What a great tracher. A youtuber scaredmonger about math classes difficulty, and the topic they saw at the math class he attended was VanKampen's theorem. To prove him wrong I was determined to learn the same topic with no previous background knowledge, and thanks to this lecture I understood the theorem. The sign of a an amazing teacher
This is so clear, conveying the brilliance of the theorems and applications in an intuitive way. Well done! The results of Van-Kampen's theorem are so incredible too
36:37 can anyone elaborate on this deformation retract im really not seeing it, i'm imagining the circle expanding onto the equator circle but then how does this leave a line through the middle?
Take the solid diameter going exactly through the center of the carved out S¹ and perpendicular. You get the first retraction to solid diameter by projecting all the points of S¹ downwards to the center, which is also a point on the diameter. Now use a small angles to project (retract) more and more down to the solid diameter, until the angle is so big that it starts to make small concentric circles on S². Thus you can fill out step by step all of S² until you come back on the other side (negative angle) down to the solid diameter again and then the same procedere as in the beginning, just from S² towards the center of S¹, retractions along the solid diameter. This way you have projected (contracted) all(!!!) points of D³ - S¹ either to the solid diameter or S², straight, without crossings of any lines. So this is what is left after contraction, as claimed.
For the second question, I would use rather the arguments from the last 10 minutes of the lecture, which are much simpler (almost trivial) to follow and much more generell. So use the polygons with vertices identified etc. and you see immediately what's going on, without need of extreme Imagination power.
Removing an S^1 can be deformation retracted to removing a torus. Now pump air into it so it fills up the sphere and you'll be left with a solid diameter running through the middle of the torus.
Starting with R^3 - S^1 choose the basepoint to be the center of the circle we removed. Now create a loop that goes around the removed circle. This creates two disjoint circles that are linked together like a chain. You could make this with some twine if you want to build more intuition. Now take a second loop that goes around the removed circle and through the same base point. First deform these loops into circles of the same radius. Then rotate one around the removed circle until it aligns with the other loop. This shows that any two such loops are homotopy equivalent. Now notice we can stretch out the basepoint into a diameter without altering the construction. We can further ballon out the diameter to fill as much of the circle as we want.
How is it that in 1:05:28 the fundamental group of A intersect with B (which is Z) is embedded in fundamental group of de disk in two dimension if this its trivial? Is there something that I missing?
Definition of "free product" (*)? I think the wedge (v) product of spaces was covered earlier, but I'm drawing a blank on that, too. Looks like a union of spaces?
The wedge sum takes one point of each space, typically the basepoint, and glues them together. Formally to construct it we take x in X and y in Y then take the disjoint union of X and Y and quotient it with the relation x ~ y. Once you've internalized the formal definition you should forget it and think gluing the base points together. The free product of two groups G and H is a lot like the free product on a set where you construct words from the generators and reduce them. However for the free product the relations on G and H will continue to hold in the product. So say G = {g | g^3=1} and H = {h | h^4=1} then G * H = {g,h | g^3=h^4=1}. Some typical element of G*H would be gh, gh^3gh^2gh, g^2h, g^2h^2gh^3, etc. So even with finite groups the free product is typically infinite just like the free group is infinite with a finite set of generators.
To see that ab =/= ba for the figure-8 space note that any homotopy of paths between a and b would have to pass through the basepoint, which means the loop would be contractible. Since the fundamental group of the circle is not trivial this is impossible.
if you are looking for topology textbooks, you can try Intro to Metric and Topological Spaces by Sutherland, which motivates topology well, or Munkres’s Topology, which is more comprehensive
Well I think that it great lecture your skills in visualiation and explaying are great, However, As a math student I feel sadness because of lack of proofs. I didn't watch every video from this course yet, as I can guess from titles you go through all the topics from book "Algebraic Topology" as I saw this book have 500 pages and use advanced math stuff lake teory category (or notation form that) so I understand why you skip some proofs in this course but I will be greatful if you can add some additional supplement with proofs!
I can’t even begin how people do this. I’m about to start mechanical engineering and I’m dreading calculus. I can do trig and algebra so I should be able to grind it out. God someone give me some motivation. Tell me it’s gonna be okay.
What a great tracher. A youtuber scaredmonger about math classes difficulty, and the topic they saw at the math class he attended was VanKampen's theorem. To prove him wrong I was determined to learn the same topic with no previous background knowledge, and thanks to this lecture I understood the theorem. The sign of a an amazing teacher
Yes He is a great teacher. I m learning a lot from his lectures ❤
Best playlist of video lectures! Really loving to follow these classes from this professor.
As a rising Grade 10 student, this video really clarified van Kampen’s theorem to me. Thank you!
This is so clear, conveying the brilliance of the theorems and applications in an intuitive way. Well done! The results of Van-Kampen's theorem are so incredible too
lovely, clear presentation!
