So this might not be the best way to look at it, but for me the concept (i.e the wrapping your head around it part) of a sheaf was simplified a lot by just considering it as the fanciest way to define the set of "well-behaved" functions on a space. Vakil's notes give a very motivating introduction with the sheaf of differentiable functions. Now the general idea is: you're given a big, global object (a topological space, i.e a curve or smth) and want to consider functions on it. Now it often happens (continuous functions, differentiable functions, etc.) that you don't need to look at the "whole". In these cases you can basically take the little pieces the whole is made of and your function will be completely determined by their values on the little pieces. The sheaf is just this. Take some big piece (some open set) and the sheaf spits out a fancy version of functions on your big piece such that if you split your big piece into smaller pieces (so an open cover) and consider the functions on those, you can glue the functions together in a nice way to get a function on the big piece. Looking at it this way makes e.g the stalk at a point just a fancy way of saying yeah these are the germs of functions. Again i highly recommend Vakil's chapter on the matter. This is (at least at the beginning) the biggest barrier towards algebraic geometry so i hope this makes sense at all (i'm sleep deprived).

what is a vector field, what is a curl and what is a divergence 😭

Addition was already known to the dinosaurs. But they were wiped out, and could anyways not communicate their knowledge to later generations. So early human hunter gatherers had to re-invent it. Certain cricket species are known to master the addition modulo a prime number, like 13 or 17. They only emerge from their larva state every 17 years - their main predators never figured it out. 😂

Is the substitution of u in the solution correct?

I love this format of video.

It seems natural to study this question using formal power series and ignoring convergence, I am not sure if this will work though and the system of equations will be much more tedious than the presented solution.

It’s decent that you admit your weakness in a branch of mathematics namely here algebraic geometry.

Or just note that n^2/(2n-3) > n^2/2n = n/2 for all n≥2. So n≥2M suffices to say that n^2/(2n-3) > M (as long as n≥2).

I need to watch this again to understand it.

The level curves of a real polynomial function in two letters of degree 1 are lines The level curves of a real polynomial function in two letters of degree 2 are (possibly degenerate) hyperbolas, parabolas, and ellipses Degree 3 level curves? What is a sheaf a generalization of?

Would be nice to see that any conic section (ellipse, parabola, hyperbola, ...) is a perspective (linear fractional) transform of the unit circle.

kzfaq.info/get/bejne/odymo6qIxMqoYXk.html The points are the vertices (vertex)

Isn't the parabola itself a degenerate case too? Of an infinitely stretched ellipse, or hyperbola.

p = 19

🙏🙏🙏 Please, please, make a video on the claim that parabolai are in the same class as ellipseis and hyperbolai. 🙏🙏🙏

n=60

I've seen a book which talks about algebraic curves and classifies conic sections into quadrolas and grammolas and maybe others. A line which is perpendicular to itself, i.e. its slope is √-1 (which is i in the complex numbers, but 8 in Z13) is called a null line. All circles are asymptotic to null lines, if the field has such things.

What if you use quaternions instead of vanilla complex numbers? Does the universe implode? Is everything a point? A sphere? Hahaha.

I'd like to ask what may seem like a "dumb question" ... to those with a better handle on this stuff than I have, but: How is the "Discriminant" of a conic section related to the discriminant in the formula for the solution to the zeroes of a parabola, if at all. Is it coincidental, or is there some connection that I might be able to visualize.

@ 13:12 X^2 and Y^2 should be reversed.

Would be fun to prove that these are all conic sections.

For starters powers of three are odd and minusing it from one would make it even. Since all even integers greater than 2 is not a prime. It is only true for n = 1

Great video, as usual. As an aside, I'm relieved to hear that my favorite on-line mathematician can't quite wrap his head around the concept of "sheaf". I've struggled, unsuccessfully, with that in connection with analytic continuation. Anyway,. this is a wonderful exposition on the fruitful general quadratic's connection to the conic sections.

Donald Knuth ... An amazing scholar. I majored in mathematics with a minor in computer science and was always fascinated with the mathematics of computer programming. To me some of his most interesting recent work deals with the nested summation of powers. He has made many contributions to the online encyclopedia integers sequences as well, a source that seems to be rarely mentioned in your videos. You are an amazing mathematician and teacher thank you

And that is two consecutive Fascinatings! from me

when the big MP opens by describing a particular area of math is "such a difficult subject", smash cut to a cabinet of urns containing the ashes of several postdocs and graduates who gave it a try. or better yet, so one of them approaching you with a book they dust off from your shelf and are like, hey can I borrow this? MP, serious suggestion: get some theater heads looking for playtime to add bits like that. For example, Sabine just does one every couple eps but it's golden. time for you to hit 1M brother

In mathematics, a sheaf (pl.: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data are well behaved in that they can be restricted to smaller open sets, and also the data assigned to an open set are equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every datum is the sum of its constituent data)

0:30 - Whenever I try to wrap my head around what a sheaf is, I end up with a sheaf wrapped around my head.

