Algebraic Topology 0: Cell Complexes
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- Опубликовано: 28 сен 2024
- Playlist: • Algebraic Topology
How do we build a space? Topics covered include gluing diagrams for torus and 2-holed torus (and more holes), Cell Complexes (also called CW complexes), and operations on spaces including product space , quotient space, and wedge product of spaces.
Presented by Anthony Bosman, PhD.
Learn more about math at Andrews University: www.andrews.ed...
In this course we are following Hatcher, Algebraic Topology: pi.math.cornel...
Thanks Andrew! I like to pause the video when you ask a question and see if I can get it right- which I do about half the time!
Great practice!
I'm just on a course of topics in topology right now, where we are learning Algebraic Topology, so much content, definitions and notation that I cant understand in the course are said to be "obvious" or "trivial". This playlist is so great that it starts with the basics, now I can understand better. Really apreciated.
I really hope this particular instructor has more math topics he covered !!! Amazing instructor
This is easier than I thought it would be. I'm prepared, thank you.
He has a special relationship with the book. That’s completely understandable. 😅
This is amazing, I've tried learning algebraic topology so many times, but this might just be the one to finally get me through it!
I am new to AT - this was a really great lecture on cell complexes, very clear also super interesting!
Thanks a lot Andrew for sharing with us these wonderful lectures !
What great teacher you are❤️Thank you for this lesson 😊👍
just a humble query , will the entire series of algebraic topology would be uploaded in this channel ?
Yes! Expect a lecture per week - typically uploaded on Wednesday evening or Thursday.
please, where are the rest of these lectures
We'll upload a lecture each week - typically on Wednesday evening or Thursday.
excellent explanation!
wow ! great lecture.
Very motivating lecture!!! 😻😻😻😻😻😻
I watched them the second time to see what I missed. Still enjoyable to watch!!! You are a GREAT TEACHER. 🙏🙏
Good lecturer
Why the RHS solid torus behavior while filling space around the LHS torus (compactification 1h.02m ) resembles the shape of the magnetic field of a coil ?
how do i get the whole lecture series?
We'll upload a lecture each week - typically on Wednesday evening or Thursday.
@@MathatAndrews where is de Rham cohomology etc.? What will be the final lecture? Where is syllabus?
@@Sidionian We are following Hatcher's text (linked in the video description) fairly closely. In the upcoming weeks, we will begin cohomology (Chapter 3).
Algebra is from Arabic aljaber which means to mend
just wanna say topos means place
Seems so clear
Great lectures. Just one doubt, at 41:13 you showed that the RP1 is the same as S1, but their fundamental groups are different, one being Z/2Z and the other is Z, so how are they the same??
RP1 is in fact diffeomorphic to S1. The first fundamental group of RP1 is Z, not Z/2Z. Perhaps you are thinking of the first fundamental group of RP2, which is indeed Z/2Z.
57:50 the boundary of S^1 isnt empty tho... it's S^1 right? The circle just consists of boundary points i think
Otherwise, a very insightful lecture! You're an extremly good teacher
Ah! You are thinking of S^1 as living inside of another space, such as the plane, in which case the boundary of S^1 with the plane is, indeed, all of S^1. However, here I was discussing the boundary of S^1 as a manifold, in which case the manifold S^1 has no boundary. This is in contrast manifolds with a boundary, such as a line segment which has as its boundary the two end points. I hope that helps!
Tried mapping a function on S1 to
Yet to convince me this has validity beyond what we can visualise in upto 4dim. n and m spaces interacting seems like a topological fiction.
Higher dimensions is a lie propagated by Big-Math to keep mathematicians employed.
Not knowlegible enough to have many examples but understand that pure maths in history did throw up work which became of practical use.
@@MathatAndrews dropped maths for many years so now loving what you guys have been working on during that time.
@@MathatAndrews we can visualise a fourth spacial dimension by thinking of how two dim slices change over time. Computers can change 4dim slices in time in extension of 3dim.
Basically tho these are only discrete iterations just like computer engines that need a clock rate of repeated instructions.
2:24
Presenting oneself in such a careless attire is disrespectful. The knowledge you possess should not lead you to believe that you can disregard social constraints.
The knowledge YOU possess is only obfuscated by your blind adherence to said "social constraints," as well as your self-righteous commitment to enforce this adherence onto others. The professor can present however he pleases.
What "social constraints" are being disregarded? I am not aware of any regulations stating that professors must dress in any certain way.
Tell me you haven’t been inside of a university in the last 60 years without telling me you haven’t been inside of a university in the last 60 years.
@@iangreenhoe6611 I'm in a top university in my country (which is a top country in math) and rightaway I noticed it. And I'm not someone particularly judgmental and I'm always being skeptical on social norms. But washing one's hair is really not that hard for what it gives in return, it just makes the day brighter for everyone. You can have a superman shirt if you want that may be understandable I guess, but at least wash your hair. I know how one can be tempted to despise everything outside of math when we're really into it (you may not even realize your hair do look greasy as you don't look at them and noone tells you), but that's a mistake actually, and it's not mean to point it out
And so you know, most math researcher do make an effort on their looks. Like really, they look good most of the time. So idk who's not put a foot in a uni here. Actually I had one math professor who had poor hygiene and people would talk in his back for it
@@holomurphy22bro please be quiet. No one cares what he’s wearing. We are here for the lecture and the knowledge. Stop judging people
Wow, someone actually trying to explain high level maths, instead of just speaking above the students and not really giving a crap whether they're following or not. Wonderful!
I dunno. The guy is scribbling on the blackboard in a large echoing chamber so we hear his voice four times, out of his mouth and then coming back off three walls. There's some give and take with students, "Help me out..." etc., so what we're getting here is a performance, not a straight commentary on or explication of the text.
