Відео

quebec

اتمنى لو لديك ترجمة عربية شرحك رائع لكن لا افهم اغلب الكلام

Just a quick off-the-cuff comment after seeing cohomology for the first time: since elements of the chain groups are Z-linear combinations of the generators (same thing as maps from the generators to Z), and elements of the cochain groups are homomorphisms from the free groups on the generators into Z (same thing as arbitrary maps from the generators to Z) and the addition operations on each coincide, aren’t they each (at least for Z coefficients) the same? I guess they might be, but the homology can differ because the boundary maps might differ. Also, I guess this has something to do with why the case of cohomology with Z coefficients is a special case and why in the torsion-free case they actually are the same.

30:48 Yes, you can check it if you like… but you can instead just remember that contravariant represented functors preserve coproducts 😉 (a fact I learnt only very recently!).

Amazing, regards from Spain, you are helping me a lot, thanks.

He has a special relationship with the book. That’s completely understandable. 😅

Very well explained, thank you!

Retraction (convergence, syntropy) is dual to inclusion (divergence, entropy). "Always two there are" -- Yoda. Attraction is dual to repulsion -- forces are dual!

Yeah, feeling bad about math, so hard and still learning. And hopefully get excited career like Sean said, just starting industry job.

What is a Union of Union to Learn more about the Ambient Spaces?

Soooo it’s 0?

how is possible that with n-2 number in a sequence get n to the power of (n-2), with (n-2) position and with n as the highest value you can have (n-2) to the power of n that is totally different.

big galaxies are knots in process ?

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 ?

at 39:00, when you said f and g are homotopy equivalent, did you mean to say homotopic?

he did not prove that GF is perpendicular to AE.

Can i use max instead of sup bcz the image set is closed and bounded.

WTF 😮

we all know it's not possible without your group theories

The only thing that the book lacks is examples. Otherwise the theoretical content is intermediate friendly.

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.

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

Dear professor Anthony Bosman, I'm a highchool student in Korea. At first, thank you for the enthusiastic video. I was really impressed by the high quality and the content of the lecture. I've already knew the concepts of fundamental group and knot theory, but I've never thought of combining both up! And also the physical visualization of abelian and non-abelian fundamental groups were intriguing. In fact, I'm trying to operate a math experience booth in the school festival, so if it's okay, can I use your instance for the experience booth?

42:31 If x^2 is rational, then we don't need it in the basis anyway so can't we skip this step?

As far as I've seen so far, this professor is a star. Love his teaching style, the topic progression, and the overall clarity. This is easily the best LA course on YT - imho it's even better than the famous MIT one ;-)

This helped me in my life. You have earned a subscriber :)

I have no idea what you're talking about but i promise i will

How can these spaces be Abelian? We are no longer describing a sequence of paths that end where they started; if the cycles can be in any order, then the map from S1 to the space can be discontinuous. You can pick up the pen and draw the cycle piecewise, so to speak. If the idea is to specify which elements constitute the cycle, then why are they Z and not Z mod 2? There's no way to make sense of this as an abelian group.

so this axiom can be used for infinity huh? I remember something like inf+x=inf but this only means that if u have 1 set of infinity points, then u can choose so that u create 2 sets each equal to that first set, from that first set. That is all you can't from that come to the conclusion that u can create 2 spheres identical to the first sphere. too many words like " create, identical" that need definition. in math: I mean, imagine u have 1 set of infinity points, how the fuk can you "create" a sphere with that in math? just by saying u "create" it? what does that even mean? I feel like u can only say that a set from all points of a sphere is equal to another set from all points of another sphere. Adding stuff like "break down a sphere", and "create a new sphere" is too out of math. those work first of all is not math, second, it make ppl think about phys, hence the so-called "paradox". but it is not a paradox if they use the correct words for it.

Euclidean spaces have n points represented by real numbers (x1, x2, ... , xn) and a dot product whereas the continuous function example is completely different but we can still find a notion of a distance function and hence a metric space which isnt just points in n-dimensional space. The example is to show its not just a concept that applies to euclidean spaces.

damn

This course is the best among all that ive ever had. Thank you

The camera man didn't do a great job

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.

Excellent lecturer and material. I find the Hatcher book detailed but uninspiring

I've noticed that the universal cover for the wedge of two copies of S_1 is literally just the cayley graph for the free group <a, b> Is this a coincidence? Because the way you drew R as a helix (with "vertices" corresponding to the base point in S_1) is technically just the cayley graph of <a>, or Z.

Great suit. Big effort on the outfit. Well done

Seems so clear

In the Z-example, ordering {0, 1 -1, ...} why is 0 the least ? Unless the ordering involved is not the usual one of the integers, but simply defines the set as starting with 0? -1 (and all other negative integers) are smaller, in the usual ordering, than 0.

That's a roundabout way of saying it 😂

Where did my comment go? I really need an explanation of how these delta complexes are groups. The fundamental group I get; it's the set of loops in a given space that are not equivalent. The Cn and delta n groups seem to be sets of vertices, but what does it _mean?_ Are we still talking about cycles? What is the group's combination operation? What is the inverse of a vertex? I don't get it, please help!

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.

Is this lecture series continuing soon? I hope so! It has been really great.

31:30 hold on a second... t=-1 gives you the colorizability determinant! because when you plug it in 1-t (where a section is on top) becomes 2 and t (where a section is underneath, on the left) becomes -1. that's really cool

If there's only one map from a simplex to a point, then why is the group Z? What are these _groups?_

Can someone please explain how these delta basis sets are groups? I get the game we are playing with them, i.e. how they are computed, but I have no clue how they are groups. What is the identity? What is the inverse of a point? What does this group act on? What is the group combinator, i.e. the *+* in those expressions? I keep seeing sets of vertices, but what does it _mean?_ I feel like we waltzed right past that and now I have no hope of understanding homology as a consequence. Also what is a kernel and an image, exactly? I can see where he's getting them, but what is their significance?

What great teacher you are❤️Thank you for this lesson 😊👍

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.

I don't understand how all these delta basis form groups. What are the points in delta 0 acting on, and what is the operation? What's the inverse and identity? What does it mean to multiply them by integers, and is that different from exponentiation in the normal group sense? Why are they all Z and not R? I'm so lost on this one.