The other day, I learnt combinatorics from Prof. Gowers, now, I am listening to Prof. Borcherds on algebraic topology. Who needs universities? Long live the lockdown!
I am sure hoping that you will continue this series on Algebraic Topology. What a great gift these teachings are to the mathematical community... and eventually, through secondary contacts and so on, the rest of the world. Thank you.
THANK YOU!! Ive spent years trying to understand a handful of things that just kept eluding me about complex functions and this lecture cleared them up!
This is incredibly perfect - was literally just looking for a course in algebraic topology and found Hatcher's book! Then this comes out the same day??
The lack of notification of your videos has been eating me up for days. I'm so relieved you uploaded! I got too used to these daily videos. I understand you must be working on something amazing right now hence the gaps. Good luck and thank you!
Thanks Dr. Borcherds, very timely as well - I had just begun reading a bit more about algebraic topology and was hopelessly lost. Will look forward to more videos soon.
Thank you for uploading this lecture! I'm currently a 3rd year undergraduate math student, what would be a good prerequisite for this class? I have strong experience in abstract algebra but not much topology background
The first four chapters of Topology, by James Munkres, would do it. If you are very comfortable with equivalence relations and the quotient topology, in particular, box diagrams and the associated constructions of the torus, cylindrical segment, Moebius band, and projective space, then you will be ready (at least from what I can tell by looking at the level of the first lecture).
@@Ken-wr7zx Thank you, I will try self studying that to try and even understand the following lectures! The material appears incredibly interesting but it feels so out of reach at the moment.
@@harisserdarevic4913 I would also recommend taking a look at Armstrong’s Basic Algebra. Armstrong included some basic Alg Topo in the second half of the book, after some essential basics of general topology.
Correct! Well, we need to assume our space is path-connected (or else "the fundamental group" isn't well defined), but then it's true. More generally, if a space X is n-connected, then H_{n+1}(X) is the abelianization of π_{n+1}(X) -- this is called the Hurewicz theorem. The special case n=0 says precisely that the first homology of a path-connected space is the abelianization of its fundamental group.
@@f5673-t1h n-connected means π_k = 0 for all 0≤k≤n. So 0-connected just means π_0 = 0 (a.k.a. path connected), and 1-connected means 0-connected + π_1 = 0 (a.k.a. simply connected). 2-connected means 1-connected + π_2 = 0. The simplest examples are spheres: Sⁿ is n-connected (but not always n+1-connected).
Let ω=2π and let z=cos(ω)+i sin(ω). Then we mapped a real line R to a unit circle in a complex plane C with every unit interval, like [0,1), or [29577,29578) covering the circle once. We mapped R onto S1, covering S1 Z times over. If we quotient our map by the integers Z, we get just one “abstract” instance of our map. We can “lift” our circle (by an “inverse” map) to any interval on the real line and an open set/interval on the circle will be lifted to an open interval on the unit segment of the real line of our choice. In both mappings we preserved the local topological structure of R and S1. The whole story started, more or less, with Fourier series and Fourier transform. If it is not so, I hope that @Richard E. Borcherds will be kind enough to correct me. This is not the only map from R to S1. The other standard one, covering the S1 only once, but in infinite time t is given i.a. in @Richard E. Borcherds’ introductory video to algebraic geometry. Btw. RUclips would do us a great favour incorporating LaTex in the comments, wouldn’t they?
Let's goooo! This is going to be epic!
indeed it is, mr quirky kirk 😳
@@gregoriousmaths266 GREG MAFFS
@@gregoriousmaths266 GREG!
No, it's going go be monic 🤪
@@incredulity that's what I'm saying
The other day, I learnt combinatorics from Prof. Gowers, now, I am listening to Prof. Borcherds on algebraic topology. Who needs universities? Long live the lockdown!
Hello ! I hope that you are doing well, do you know any other channel like thses
I am sure hoping that you will continue this series on Algebraic Topology. What a great gift these teachings are to the mathematical community... and eventually, through secondary contacts and so on, the rest of the world. Thank you.
THANK YOU!! Ive spent years trying to understand a handful of things that just kept eluding me about complex functions and this lecture cleared them up!
This is incredibly perfect - was literally just looking for a course in algebraic topology and found Hatcher's book! Then this comes out the same day??
The lack of notification of your videos has been eating me up for days. I'm so relieved you uploaded! I got too used to these daily videos. I understand you must be working on something amazing right now hence the gaps. Good luck and thank you!
Excellent as always Dr. Borcherds!
