@@TheDavidlloydjones Ah, thank you for the correction! English is not my native tongue so it's always good to be made aware of some recurring mistake I've probably made a million times without being corrected :)
Yes. Imagine if he transferred his intelligence to making a TH-cam video! But the back of his head as he scribbles on the blackboard is pretty much like that of every other idjit out there who thinks that's an activity worth recording and rebroadcasting. Just nuts!
Outstanding lecture! Thanks for providing it. At 31:30, one may note that removing any point from the circle leaves a contractible space, while this is not the case with the annulus. Or more simply, the annulus has a nonempty interior, whereas the circle has empty interior.
I really enjoyed your lecture! Thank you for what you are doing. I'm from Russia, and in my university I don't have such a course. I have to work through a lot of material on my own, and your lectures do help me.
Timestamp 1:02:27 this is really nitpicky, but I think Freedman's result was that there are uncountablely infinitely smooth structures on R4, but it was known that for any n other than 4, there's only 1 unique smooth structure. He said 4 and higher though. The reality is much weirder: R4 is the only exception to the rule that "classifying Rn topologically or smoothly is essentially the same task". (And it's an exception in an unexpected, dramatic, big way) Really great lecture! I'm hooked
@Mohammad Maruf Mamun the same thing happened to me. I was a pure algebra and topology (and category theory) guy, but turns out, introducing differential structure just makes certain things mysterious and interesting
It was an awesome lecture with you sir on this channel here, just superb , the funny elements you added in the class were the best part as most professor just focuses only on giving an 1 hour boring lecture just by filling up the board the whole time. We need professors like you in our institutes.
This is a great lecture! I really like this approach to topology as opposed to other videos I've seen. And the way he mispronounces all these mathematicians' names is kinda adorable.
I wish I could ask questions! Does anyone know about the fundamental relationship between chaos theory and algebraic topology? I never knew chaos theory was necessary for forward movement in topology! Fascinating.
For me these are the most inciting lectures in the subject. Thank you, Professor Pierre Albin! And I am really eager to ask, are there videos of the second semester yet?
Could you please clarify this doubt? You said that the maximum number of closed curves we can remove without disconnecting the surface is the genus. You also said that it is the first betti number. But, isn't the first betti number of the double torus 4?
Is this guy part Greek or something? He said "γεωμετρία" (geometria) with perfect Greek accent and it really startled me because non-native speakers struggle with pronouncing words like it right, and he kinda looks Greek too.
I haven’t taken a college math course in about 5 years and even then didn’t get in to much conceptual math. What kinda prerequisites would someone need to know to understand topology? I’m very curious and this first lecture seemed fun even, I’ll attempt to keep following along but could anyone describe what fundamentals someone should know going in to this course that might make it easier to understand?
You will need algebraic and topological concepts. Good books in Algebra are Topic in Algebra by Herstein or Contemporary Abstract Algebra by Gallian. For topology, I can advise Munkers and Willard. Hope this helps.
Perhaps you should look up abstract algebra lectures by Benedict Gross here on youtube. They are also very fun! (sometimes could seem like its going way too fast tho)
Also, maybe you should also look up linear algebra, and for that I recommend lectures and book by Gilbert Strang on that subject. His lectures on linear algebra are lovely.
What are the mathematical requirements that I must study before I learn topology as a beginner, given that I did not study mathematics after high school
Dumb question. How does the given definition of continuous (23:49) capture the meaning of “continuous”? Functions from an open domain to an open range have open BOUNDARIES (as in the open range (0,1) in contrast to the closed range [0,1]), but what continuity ACTUALLY means is continuous INSIDE the domain. A definition constraining the boundaries of set would not evidently constrain the interiors thereof. Maybe somebody could clarify, since the teacher didn’t. Thanks.
