16:21 - small typo - should be "p in Y" instead of "p in R"

Note that Wirtinger derivatives work for any automorphisms. For normal perfect field extension L over K, every linear function on L can be written uniquely as linear combinations of the automorphisms on L over K. The Wirtinger derivatives serve as decomposition operators.

This lecture really gives nice geometric intuitions rather than rigorous algebraics, thanks!

25 is also a 2-adic number

Interesting, for the infinite dihedral group, I was expecting a connection between the symmetries of a circle (imagining a "polygon with infinite sides").

Hello, Can i use sigma function to prove the infinite product of lemniscate sin ? Thank you !

Very good ⚡👍

29:00 mandatory project euler shoutout for anyone wanting to challenge themselves with pure, recreational math. most of it is number theory but theres plenty graph theory/pathfinding, game theory (iirc), geometry, algebra, and some other pretty niche topics.

Sir, I have a doubt, at many places group is represented by G how can it be possible, G may be a set not group, group should always denoted by (G,*). Please clear it🙏

The QNR*QNR = QR proof felt like trickery!

Ehrenmann

So if an equation holds in the integers, it holds in the integers-modulo-x ?

The intro on stacks and topoi is so funny

I guess around minute 26, with m=9, 8 is congruent to -1, but its order is 2 (as is always the case for -1). Right?

I might be a little late but this is exactly what I was looking for. Homological algebra and its connections to algebraic topology are so interesting to me and these videos are very helpful for making the ideas more concrete. Thanks for posting them😊

This is the saddest case of my mysophonia acting up I've come across so far. I can't listen to this man despite the obvious quality of the lectures. He keeps smacking into the microphone constantly...

In which book can I find today's content?

Prof thank you so much for your contributions for ppl who can't afford expensive courses.

Great explanation. Would have been great to have burnsides complement lemma as well...

haha ur part abt the word "normal" being everywhere was funny awesome lecture

Dear Prof. Borcherds. Thanks for all these great contents you provide for math enthusiastics. Looking forward to learn more from you and as a suggestion, I think algebraic number theory would be great!

Very nice ! By the way, the sign in the wreath product G \wr H is read "wreath".

Thank you for this video! Monster group lives in 196883 dimensions, meaning it has a 196883-dimensional faithful representation. (Isn't this more precise?)

yazın çok kötü reis

Two to the zeroth power is one.

Sir I envy your library....

간단한 해석기하학 방정식 계산에서 출발했다.

friends, are there any exams that is appropriate for this lecture?😂😊

This series is so amazing!!!😃 It would be so awesome to see the rest!

Thank you very much professor. Please keep uploading videos and lessons

Thank you, professor. note : in PID, an element that is irreducible is prime; and using this, we can show that PID is UFD.

Many of the figures in this video came from a book Is that book just called table of functions by jankhe and edme? I can't exactly tell what the title was but this book came up after a search and I'm wondering if someone could verify

With Mary Tyler Moore

just want to say thank you , im a high school student desired to learn NT but didnt know how to begin and felt miserable , your lecture brought me back to the simple joy of math , its great.

Hey professor, if you're reading this, do you plan on covering Graph Theory at all? If you have any insights on this topic I'd love to hear them.

Thanks a lot for these great lectures! I think on the slide at 10:40 it must be M = submodule of R \plus R, not R \plus R \plus R. Am I right? At 21:08 in D(2) it must hold (2,1+wqrt(-5))) = (1) because 2 is invertable in the ring.

i love how it ends with "why this thing should exists"

In the proof for the surjectivity (after 22:49) maybe it is better to say that if m is in F(Df_1) we can extend it such that mf_1^n is in all the F(Df_i) for a large n hence (because of the sheaf property) also in F(X) which means that m lies also in M[f_1^-1]. Is this clearer?

"purely inseperable extensions will... frankly the less said about them the better" lmao

Wow! I read about Caley's theorem in a textbook but I really had no idea what I was reading about. This lecture spells it out so well!

I have mixed feelings about how you write Z for the integers

In 5:20 maybe it should be 'points of the group with values in the scheme T', not S? I can not figure out that map in 8:05 from Z[X] to Z[X] tensored Z[x] to be right...

I don’t get what’s going on in the last slide does anyone has some explanation to that? Why should the functor take any scheme S to a single point? What does he mean with 'families of two point sets over S'?

Gibberish

Thank you Mr. Borcherds for these amazing lectures! I needed some time at 24:03 to understand, it is a little bit misleading, maybe it would be good to add that q can only be n^j/m^k for some j>0, and then it gets clearer (because we can choose the (coprime) (m,n) pair freely) that q can be any nonzero rational number.

우주너머에서는 서로의 우위가 없음을 알았어요.

"Generic" isn't hard to define. Just use the magic words 'trancedental extension' and everything will be alright. Basically all properties of R transfer over to R[X], like ideals, and you can quotient from here with the ideal X−α, and quotienting basically preserves all properties.

Just a comment about the history. I have NOT read any of the original sources at all, so definitely 'fwiw', but: I had always been under the impression that Mordell had 'only' dealt with the rational points on an e.c. over the rationals, and that Weil introduced the machinery to handle a.v.s over number fields - this seems to match Wikipedia's description of the history, and does match Wolfram World's "... For elliptic curves over the rationals Q, the group of rational points is always finitely generated [...] was proved by Mordell (1922-23) and extended by Weil (1928) to Abelian varieties over number fields." Meanwhile, Manin's Appendix II of Mumford's AV's claims that Lang's contribution was to deal with the case of the base field being of finite type over the prime field. On the other hand, some of the internet believes that Neron did this...

It's a brilliant course! It's straightforward for the essence of complex analysis.

The definition of general position here is incorrect. For example, the lines AB, CD, and EF are not allowed to be concurrent (these lines don't share a common point) is the simplest of an infinite amount of relations these don't satisfy. It actually means that you need to take a transcendental extension for every new point. At least, that would be the rigorous way to explain older texts. If there were only two relations, they would have mentioned this.