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You keep talking of "closures of points" and "closures of singletons" as in 22:30, but Spec(R) is always Tychonoff, or T1 space, so all singletons are closed sets.

If anyone knows: at 10:56, how do we know L even exists by only doing a countable number of iterations?

3:40 - Shouldn't Spec(k[x]/(x^2)) have two points? (0) and (x) := (x+(x^2))?

Did we just skip over why x is transcendental at 6:00 but spent an entire 3 minutes explaining the trivial case of cos 2pi/7? :D

at 1:27 Mr Borcherds says "...1 is a unit..." Can someone expand on this idea?

The best

Would have been a great talk, but the slides are out of focus. It was in focus for the first few seconds and then the focus suddenly changes and nothing can be read.

Borcherds concludes the proof of Hilbert 90 at about 15:17 by saying that θ + α•σ(θ) + α^2•σ^2(θ) + … = β. But this looks incorrect to me. The terms of β are (α•σ)^k(θ), which are not equal to (α^k)•σ^k(θ). E.g. (α•σ)^n = N(α)•σ^n = id ≠ (α^n)•σ^n unless α^n = 1. The proof still works, but we need to show that 1 + α•σ + (α•σ)^2 + … + (α•σ)^(n-1) is not identically = 0. It is a linear combination of the characters σ^k, but with coefficients 1, α, α•σ(α), etc.

The circle of the Fano plane shouldn't be outside the triangles, it is the line containing the three midpoints of the sides of the triangle.

My general philosophy here is that definition is irrelevant, you can always define yourself right, but this is meaningless when you can't use this in a mathematical proof. The only right thing to do mathematically is to find the places where you would intuitively use the dimension in your argument, and construct a "measure" with the properties you would want here. In order to construct the Krull dimension, or any dimension with ordinals, you would already want to know everything you can proof with it, making them completely useless.

Great 👍

3:57 Anyone can explain more about the proof for K is uncountable using Aoc?

I dunno y im watching this but learning this interesting Number Theory even though i passed my graduation from College lol.

I don't understand what you mean when you say "the closure of a point" or "this point is not closed", since Spec(R) is always a T1-space, so singletons are always closed.

Hi, professor, I wonder if there is a book which your lectures are mainly based on. I just want to find such a book so that I can keep watching your lectures and learn well. THANK YOU!

took abstract algebra with him many years ago at Berkeley.

First lecture goes in Riemann hypothesis lol

i was thinking about this lately (the permutation of letters in the alphabet) turns out its a real established concept

I really enjoyed that. I love those "bizarre" occurrences in mathematics. I believe that the 1..24, 70 coincidence is one of the "bizarre" coincidences that is key to constructing the Monster group, too. Does a similar computer-graphics type video exist for the Monster, too?

19:37 - What is N? Is it supposed to be the same as K?

At 23:57 he says "with roots in this field". Shouldn't it be "with coefficients in the field K(a0, ..., an-1)"?

Some context for group action on functions: If you switch every complex number with its complex conjugate, all mathematics would stay the same. But suppose you have a function f, that sends 1+i to 2-i. In the conjugate world, this function would send 1-i to 2+i. This means that for the conjugate function f*, f*(1-i)=2+i, if f(1+i)=2-i, or f*(x*)=y*, if f(x)=y. Basically all possible properties of the function, like being holomorphic will be preserved. And a similar operation works on function operators, like the Fourier transform. It will be replaced by the inverse Fourier transform, if you define things right.

"Roots of Unity" ...so much potential for an album name.

Please do not leave us alone!

I enjoy all of your lectures and courses, thank you for your generosity and god bless you. Wishing you the bests. Farhad Abdi

LYSDEXIC'S NIGHTMARE!

At 8:16, shouldn't the right hand side be 0 and not 1?

One stop shop for algebra lectures _and_ plastic bag ASMR

8:50 its GL_3(F_2)

1897 Math Tripos question is scary, so is to comprehend the intelligence that existed at that time!

What a fantastic way to simplify collinearity! I love your method of exposition to understand group structure of $\wp$. Thank you prof.

14:40 - Either the integral should be from 0 to 2pi or from -pi to pi, for you should add a factor of 2pi to inx and imx in the exponent. Otherwise this equality is not true.

14:52 p is prime so n=2^k, and it is Fermat prime

3:56 algebraic object? what he means are complex and real are not algebraic extension of rational?

At 8:11, why is d(at+b/ct+d) = dt/(ct+b)^2 ? And then why is (ct+d)^2 and not -2 (since it's in the denominator?

As a non-native English speaker who has experienced reading lots of translated books and articles in the past, if something doesn't add up in the text and the text is a translation, my first gut instinct would be that either I misunderstood something, or that it's more likely the translator's error than the author's error. It must be extremely difficult to convey all the nuances of the original text in a faithful way and sometimes there just isn't enough context to compensate for the language differences while fully preserving the meaning. This only gets exponentially worse if the text is a translation of a translation...

This is the best discussion of the problem. I thought I understood it, but then turns out my intuition of the winning probability when you switch was wrong! My brain went, "well duh, if you don't switch, you win 1/3 times but if you do switch you win 1/2 times, because now there are 2 doors." 😂 (Also, rightfully noted that goats are much more precious than cars overall!)

Prof. Borcherds claims at around 14:42 that we saw earlier what the poles and zeros of the meromorphic differential dx are for a plance curve f(x,y) = 0, namely the claim is that dx has poles of order 2 at the d points at infinity and zeros of order 1 at the zeros of df/dy. I couldn't find a video for this? Can someone point me to a calculation?

Around 9:23, what does "Improperly equivalent but not properly equivalent" mean? Isn't that trivial?

Unlike most covering Weil here on YT, he writes more legibly for some reason of effort or natural style.

Truly delightful explanation as explained clearly and simplistically. If this sunny morning I was looking for God equalling one then the answer has been given. Thank you.

We are grateful Professor.

Absolute cinema

Should'nt it be T^3 instead of S^3 for z?

Sorry I don't get the point why I should expect that the coefficients of a Theta series give me the number of certain vectors in a lattice. Could someone explain.

🐢😎

why did we define sine with period of 2 pi? It seems more fundamental with period 1, and only circle yields the period of pi.

"It's extremely implausible that Aristotle would have said this solely based on a preconceived idea of women's inferiority or because he "didn't check Mrs. Aristotle's teeth". For me, the most probable and logical explanation should be related to pregnancy. It's very likely that women were mothers at a relatively young age, and pregnancy would have been quite harmful to their teeth. So, it's very likely that the women he inspected had fewer teeth."

"We can see that it is onto, because we know every number has polar coordinates" Actually, showing every complex number has polar coordinates isn't trivial. It's a really good challenge, since there are multiple ways to proof it.