Ben Green's lectures on nilsequences
Go back to the program
https://motivichomotopy.hatenablog.jp/entry/2020/10/19/171757
Green's page about the videos
http://people.maths.ox.ac.uk/greenbj/videos.html
Lecture 1
http://www.youtube.com/watch?v=HM3jR0b4VHY
1:18:00 Explains the von Neumann theorem for Gowers uniformity norms
Lecture 2
http://www.youtube.com/watch?v=27Ljd_M--Qw
He mentions GI(s)
https://youtu.be/27Ljd_M--Qw?t=230
And its converse directly after that.
t=660
Introduces the Gowers norms
t=4000
Rigorous statement of GI(s)
Lecture 3
http://www.youtube.com/watch?v=u4inkf0VJAU
This lecture is about the converse to GI(s). Along the way he explains useful notions and techniques about polynomial sequences in nilmanifolds.
The lecture starts with recap of the statement of GI(s)
https://youtu.be/u4inkf0VJAU?t=28
The notion of polynomial sequences revisited
https://youtu.be/u4inkf0VJAU?t=900
He says he can send typed notes
https://youtu.be/u4inkf0VJAU?t=1260
Derivatives of nilsequences $\Delta _h \chi $
https://youtu.be/u4inkf0VJAU?t=1289
which enables one to use induction on $s$ in some situations. The final statement of this part is stated at:
https://youtu.be/u4inkf0VJAU?t=3110
Be careful that the derivative $\partial _h p$ of a polynomial sequence $p\colon \mathbb Z \to G $ and the derivative $\Delta _h \chi $ of a nilsequence $\chi \colon \mathbb Z \xrightarrow p G \xrightarrow{\varphi } \mathbb C$ are two different things!
Proof of converse to GI(s) begins
https://youtu.be/u4inkf0VJAU?t=3347
At 1:14:00 (t=4440) the proof is done.
Lecture 4
Equidistribution of polynomial sequences.
https://youtu.be/yFdoQwDBMRE?t=28
First he talks about the general motivation.
Then he introduces the equidistribution phenomenon in the $\mathbb R/ \mathbb Z$ case.
https://youtu.be/yFdoQwDBMRE?t=345
Theorem of Leon Green
https://youtu.be/yFdoQwDBMRE?t=494
which reduces some problems to the abelianized case $G/[G,G]$.
Definition of "δ-equidistributed"
https://youtu.be/yFdoQwDBMRE?t=889
and the equidistribution theorem.
Announcement of the main result
https://youtu.be/yFdoQwDBMRE?t=1245
Explanation of the symbols in the statement lasts until 34:00 (t=2040).
Idea of the proof of the equidistribution theorem
https://youtu.be/yFdoQwDBMRE?t=2110
The van der Corput lemma is part of the machinery which enables induction on $s$.
Its proof lasts until 44:00 (t=2640).
Next he explains Weyl's idea
https://youtu.be/yFdoQwDBMRE?t=2673
which employs Fourier analysis to get vertical frequency.
He uses the induction technique from Lecture 3
https://youtu.be/yFdoQwDBMRE?t=3254
and spend some minutes to explain how it works.
Idea of proof finished at 1:07:20.
"Why do I care about equidistribution of polynomial sequences on G/Γ?"
https://youtu.be/yFdoQwDBMRE?t=4200
including the decomposition theorem
https://youtu.be/yFdoQwDBMRE?t=4753
Why one has to consider polynomial sequences even if one is primarily interested in linear sequences $n\mapsto g^n x$
https://youtu.be/yFdoQwDBMRE?t=5163
Lecture 5
Lecture 6
http://www.youtube.com/watch?v=ZwHsSiB-Pl8
0:54--11:15
The statement of the generalized Hardy-Littlewood conjecture