Thanks Dr. Borcherds, very timely as well - I had just begun reading a bit more about algebraic topology and was hopelessly lost. Will look forward to more videos soon.
so excited about this series!!!
wtf this is the 3rd time i've encountered you on youtube haha
Excellent lecture. Very interesting, informative and worthwhile video. Many thanks.
Incredible Prof Borcherds, I should have taken some of your classes while I was at Berkeley!
I remember hearing about fundamental groups and such at topology2 class. The teacher was kind and good.
He just won't stop
Thank you for providing this content and giving back to the mathematical community. 😊
been wanting a supplement for this topic, looking forward to more, thank you.
Thanks! I look forward to the next lectures, :)
Exciting ! We do it for the fun of it 🤘🏻 Thanks professor 😊
Bro you've been making so much!
I missed it. So glad the videos are back and better than ever!
Thank you for uploading this lecture! I'm currently a 3rd year undergraduate math student, what would be a good prerequisite for this class? I have strong experience in abstract algebra but not much topology background
The first four chapters of Topology, by James Munkres, would do it. If you are very comfortable with equivalence relations and the quotient topology, in particular, box diagrams and the associated constructions of the torus, cylindrical segment, Moebius band, and projective space, then you will be ready (at least from what I can tell by looking at the level of the first lecture).
@@Ken-wr7zx Thank you, I will try self studying that to try and even understand the following lectures! The material appears incredibly interesting but it feels so out of reach at the moment.
@@harisserdarevic4913 I would also recommend taking a look at Armstrong’s Basic Algebra. Armstrong included some basic Alg Topo in the second half of the book, after some essential basics of general topology.
Basic Topology*
this introduction was already amazing
I remember reading that the first homology group is the Abelianization of the fundamental group. (someone please correct me)
Correct! Well, we need to assume our space is path-connected (or else "the fundamental group" isn't well defined), but then it's true.
More generally, if a space X is n-connected, then H_{n+1}(X) is the abelianization of π_{n+1}(X) -- this is called the Hurewicz theorem. The special case n=0 says precisely that the first homology of a path-connected space is the abelianization of its fundamental group.
@@BenSpitz what does n-connected mean?
@@f5673-t1h n-connected means π_k = 0 for all 0≤k≤n. So 0-connected just means π_0 = 0 (a.k.a. path connected), and 1-connected means 0-connected + π_1 = 0 (a.k.a. simply connected). 2-connected means 1-connected + π_2 = 0. The simplest examples are spheres: Sⁿ is n-connected (but not always n+1-connected).
@@BenSpitz The higher homotopy groups are abelian anyway
It would be great to see some discussion on spectral sequences :)
What's R/Z, real numbers with the integers removed?
That is more like equivalence classes, with x and y are in same equivalence class iff x-y is an integer
One defines an equivalence relation on R by saying x~y if x-y is an integer. The resulting set of equivalence classes is denoted by R/Z
It's a quotient group (also called factor group in some languages).
Let ω=2π and let z=cos(ω)+i sin(ω). Then we mapped a real line R to a unit circle in a complex plane C with every unit interval, like [0,1), or [29577,29578) covering the circle once. We mapped R onto S1, covering S1 Z times over. If we quotient our map by the integers Z, we get just one “abstract” instance of our map. We can “lift” our circle (by an “inverse” map) to any interval on the real line and an open set/interval on the circle will be lifted to an open interval on the unit segment of the real line of our choice. In both mappings we preserved the local topological structure of R and S1.
The whole story started, more or less, with Fourier series and Fourier transform. If it is not so, I hope that @Richard E. Borcherds will be kind enough to correct me.
This is not the only map from R to S1. The other standard one, covering the S1 only once, but in infinite time t is given i.a. in @Richard E. Borcherds’ introductory video to algebraic geometry.
Btw. RUclips would do us a great favour incorporating LaTex in the comments, wouldn’t they?
Set difference is backslash, quotient is forward slash. Can be confusing sometimes.
This is really good!
K theory. So cool name.
awesome!
Thankyou
thanks! 🙏🙌
YEEEEEEEEEEEEEEEE
What is the difference between Z and the free group on a,b? It seems like they should be isomorphic since Z also has two generators.
Oh, I guess the free group is not abelian
@@pmcate2 yes you are right.
Instant classic
Yaaaay!!
yay!
Sir please do a lecture series on Arithmatic Geometry
yeeeeeeeeeeeeeeee
Hello :)
Hello!
antioxidants deoxidize where the lungs are
The lack of motivation is unfortunate.
Rika: finally! A happy timeline 😁
Satoko:
👁👄👁
➖👄➖
🔫🛑👄🛑
Higurashi?