"The other good thing about saying something is 'locally uniformally convergent' is that no one can quite remember the definition of it, so they'll probably give you a pass for whatever you want to claim for it" :)
at 8:32 I was feeling guilty about not knowing the conditions for this property, then felt a lot better about my ignorance within the following 30 seconds :)
You can bound the terms |1/(mω₁ + nω₂)|² from below by 1/((m+n)²r²), where r = max{|ω₁|, |ω₂|}. Summing that over m,n ∈ ℤ, you should be (if I didn't make any mistakes) able to bound the series from below by the Sum (m-1)/m², which diverges
I think that the @Jussari approach is great, but you can also apply the same sort of informal reasoning that Borcherds uses in the previous video (Elliptic Functions). For an integer N >0 look at the circle of radius N centered at z, and consider the concentric circle of radius N+1. The area of the disc contained between these two is proportional to N, and the number of lattice points (which are, at large scale, uniformly distributed) is therefore about a constant times N. Now if you look at that sum you'll get that this disc contributes a constant times N*(1/N^2) = constant/N. Summing over all such discs gives you a constant times the harmonic series, which diverges.
This is possibly a very dumb question and has a simple solution, but is there any reason why the number of poles has to be an integer? Couldn't there be some function that effectively behaves as having a noninteger pole? (whatever that would mean). If so couldn't we use that to get an elliptic function of the same order?
We are dealing with holomorphic functions, where everything can be written as power series. The order of a pole is exactly the minimal degree of non-zero coefficient in your expansion near the pole. It has to be an integer. In real analysis we have some weird functions like 1/sqrt(x) which have " a pole of order 1/2" at 0, but these formulas are not legitimate holomorphic functions (since sqrt(z) is not really well defined as a holomorphic single valued function for z near 0).
@@lordabizi5807 I agree with your logic about the power series expansion, that's why I framed the question as "effectively behaves as having a noninteger pole". I was hoping that there was something like this but I guess being multivalued like the 'sqrt' is the closest people have come to this.
There are indeed (multivalued) generalizations of elliptic functions with "poles of non-integer order". However these are excluded from the discussion in the lecture by the condition that the functions are meromorphic.
Angular momentum is quantized in physics L= nh. Angular momentum is a fundamental. E = fh for photons or pure energy. Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. "Always two there are" -- Yoda. "Science without religion is lame, religion without science is blind" -- Einstein. Science is dual to religion -- Einstein.
Thank you Professor, these lectures are real gems
"The other good thing about saying something is 'locally uniformally convergent' is that no one can quite remember the definition of it, so they'll probably give you a pass for whatever you want to claim for it" :)
jesus christ ! that is the only thing I can remember the definition of at the moment
Good sense of humor combined with deep knowledge is a blessing
Watching your interview video, I became a big fan of you.
Does anyone know any good lectures on Riemann geometry like this? I love watching lectures like these.
Just looked it up: In LaTeX the Weierstrass P is \wp ($\wp$ in math mode).
at 8:32 I was feeling guilty about not knowing the conditions for this property, then felt a lot better about my ignorance within the following 30 seconds :)
Wow that bit about the Tripos example questions, so darn glad that those kinds of questions are out of style now! Yikes
Can anyone explain me more why the series in 2:42 does not converge absolutely?
You can bound the terms |1/(mω₁ + nω₂)|² from below by 1/((m+n)²r²), where r = max{|ω₁|, |ω₂|}. Summing that over m,n ∈ ℤ, you should be (if I didn't make any mistakes) able to bound the series from below by the Sum (m-1)/m², which diverges
I think that the @Jussari approach is great, but you can also apply the same sort of informal reasoning that Borcherds uses in the previous video (Elliptic Functions). For an integer N >0 look at the circle of radius N centered at z, and consider the concentric circle of radius N+1. The area of the disc contained between these two is proportional to N, and the number of lattice points (which are, at large scale, uniformly distributed) is therefore about a constant times N. Now if you look at that sum you'll get that this disc contributes a constant times N*(1/N^2) = constant/N. Summing over all such discs gives you a constant times the harmonic series, which diverges.
The Riemann zeta function is locally uniformally convergent, therefore all it's non trivial zero's have real part 1/2
℘-function
I've found an interesting animation of the function ruclips.net/video/WnaUZrPnZ30/видео.html , I guess this comes from Eisenstein series
This is possibly a very dumb question and has a simple solution, but is there any reason why the number of poles has to be an integer? Couldn't there be some function that effectively behaves as having a noninteger pole? (whatever that would mean). If so couldn't we use that to get an elliptic function of the same order?
We are dealing with holomorphic functions, where everything can be written as power series. The order of a pole is exactly the minimal degree of non-zero coefficient in your expansion near the pole. It has to be an integer. In real analysis we have some weird functions like 1/sqrt(x) which have " a pole of order 1/2" at 0, but these formulas are not legitimate holomorphic functions (since sqrt(z) is not really well defined as a holomorphic single valued function for z near 0).
@@lordabizi5807 I agree with your logic about the power series expansion, that's why I framed the question as "effectively behaves as having a noninteger pole". I was hoping that there was something like this but I guess being multivalued like the 'sqrt' is the closest people have come to this.
There are indeed (multivalued) generalizations of elliptic functions with "poles of non-integer order". However these are excluded from the discussion in the lecture by the condition that the functions are meromorphic.
@@richarde.borcherds7998 thanks for the clarification.
Angular momentum is quantized in physics L= nh.
Angular momentum is a fundamental.
E = fh for photons or pure energy.
Energy is dual to mass -- Einstein.
Dark energy is dual to dark matter.
"Always two there are" -- Yoda.
"Science without religion is lame, religion without science is blind" -- Einstein.
Science is dual to religion -- Einstein.
Congrats to the senior wrangler in 1912. You earned it. 🤣
yeeeee eeeeeeeeeeeeeee