Great video, thank you ! I have a question, why isn't the sign of the edges of a 2-simplex alternating ? I thought that the border maps where unique. del_n( [a_0,...,a_{n-1}] ) = sum_{i=0,...,n-1} (-1)^i [a_0,...,a_{i-1},a_{i+1},...,a_{n-1}] ?

Thank you Daniel.

Thank you for walking us through it!

Maybe homological algebra is the study of correcting the failure of a map seeing two objects as equal (0->A->B->0), by using various notions of equivalence (quotients, weak equivalence) to derive linear approximations (resolutions, cohomology rings) to your original setup. You give names to your inability to solve a problem (eg: H^1) and just keep computing elaborate constructions (eg: spectral sequences) until something gives way.

Excelent! Greetings from Peru

Valeu amigão!

1:34 goodbye housed off property

thanks for the video - really well done ! super helpful ❤️

How are these differ with fiber bundles

Please keep this series going!! I really enjoy your videos. I appreciate you taking the time to make them.

Very clear!

too good

thanks Daniel Chan, the presentation of so much knowledge was super!

thanks for uploading

You are the professor of my dreams! Your explanation was so clear, I could listen to you all day. Thank you so much for this incredible video.

Thank you so much for this, this was a life saver.

Turns out, this video happens to be me 5000th like. I like that.

A real example would have been nice!

A real example would have been nice!

@11:10, if we are taking a germ f at p, then for any other germ at p, say g, they agree for some small enough neighborhood of p, so the derivative D_gamma f = D_gamma g, is that correct? If that is true, then for the definition of derivation at a point p, doesn't that mean f(p)=g(p), and δ(f)=δ(g)?

Greatly explained! Thx alot, Daniel!

What happened to subsequent lectures?? Is it coming?

Very nice, explained the topic very well

Thank you. A great intro to the topic

Would it not be better to say that "There are only an even number of zeros" vs "There are only an odd number of zeros" and then do the description of the classes of zero crossings describe an invariant of the manifold?

Meanwhile Mr. Best has 42874928795853242309872495 subscribers.

Could you do a video on books you think are good for various mathematical topics? Books that you found useful to your own development.

Is there a simple way to see directly in our example why the intersection product of p^*(C = nodal cubic) and the exceptional divisor E is zero without replacing C by a an equivalent divisor that doesn't intersect the point we are blowing up? How do I read of some of the intersection points in the chart X = 1 in the blowup X^~ from the defining the pullback divisor x^2 * ( (y/x)^2 - x + 1) ? DO I have to replace the pullback divisor x^2 * ( (y/x)^2 - x + 1) by something equivalent because it contains the x^2 factor corresponding to the exceptional divisor x = 0 in the patch?

This video should be added to the "A User's Guide to Coherent Sheaves" playlist (or the one on cohomology of coherent sheaves).

Thanks for making these videos! They're fantastic, I'm really enjoying the series so far! I'm probably missing something, but the computation of H^d(P^d, O(-n-d-1)) for n = 0 seems slightly wrong to me. When considering the basis of C^d(O(-n-d-1)H) don't we have to restrict ourselves to rational functions without a positive power of x_0 in the denominator to avoid cancelation of x_0s, which would result in a an order of vanishing on H of less n+d+1? Then the Laurent monomials should not have a pole at H (so the Laurent polynomial point of view does not actually seem symmetric w.r.t the choice of the coordinate x_0 defining the hyperplane H). And when one multiplies the Laurent monomial 1/(x_0...x_d) by x_0^(d+1) the order of vanishing on H is d, which is one less than we want. I think dividing the Laurent monomial by another x_0 should work.

Where do we use k is algebraically closed?

Just found your channel, time to binge all your videos 😁😁

Historically was the Zeta function something that was just "arbitrarily defined" or did it come about through ring theory in the sense of the video? I've seen the Zeta function presented in several ways. The usual way is simply to poof it out of thin air of course. One simply defines it as the typical sum or product. The second way I've seen it presented is through the generalization of the Gamma function typical of Riemann. The third way is through Dirichlet series which obviously begs the question. The fourth way is given in the video but likely not the origin of the Zeta function. Obviously it is very likely people "discovered" the zeta function by simply toying around long before anyone knew what to call it but I'm curious if the "ring theory version" was known long before or if it is relatively recent. My guess is that it is 20th century and that, in some sense, some of the ring theory mathematics were constructed "around" the zeta function in trying to formulate a solution to the RH. Euler in the 1700's obviously was toying around with the Zeta function as were some of those before him but these were generally for specific values and without any real understanding of any deeper connection to number theory. From best I can tell is that the "ring theory" version simply fits ring theory to the zeta function to connect it. Since the zeta function is defined in terms of primes there is the obvious connection to finite fields. That is the link that brings the Zeta function under the umbrella of ring theory which enables one to view the Zeta function in a much greater light.

Very well done !

Served well

your videos are amazing

Thank you!!

Great video !

Is there some very general law that relates subojects symmetries of an object to the objects symmetries? That is, where we have a composition of objects that form a super object and the composition somehow acts on the symmetries(or even more general stuff) of the subojects to get the objects symmetries. e.g., something like Sym(A o B o ...) = Sym(A) * Sym(B) * ... . I'm thinking though in the most general categorical context.

gm thanks for making this!

Please give some name of reference books to study ❤

God this is so helpful and stuff.

@1:50 His observation of the ‘canonical’ divisor’s place in relation to the entire space of divisors is the attitude of an Abstract Mathemetician. More practically, there is very glaring difference between the arbitrarily chosen divisor and the canonical divisor, which would invoke a different attitude entirely. In particular, one’s relationship with the Reimann Roch theorem wouldn’t be one of trust: Rather, the discipline of proving it would be nothing other than seeing this difference in nature take form. The vocabulary of sheaves is the technique of the topologist, …, and this equivalence in homology he wrote down is something that makes his attitude a part of the content. This isn’t naturally relatable to the Reimann Roch theorem in particular. Sheaves reconstruct one’s understanding of analysis in a fundamental way: for me.

Wow this is some advanced stuff. Im looking forward to one day learning this. I am currently studying modules!

You are incredible

This channel is incredible. So glad I found it

omg, did I say that I love you?

love your videos <3

thank you for this wonderful video

I just want you to know, that unlike vector bundles on open covers, you're never trivial.