I’m not exactly sure what you mean by “open boundaries”, but the intuition for this definition comes from analysis. In analysis, you can prove (from the epsilon - delta definition of continuity) that a function is continuous if and only if preimages of open sets are open sets. I believe Allen Hatcher also provides some intuition in his notes on point set topology, which you can find here: pi.math.cornell.edu/~hatcher/Top/TopNotes.pdf
Another dumb question. At 32:30 he explains a “continuous family” of maps and writes F:Xx[0,1]->Y “so” (X,t) |-> f_t(X) is continuous. I find this uninterpretable, unreadable. 1) clearly f_t(X) is a map, since it is a function of X,that is, *maps* values of X to f_t(X). But in what if any sense is (X,t) a map; it is NOT a map, but rather a continuous open range of numbers from X to t. An open line segment is no map at all, more like a dead thing, a mere piece of space,the input to a map perhaps but not a map. 2) Does the scope of this particular F end after X or ] or Y? Y, I’ll guess, but then how can Y be the output if it’s part of the input? 3) Is the idea of a family that members map from one to another, so both the inputs and outputs are members of the family? If so Xx[0,1] must be a continuous map which it isn’t since it is a subregion of R^2. And Y must be a continuous map which it could be but not necessarily by anything written or said here. And (X,t) must be a continuous map which it isn’t since it is an open line segment. And f_t(X) must be a continuous map which it could be by assumption from Y being a continuous map but then we don’t really have reason to believe that yet, do we? So in what sense is this a family of continuous maps? Apparently none. 4) Maybe the map is BETWEEN things in Xx[0,1] and things in Y, and as an example (X,t) which is in Xx[0,1] seems to map according to what he said to f_t(X) which could only be an element in Y, if Y were some kind of universe of FUNCTIONS and f_t(X) an element in that universe, which I wish he would explain since I never heard of a domain or a range being a world of functions, since functions take domains to ranges, that would be a mixing and a confusion of categories, where I grew up. I mean, you could MENTION that Y is not a normal numerical range, some subset of R^n, but a “space” of functions or continuous maps or something, just you know, so I (we?) could understand you. I’ll have to wrap my head around that. Then what would make this map to a function in Y a continuous map, is something also evidently unstated. Is it the openness of the segment (X,t)? If so, that could be made explicit and then be explained, why that follows, too. Please help this struggling, concrete minded follower understand what you are saying.
He wrote that the function acts like (x,t) |-> f_t(x) (x here is a small and represents a variable, not the space X), i.e. F is a function that maps Xx[0, 1] (with the product topology) into Y. Basically, you don't think of t as a parameter, but as another argument of the function F. The (x, t) |-> f_t(x) notation is the same as saying that F(x, t) = f_t(x).
I know this is an old comment, but I'm not totally sure what you mean. An n-dimensional manifold is defined by its having a neighborhood around an arbitrary point homeomorphic to Euclidean n-space. Curvature is only defined on certain smooth manifolds.
11:47 this is nonsens what he saying -- saying first something else then something different and then that that answer to other question but to first is NO very twisted why to say something and try not say what realy neeed to say-- but still something new
The backs of people's heads as the write n blackboards are not a fit subject for TH-cam. It's been done already. Two of them was too many. If you want to teach a class, teach a class. If you want to make a video, sure, make a video. Just try to get it straight in your head that they are two different things, OK?
Algebraic topology is one of the most beautiful courses in mathematics.
Really ? I have only general ideas what this of Mathematics is about ! But, u give me motivation to start this course ! Thank u!
This guy is so charming and witty. It's like they took a slight-of-hands magician to give a math talk.
sleight
Completely different concept.
@@TheDavidlloydjones Ah, thank you for the correction! English is not my native tongue so it's always good to be made aware of some recurring mistake I've probably made a million times without being corrected :)
This guy is one of the rare professors whose classes are enormously exciting
Yes. Imagine if he transferred his intelligence to making a TH-cam video!
But the back of his head as he scribbles on the blackboard is pretty much like that of every other idjit out there who thinks that's an activity worth recording and rebroadcasting. Just nuts!
Outstanding lecture! Thanks for providing it. At 31:30, one may note that removing any point from the circle leaves a contractible space, while this is not the case with the annulus. Or more simply, the annulus has a nonempty interior, whereas the circle has empty interior.
I really enjoyed your lecture! Thank you for what you are doing. I'm from Russia, and in my university I don't have such a course. I have to work through a lot of material on my own, and your lectures do help me.
me too,i'm from china and my University didn't have these course too:) but my english is very poor......
Best course in algebraic topology so far
Can you upload the next semester's Algebraic Topology?Thank you a lot.
THIS
The same request.
Timestamp 1:02:27 this is really nitpicky, but I think Freedman's result was that there are uncountablely infinitely smooth structures on R4, but it was known that for any n other than 4, there's only 1 unique smooth structure. He said 4 and higher though. The reality is much weirder: R4 is the only exception to the rule that "classifying Rn topologically or smoothly is essentially the same task". (And it's an exception in an unexpected, dramatic, big way)
Really great lecture! I'm hooked
@Mohammad Maruf Mamun the same thing happened to me. I was a pure algebra and topology (and category theory) guy, but turns out, introducing differential structure just makes certain things mysterious and interesting
These are super cool! Waiting for semester 2! :)
Still waiting
I'm another one who is waiting. Please someone upload semester 2 :)
It was an awesome lecture with you sir on this channel here, just superb , the funny elements you added in the class were the best part as most professor just focuses only on giving an 1 hour boring lecture just by filling up the board the whole time. We need professors like you in our institutes.