I guess I wonder what the hell you've been smoking, RedBar.
My feeling is that RUclips could be a useful way of conveying teaching -- and pointing a camera at lecturers shouting in classrooms is not one of those ways.
eachers need to learn that teaching students in a classroom and making a RUclips program are two very different activities; you don't get the second without any effort by letting somebody point a camera at you doing the first.
It really, really is a different activity. You actually have to do the work of making a video.
Surprise!
@@TheDavidlloydjones What have *you* been smoking? The first half of your comment seems AI generated and the second half seems delusional.
for future reference:
21:35 finishes review of sticking the edges of polygons together to make surfaces
22:00 how to build a cell complex
46:00 euler characteristic of RPn, calculated by building it up as a cell complex
46:33 product of two complexes
54:25 boundary of a product
1:03:15 quotients
Thanks!
@@MathatAndrews
Yup: He's given you the table of contents for the video you need to make.
We're very lucky to have a whole algebraic topology course on here for free. Just started to go through Hatcher's book myself, these lectures are great.
Yeah I'm grateful for this. This course is actually easier than I thought it would be. Interested in how Algebraic topology 2 would be. Maybe a bit tougher. Have you taken Topology 2? If so how is it?
I’m not sure how the courses work in the US, but the topology courses I’ve taken have only covered up to things like the fundamental group and the classification theorem, albeit in depth. Beyond that would be stuff I haven’t taken courses on. Actually I was reading Hatcher to get a head start on that stuff for next year.
@@yeast4529 The program for a PhD mathematics here in the states typically requires you to take Topology 1 and 2. Okay; I haven't gotten too much into fundamental group as of yet, but I do know it deals with shapes and holes within topological spaces, right? I hope I'm prepared for topology 2 honestly. Oh and Allen Hatcher is awesome! I thank you for your prompt response, I'll do more research. Thanks.
Thanks @MathatAndrews a lot!!!🎉 we also expect differential geometry lecture!
There is a whole playlist on this channel with Differential Geometry lectures
I dont get how the 2-skeleton map phi gives us the torus.
I wish he would have visualized a bit more that step
Hatcher's book is great, nice choice!
Thank you Andrews University and Professor Bosman for making such a hard taught subject very simple to understand with high quality video and audio. Thank you so much. waiting for new courses like Algebra(Abstract), Number theory, Lattices, Real, Complex and functional analysis.
Thank you very much!
These classes are wonderful. My respects to you from Spain.
These are amazing, thank you so so much!!
Thanks for the great lectures! I do really learn algebraic topology from your lectures! 👏
Thanks ill take memory here to watch again 27:44
You're a great lecturer, thank you for making these available! I've always struggled with this subject and this has made it much more accessible to me.
The only thing that the book lacks is examples. Otherwise the theoretical content is intermediate friendly.
i am having trouble moving past the point in the lecture where 2-skeletons are introduced.
would it be nonsense to try and draw a connection between the need for *two* maps in a stereographic chart for S2, and the need for the D2 attachment map to wrap around each edge *twice* (once along the direction, once against; i.e, 27:02) in a 2-skeleton? it could just be a bad analogy, since two *separate* points are used in the stereographic projection maps.
i see how D2 could *locally* generate the 'skin' onto a 1-skeleton, but i don't understand how it correctly 'covers' the shape (as opposed to just introducing a local coordinate system in a small region around each of the lines in the 1-skeleton). using the example at the time stamp above, my intuition is that the attachment map would leave the **opposite** side of the torus from b (i.e, the area antipodal to the 'b' edge) without any 'skin'/D2 attachment.
i would appreciate anyone's insight.
Is glueing commutative? Can I glue them in any order? Even if the sides are symmetrical, I think there is more than one way to glue them. The cylinder can curl in instead of bend around, even if it starts and ends the same. I think means matters, even if the start and end are the same.
What a charming professor, their passion for algebraic topology really radiates from the screen.
For the S1x[0,1] @49:14 the first picture you have each point going into a third dimension, is that intensional or just the appearence of the dipiction? I'm assuming its nothing given the pinwheel looks like its planar. Oh, I guess the interval would have to go somewhere thats not on S1 as those would be other points.
Right, you either stay in the same plane and get the annulus or go to a 3rd dimension and get some sort of cylinder.
ALHAMDULILLAH
Isn't the boundary of a circle is itself? as any neighborhood of a point on the circle intersects both circle and its complement.
It makes sense that the boundary of the circle is empty if we define the boundary of a set to be the boundary of its interior.
The circle is one-dimensional, and the neighborhoods of its points are too, and none of them contain any points not on the circle
A question! At 49:14 he says that S1 x S1 (circle times circle) is just a torus, but if torus and klein bottle are both structurally the same expect the surface orientation or how do you "glue" the D2 to the 1-skeleton, what in the product makes it become a torus and not a klein bottle?
I think they might be equivalent.
They're not equivalent. Basically the twist you introduce by identifying the sides of the rectangle differently means that you can't represent the Klein bottle as a product (it's a nontrivial fibre bundle). The case is similar for the Möbius strip as contrasted with an untwisted strip. The untwisted strip can be represented as the product of a circle and an interval, while the Möbius strip can't be.
This book is not very friendly for a non-native English speaker,I think😂
Nor for many English speakers! Quite a bit of Greek...
Not easy to learn pictograms when only trained in the Roman alphabet ,either.
it’s not too friendly for native speakers either! John Lee’s “Introduction to Topological Manifolds” is easier to follow, I think, especially when accompanied by this lecture series.
Great lecture! Just a quick question, throughout the video you use gluing to explain topics but the basic idea of topology is to avoid tearing and gluing, that’s a bit contradictory don’t you say?