I finally got it! Its actually pretty easy to understand if you get the hang of it
Very clear and neat quality
Useful, Clear, and Beautiful. Thanks.
Amazing, regards from Spain, you are helping me a lot, thanks.
36:37 can anyone elaborate on this deformation retract im really not seeing it, i'm imagining the circle expanding onto the equator circle but then how does this leave a line through the middle?
similarly for 1:06:51
Take the solid diameter going exactly through the center of the carved out S¹ and perpendicular.
You get the first retraction to solid diameter by projecting all the points of S¹ downwards to the center, which is also a point on the diameter.
Now use a small angles to project (retract) more and more down to the solid diameter, until the angle is so big that it starts to make small concentric circles on S². Thus you can fill out step by step all of S² until you come back on the other side (negative angle) down to the solid diameter again and then the same procedere as in the beginning, just from S² towards the center of S¹, retractions along the solid diameter.
This way you have projected (contracted) all(!!!) points of D³ - S¹ either to the solid diameter or S², straight, without crossings of any lines. So this is what is left after contraction, as claimed.
For the second question, I would use rather the arguments from the last 10 minutes of the lecture, which are much simpler (almost trivial) to follow and much more generell.
So use the polygons with vertices identified etc. and you see immediately what's going on, without need of extreme Imagination power.
Removing an S^1 can be deformation retracted to removing a torus. Now pump air into it so it fills up the sphere and you'll be left with a solid diameter running through the middle of the torus.
Starting with R^3 - S^1 choose the basepoint to be the center of the circle we removed. Now create a loop that goes around the removed circle. This creates two disjoint circles that are linked together like a chain. You could make this with some twine if you want to build more intuition. Now take a second loop that goes around the removed circle and through the same base point. First deform these loops into circles of the same radius. Then rotate one around the removed circle until it aligns with the other loop. This shows that any two such loops are homotopy equivalent. Now notice we can stretch out the basepoint into a diameter without altering the construction. We can further ballon out the diameter to fill as much of the circle as we want.
How is it that in 1:05:28 the fundamental group of A intersect with B (which is Z) is embedded in fundamental group of de disk in two dimension if this its trivial? Is there something that I missing?
Damn, 3000 views for this quality, math is crual. Ty !
Amazing currently learning
@13:58 Is it true we can assume if two spaces are homotopic equivalent they have the same Fundamental group ?
yes, the fundamental group is a homotopy invariant
Yes, this was one of the main topics towards the end of the last lecture (lecture number 5)
thank you so much, super intuitive explanations, wich i often miss in topology
Definition of "free product" (*)? I think the wedge (v) product of spaces was covered earlier, but I'm drawing a blank on that, too. Looks like a union of spaces?
The wedge sum takes one point of each space, typically the basepoint, and glues them together. Formally to construct it we take x in X and y in Y then take the disjoint union of X and Y and quotient it with the relation x ~ y. Once you've internalized the formal definition you should forget it and think gluing the base points together.
The free product of two groups G and H is a lot like the free product on a set where you construct words from the generators and reduce them. However for the free product the relations on G and H will continue to hold in the product. So say G = {g | g^3=1} and H = {h | h^4=1} then G * H = {g,h | g^3=h^4=1}. Some typical element of G*H would be gh, gh^3gh^2gh, g^2h, g^2h^2gh^3, etc. So even with finite groups the free product is typically infinite just like the free group is infinite with a finite set of generators.
What is a Union of Union to Learn more about the Ambient Spaces?
Very helpful! :)
To see that ab =/= ba for the figure-8 space note that any homotopy of paths between a and b would have to pass through the basepoint, which means the loop would be contractible. Since the fundamental group of the circle is not trivial this is impossible.
Brilliant !!!! Thank you !!!
1:05:00 Seifert was a German mathematician*
lovely presentation
Great video!
What do you think about the textbook introduction to topology from UCLA?
if you are looking for topology textbooks, you can try Intro to Metric and Topological Spaces by Sutherland, which motivates topology well, or Munkres’s Topology, which is more comprehensive
Amazing
Is this the doctor who revealed the magician's lies using the knot theory?
Yep 😅
@@MathatAndrews ❤️
Could we use algebraic topology to multiply matrices of incompatible dimension?
No
B^3 is better b/c we need the ball to be open, while D^3 is closed. But that doesn't matter!
Well I think that it great lecture your skills in visualiation and explaying are great, However, As a math student I feel sadness because of lack of proofs. I didn't watch every video from this course yet, as I can guess from titles you go through all the topics from book "Algebraic Topology" as I saw this book have 500 pages and use advanced math stuff lake teory category (or notation form that) so I understand why you skip some proofs in this course but I will be greatful if you can add some additional supplement with proofs!
I can’t even begin how people do this. I’m about to start mechanical engineering and I’m dreading calculus. I can do trig and algebra so I should be able to grind it out. God someone give me some motivation. Tell me it’s gonna be okay.
nice
Got homeomorphisms in legal concepts?