaths]

I take that as a compliment :D

[@stevenzheng5459]

Topology; studying surfaces in reference to holes Bottomology; studying holes in reference to surfaces

[@smftrsddvjiou6443]

useless.

[@brightsideofmaths]

Einstein disagrees :D

[@oni8337]

isnt the closure of S the same as all its limit points?

[@brightsideofmaths]

This depends how you define "limit point".

[@oni8337]

@@brightsideofmaths Checked on wikipedia and the closure is actually the union of the set and its boundary as well as the union of the set and the set of all limit points, which is x in X such that for all neighbourhoods of x, U, (U\{x}) intersecting S is not an empty set, and not all points of closure are limit points because S may be the union of some open ball and an isolated point far away from the ball

[@brightsideofmaths]

@@oni8337 Still you should say what your definition of a limit point is :)

[@oni8337]

@@brightsideofmaths It is the one mentioned in your video for an accumulation point, x is a limit point if for all members of the topology containing x; all U in T where x is in U, The intersection U\{x} and S is non-empty. In other words U without x and S are not disjoint

[@brightsideofmaths]

@@oni8337 Thanks. I really prefer the term accumulation point for a lot of reasons. However, I wanted to be sure because you are commenting below the first video and not the second :)