So you've been doing nothing your whole life?

I handled the 2nd problem a little differently. I first used the fact that 2^k mod(9) takes on the pattern 1, 2, 4, -1, -2, -4, ... This makes it obvious that 1 pairs with -1, 2 pairs with -2, and 4 pairs with -4 to get 0 mod(9). Second, I simply made a 20x20 table going from 0 to 19. The columns represent 2^0, 2^1, 2^2, ... which is 1, 2, 4, ... mod(9) respectively. Same idea for the rows. This table will have 4 columns and rows corresponding to each of 1 mod(9) and 2 mod(9). There will be 3 columns and 3 rows corresponding to each of 4 mod(9), -1 mod(9), -2 mod(9), and -4 mod(9). The probability we're interested in is just 2*P(picking from a 1 mod(9) column and -1 mod(9) row) + 2*P(picking from a 2 mod(9) column and -2 mod(9) row) + 2*P(picking from a 4 mod(9) column and -4 mod(9) row). Note, my way needs to multiply by 2 since I didn't restrict the probability space by asserting n<m so I also need the symmetrical cases where m<n. For example, a case like picking a -1 mod(9) column and a 1 mod(9) row would need to be included. Thankfully, it's easily handled via the multiplication by 2. 2*P(picking from a 1 mod(9) column and -1 mod(9) row) = 2*(4/20)*(3/19) = 12/190. Note that you use 19 and not 20 because my probability space simply has the restriction m <> n. So I only have 19 rows to pick from once the column is chosen. 2*P(picking from a 2 mod(9) column and -2 mod(9) row) = 2*(4/20)*(3/19) = 12/190. 2*P(picking from a 4 mod(9) column and -4 mod(9) row) = 2*(3/20)*(3/19) = 9/190. Answer = (12+12+9)/190 = 33/190.

Something curious to note with respect to what is said here is sqrt(|1 - x^2|) = sqrt(|x^2 - 1|). Now if we integrate either of these... :>

17:40 Marty, you're not thinking four-dimensionally

All quadratic compositions are one and the same, a thing you can't say for cubics, for instance.

This, conveniently, is directly relevant to some work I'm doing.

I d like to see more about these curves in the complex plane

Kind of disappointed that the first 15 minutes of the video mostly consisted of mindlessly plugging numbers into stuff, and that we then jumped at 15:40 into the main result without even a sketch of its proof. Same thing about the invariance of the 3 curve classes under affine transformation. All exploration and no theorems leaves me with mathematically blue balls

Inappropriate title. I will call electric and magnetic fields and the direction of propagation the holy trinity. Borrowing terms for religion is inappropriate.

@6:00 : name of points is "hyperbola vertices"

fascinating! great presentation

Great video! Conics are one of my favorite topics. I never realized that a determinant could be also be used to determine the type of conic. Thinking about these objects in CxC blew my mind. Is there some sort of 4-D saddle happening in the last example? There is one more degenerate case: b=r=s=t=0 and sign of a = sign of c. This yields a degenerate circle/ellipse, which is a point (the radius/axes are zero). This occurs when the cutting plane of the double cone passes through the point where the tips of the two cones touch. If you keep the plane passing through that point but tip it up until it just touches the surface of the cones you get a degenerate parabola (the single line). Keep tipping the plane inside the cones and you get a degenerate hyperbola (the two lines).

15:56 There's also a very nice etymological harmony of the sign of this "discriminant" b^2 - 4*a*c with the corresponding curves. A parabola comes from the Greek παραβάλλειν: to compare, to be side by side, to be equal - and the discriminant is equal to zero. Likewise, an ellipse comes from έλλειψις, "deficiency", and the discriminant is < 0; finally, a hyperbola comes from υπερβάλλειν, "to be in excess", and the discriminant is > 0. These in fact are precisely the origins of these terms in the original conception of quadratic curves as sections of a cone by a plane. Namely, the angle the plane forms with the vertical is either the same as that of the cone generator (parabola), larger (hyperbola) or smaller (ellipse).

Yes, we would like a sequence, the trinity deserves two more videos. (vertices of the hyperbola)

this is shocking, In india we are taught this in 12th standard under Coordinate Geometry and we solve a ton of questions with varying difficulties manipulating the same equation

Sure! we want to see that space where parabolas are also included in the equivalence class

At 6:02 those two points are called the vertices of the hyperbola. Btw, all these creatures are also called conic sections because they live inside a double cone.

z^2 + w^2 = z^2 - (iw)^2

I feel the same about modern algebraic geometry. I'd much rather the classical way of ideals and varieties. I don't understand the sheaf and scheme stuff.

Good refresher on complex numbers and the interesting results you can get with them. Tank you!

The two line case an easy example is x^2 -y^2 = 0, factors to (x+y) (x-y) = 0; which gets 2 lines

23:08