This is a great lecture! I really like this approach to topology as opposed to other videos I've seen. And the way he mispronounces all these mathematicians' names is kinda adorable.
A GREAT lecture! Most impressed by his knowledge and wit.
Pierre Albin was right to have some doubts, Mittag-Leffler is from Sweden not Norway.
Up to 1905 Norway and Sweden were the same kingdom, so he was right
thanks from India please upload some more topics like complex analysis, functional analysis
I wish I could ask questions! Does anyone know about the fundamental relationship between chaos theory and algebraic topology? I never knew chaos theory was necessary for forward movement in topology! Fascinating.
In 31:52, I think we don't need the dimension theorem. If we subtract two points on S^1 becomes disconnected. However, This isn't the case in Ann.
when you're enjoying the nice introduction and then 8:38 arrives
Am I watching this because a guy on tiktok said it was hard? Yes💀
Bro did u watch the topology edit also? 😭😭
@@intrinsic9585 yep💀
no shot 😂
I’m here from the edit lmao
Same here🗿
This is a wonderful presentation
Why hasn't Semester 2 been uploaded!?
For me these are the most inciting lectures in the subject. Thank you, Professor Pierre Albin! And I am really eager to ask, are there videos of the second semester yet?
At 41:30, wouldn't the deformation retraction be f(t): x => (1-t)x so that, according to the definition of a deformation retraction, f(0)=IdX ?
Could you please clarify this doubt? You said that the maximum number of closed curves we can remove without disconnecting the surface is the genus. You also said that it is the first betti number. But, isn't the first betti number of the double torus 4?
Anywhere online to find the supplements he's referring to?
ابدع بمعنى الكلمة
اتفققق
This is fun to hear. But I don't understand a thing. Where do I do a foundation course?
Is this guy part Greek or something? He said "γεωμετρία" (geometria) with perfect Greek accent and it really startled me because non-native speakers struggle with pronouncing words like it right, and he kinda looks Greek too.
He probably speaks Spanish well (mentioned that he was brought up in Mexico). Greek and Spanish phonetics are almost identical.
I haven’t taken a college math course in about 5 years and even then didn’t get in to much conceptual math. What kinda prerequisites would someone need to know to understand topology? I’m very curious and this first lecture seemed fun even, I’ll attempt to keep following along but could anyone describe what fundamentals someone should know going in to this course that might make it easier to understand?
You will need algebraic and topological concepts. Good books in Algebra are Topic in Algebra by Herstein or Contemporary Abstract Algebra by Gallian. For topology, I can advise Munkers and Willard. Hope this helps.
Perhaps you should look up abstract algebra lectures by Benedict Gross here on youtube. They are also very fun! (sometimes could seem like its going way too fast tho)
Also, maybe you should also look up linear algebra, and for that I recommend lectures and book by Gilbert Strang on that subject. His lectures on linear algebra are lovely.
4:44 gauss bonet theorem
until 21:00 , history of topology
6:42 riemman wants to go further
Does anyone have a solution manual for the Allen Hatcher textbook?
Here from Aleph 0's vid, wish me luck 🤞
what book did he mention when he said chapter 0,1,2 ?
thanks in advance !
Allen Hatcher’s “algebraic topology “. Go to his website to get it.
@@bellfoozwell thank you!
God how can so much cool fit in one person?
what is the lecture notes or textbook?
Allan hatcher , algebraic topology
What is the textbook he mentioned? Thanks.
how did no one notice the typo at @53:06
f to X, it means no one is getting what's goin on :D although no i'm satisfied that on the right track
I did not understand Poincare approach
What are the mathematical requirements that I must study before I learn topology as a beginner, given that I did not study mathematics after high school
Set theory, proof writing, abstract algebra and point-set (standard) topology. This is a very advanced class, but also very enjoyable.
do u know where we can find exercises/ book to accompany these lectures?
Dumb question. How does the given definition of continuous (23:49) capture the meaning of “continuous”? Functions from an open domain to an open range have open BOUNDARIES (as in the open range (0,1) in contrast to the closed range [0,1]), but what continuity ACTUALLY means is continuous INSIDE the domain. A definition constraining the boundaries of set would not evidently constrain the interiors thereof. Maybe somebody could clarify, since the teacher didn’t. Thanks.
I’m not exactly sure what you mean by “open boundaries”, but the intuition for this definition comes from analysis. In analysis, you can prove (from the epsilon - delta definition of continuity) that a function is continuous if and only if preimages of open sets are open sets. I believe Allen Hatcher also provides some intuition in his notes on point set topology, which you can find here: pi.math.cornell.edu/~hatcher/Top/TopNotes.pdf
@@gateronblackinksv2173 edited question. Thank you!
I want to show that Show O(p,q) is homotopic to O(p)×O(q) for p and q positive integers. Is there any hint or idea how to define homotopy?
Another dumb question. At 32:30 he explains a “continuous family” of maps and writes F:Xx[0,1]->Y “so” (X,t) |-> f_t(X) is continuous. I find this uninterpretable, unreadable. 1) clearly f_t(X) is a map, since it is a function of X,that is, *maps* values of X to f_t(X). But in what if any sense is (X,t) a map; it is NOT a map, but rather a continuous open range of numbers from X to t. An open line segment is no map at all, more like a dead thing, a mere piece of space,the input to a map perhaps but not a map. 2) Does the scope of this particular F end after X or ] or Y? Y, I’ll guess, but then how can Y be the output if it’s part of the input? 3) Is the idea of a family that members map from one to another, so both the inputs and outputs are members of the family? If so Xx[0,1] must be a continuous map which it isn’t since it is a subregion of R^2. And Y must be a continuous map which it could be but not necessarily by anything written or said here. And (X,t) must be a continuous map which it isn’t since it is an open line segment. And f_t(X) must be a continuous map which it could be by assumption from Y being a continuous map but then we don’t really have reason to believe that yet, do we? So in what sense is this a family of continuous maps? Apparently none. 4) Maybe the map is BETWEEN things in Xx[0,1] and things in Y, and as an example (X,t) which is in Xx[0,1] seems to map according to what he said to f_t(X) which could only be an element in Y, if Y were some kind of universe of FUNCTIONS and f_t(X) an element in that universe, which I wish he would explain since I never heard of a domain or a range being a world of functions, since functions take domains to ranges, that would be a mixing and a confusion of categories, where I grew up. I mean, you could MENTION that Y is not a normal numerical range, some subset of R^n, but a “space” of functions or continuous maps or something, just you know, so I (we?) could understand you. I’ll have to wrap my head around that. Then what would make this map to a function in Y a continuous map, is something also evidently unstated. Is it the openness of the segment (X,t)? If so, that could be made explicit and then be explained, why that follows, too.
Please help this struggling, concrete minded follower understand what you are saying.
He wrote that the function acts like (x,t) |-> f_t(x) (x here is a small and represents a variable, not the space X), i.e. F is a function that maps Xx[0, 1] (with the product topology) into Y. Basically, you don't think of t as a parameter, but as another argument of the function F.
The (x, t) |-> f_t(x) notation is the same as saying that F(x, t) = f_t(x).
What university is this?
i think University of Illinois at Urbana-Champaign.
@@martinmartinez4685 it is.
I need help with some typical questions can u help
Timestamp 54:15
Library doesn't have Hatcher. I like physical copies :(
luckily hatcher is cheap as dirt. it's sub-30 pounds.
@@NotLegato oh cool I'll order it then. Fascinating subject so far.
What book is the lecture using?
My impression is that he is referring to the book Algebraic Topology by Allen Hatcher, even though he mentions it as Patrick's book.
great!
This is one of the easiest classes I took at my university. I don't understand how people struggle with this subject
The algebraic topology Feynman
7:00
what gives the "locally resembles a plane" characteristic is curvature, not homeomorphism to planes. otherwise, thanks for the great lecture
I know this is an old comment, but I'm not totally sure what you mean. An n-dimensional manifold is defined by its having a neighborhood around an arbitrary point homeomorphic to Euclidean n-space. Curvature is only defined on certain smooth manifolds.
The answer is always no if you use infinity-categories ;)
I couldn't help mistake him for Jimmy Carr
Jimmy Carr is his evil twin
I wonder where he learned to pronounce poincare this way
I am from all over place
I don’t think organic chemistry tutor will save me this time
I'm from all over the place 😆
"17."
Please buy a better eraser
So many holes to fill in......
Why do they still use chalk boards. I cringe and get goosebumps hearing chalk or erasers on them
"STEVE JOBS IS ALIVE."(AYANA)
11:47 this is nonsens what he saying -- saying first something else then something different and then that that answer to other question but to first is NO very twisted why to say something and try not say what realy neeed to say-- but still something new
The backs of people's heads as the write n blackboards are not a fit subject for TH-cam. It's been done already. Two of them was too many.
If you want to teach a class, teach a class.
If you want to make a video, sure, make a video.
Just try to get it straight in your head that they are two different things, OK?
Why you are laughing
Math is Nice but totaly unuseful Profs are paid juste for fun 🎉
laughing should not be allowed in a math class. This should be a fundamental axiom of all math classes.
not funny
Not allowing things should not be allowed in a math class.